CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the priority of PCT/US2010/042192, filed on Jul. 15, 2010, which claims the priority of U.S. Provisional Application No. 61/225,672, filed Jul. 15, 2009, the entire contents of which are hereby incorporated fully by reference. 
    
    
     FIELD OF THE INVENTION 
     The invention pertains to the detection of network performance anomaly events based on Network Performance Indicator (NPI) Operational Measurements (OM). 
     BACKGROUND OF THE INVENTION 
     As communications technology has evolved, communications technology users have become increasingly reliant on the ability to communicate almost instantaneously with others all over the globe. With this technology seemingly available everywhere, users of network resources have come to perceive performance delays of as little as 2-3 seconds as unacceptable. Time delays in data transfers and dropped phone calls in mobile telephone systems irritate and alienate customers and thus, service providers try to pay close attention to performance problems and correct them as quickly as possible. 
     Operational Measurements (OM&#39;s) in the context of network performance are network parameters that are measured and used as Network Performance Indicators (NPI&#39;s). These measurements can include call success rates, call termination rates, Quality of Service (QOS) measurements, traffic and routing measurements, network outage statistics, and the like. These OM&#39;s are typically measured over a fixed period of time, referred to as “OM transfer periods”. 
     Early detection of network performance anomalies could help avoid network outage events. A slow and persistent degradation of NPIs can indicate an issue such as memory leak. Additionally, simultaneous large abrupt and sudden changes in, for example, the call success rates from multiple NPIs can indicate the onset of outage events (the outage can be partial, i.e. losing &gt;10% of capacity, or total outage). Therefore, it would be desirable to utilize the NPI process to help avoid or reduce the outage downtime of the network and other problems such as memory leak by devising a way to automatically process the NPIs to detect the occurrence of slow and persistent NPI OM degradation, severe and sudden degradation in NPI OM, and potential outage events and raise an appropriate log or alarm to alert the operator of the observed performance anomaly so that they can be investigated and dealt with in a timely manner. 
     There are many relevant existing stochastic process control algorithms that are routinely used in various industries to monitor product quality such as Shewhart, EWMA, and Page&#39;s CUSUM control charts. However, these standard quality control algorithms only deal with detecting deviations of the monitored quality metric from a fixed (known or unknown) mean value that is constant over time. In the NPI performance anomaly detection problem, the mean value of success rates can fluctuate slowly over time in normal operation (e.g., due to the change in traffic level or services usage pattern during the day), and thus only a statistically significant large and abrupt degradation, or a slow but steady degradation, from the most recent average success rates would indicate a possible onset of a new outage. This time-varying statistical characteristic of the NPI prevents direct application of these traditional stochastic process control algorithms. 
     SUMMARY OF THE INVENTION 
     In order to provide an early and more accurate determination of network problems, current NPI OMs are compared with samples of recent historical NPI OMs so that changes in the NPI OM are detected based on current overall network conditions rather than on conditions that may have existed at statistically insignificant earlier operational periods. By constantly adjusting a performance threshold, against which the current NPI OM is compared, by using a smaller and very recent sampling of NPIs (in the case of sudden and abrupt performance-NPI degradation detection) or a larger and greater number of NPIs over a wider time priod (in the case of slow and persistent NPI degradation detection) to establish the threshold, detection results are more accurate and meaningful. 
     Other aspects and features of the present invention will become apparent to those ordinarily skilled in the art upon review of the following description of specific embodiments of the invention in conjunction with the accompanying figures. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates a network environment that incorporates the claimed network degradation processor that is configured to perform the claimed steps of the invention; 
         FIG. 2  is a flow diagram illustrating operation of a severe &amp; abrupt NPI degradation detection node and rule in accordance with the principles of the present invention; 
         FIG. 3  is a flow diagram illustrating the severe performance degradation alarms used to indentify a probable network outage; and 
         FIG. 4  is a flow diagram illustrating the operation of an NPI Slow &amp; Persistent Degradation Detection node based on NPI Monitoring. 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     The present invention will now be described in connection with an exemplary embodiment for a mobile (cellular) telephone network. However, it should be noted that the present invention is broadly applicable to many other types of network degradation detection schemes and to networks other than mobile telephone networks. 
       FIG. 1  illustrates a network environment that incorporates the claimed network degradation processor that is configured to perform the claimed steps of the invention. As shown in  FIG. 1 , a network environment  100  includes a core network  101  coupled to an internet network  103 , a PSTN network  105 , and a cellular/radio network  107 . Various communication interfaces are coupled to the networks to enable communication between users of the network. For example, a VoIP phone  103 - 1 , and laptop computer  103 - 2 , and a desktop computer  103 - 3  are coupled to the internet network  103 ; a landline telephone  105 - 1  is connected to PSTN network  105 ; and a laptop computer  107 - 1 , a mobile telephone  107 - 2 , a machine-to-machine device  107 - 3  and a fixed wireless access device  107 - 4  are coupled to cellular/radio network  107 . 
     Core network  101  includes a mobile switching center  101 - 1 , a data support node  101 - 2 , a home location register  101 - 3 , and other network functionality  101 - 4 . In addition, however, core network  101  also includes network degradation detection processor  101 - 5 , which can comprise a severe and abrupt NPI degradation detection node, an NPI slow and persistent degradation detection node, or a combination of both. Network degradation processor  101 - 5  is a processor that is configurable to perform the steps described in connection with  FIGS. 2-4  and to perform the processes described herein. 
       FIG. 2  is a flow diagram illustrating operation of a severe &amp; abrupt NPI degradation detection node and rule in accordance with the principles of the present invention. In this example, the degradation detection is taking place in a mobile telephone network, but it is understood that the invention is not limited to this example. Specific exemplary algorithms for performing the steps described in connection with the figures are provided in later sections of this application. 
     As shown at step  201 , processing begins by reading and storing the current (most recent) NPI OM, e.g., an operational measurement related to call success rate for the telephone network. In a typical large telephone network operating normally, call failures, SMS failures, handover failures etc, will be occurring every hour, but on a per-second basis, there typically will not be very many such failures. Thus, in such a system, the NPI OMs will be taken every few minutes, every 10 minutes, etc. It is understood that the choice of how often to take such measurements is within the discretion of the network operator. 
     At step  203 , the mean (average) value of the last n immediately preceding NPI OMs is calculated by adding up the NPI OM values of the last n immediately-preceding NPI OMs and dividing the sum by n. For the purpose of detecting severe and abrupt degradation, the value of n is small, e.g., 2 or 3, so that the current NPI OM value is being compared to only the last few NPI OMs rather than a larger window spanning a larger time period. Thus, for example, if the network has NPI OMs taken every 10 minutes, and the current NPI OM value is taken at time t and if n is decided to be 3, then the NPI OM values at t-10 minutes, t-20 minutes, and t-30 minutes would be combined and then divided by 3 to determine the mean value for the purpose of step  203 . In the example given above, it is suggested that the value of n should be small, e.g., 2 or 3. However, it is understood that the value of n can be changed depending on the needs of the network operator, and an n value of 10 could, for example, still be considered “small” for the purpose of this invention. 
     At step  205 , the variance value of the last n NPI OMs is calculated by taking the standard deviation of the last n NPI OMs (the NPI OMs taken at t-10 minutes, t-20 minutes, and t-30 minutes in this example). As is clear, the processes of steps  103  and  105  are calculations of a moving average for the mean and variance values of the NPI OMs within the moving average window defined by the value of n. 
     At step  207 , a Severe and Abrupt Performance Degradation Threshold (SAPDT) is calculated using the moving averages calculated in steps  203  and  205 . The threshold essentially identifies what was, in this example, a “normal” rate of call success (and thus call failures) over the last n (3 in this example) NPI measurement periods and establishes a predetermined rate of call success (and failure) that will be considered as acceptable. Conversely, this also establishes the point at which the rate of call success (and failure) has become unacceptable. This enables a comparison of what was “normal” degradation over the previous n sample periods with what the current level of degradation is (described below with respect to steps  211  and  213 . Specific examples of algorithms for performing the calculation of the SAPDT are provided later in this application. 
     At step  211 , a comparison is made to determine if the current NPI OM has crossed the SAPDT, indicating the existence of a severe performance degradation relative to the current moving average window. If the comparison indicates the existence of a severe performance degradation, the process proceeds to step  213  where a severe performance degradation alarm is triggered, and any action desired can be taken by the network operator or other monitoring entity. If the comparison indicates no existence of a severe performance degradation, the process proceeds back to step  201 , where the moving window is “moved” and the process begins again on the next current NPI and on the new set of n NPI OMs. Using the process of the embodiment described with respect to  FIG. 2 , the comparison is made based on a changing and up-to-date threshold, in contrast to prior art systems which use fixed and potentially out-of-date thresholds 
     In another embodiment of the invention, described now with respect to  FIG. 3 , the severe performance degradation alarms are used to indentify a probable network outage. In the everyday operation of a network, certain conditions, for example, periods when heavy network traffic is experienced, may increase the number of certain NPI alarms for a particular NPI, but not for others. However, if within a particular NPI OM measurement period there are multiple NPI indicating severe performance degradation simultaneously, this could indicate a partial or complete network outage. In accordance with the operations described in connection with  FIG. 3 , the severe performance degradation alarms triggered using the process of  FIG. 2  can be used to identify such network outage conditions and trigger a potential outage alarm so that measures can be taken to investigate and/or correct any problems that may be occurring. 
     Referring to  FIG. 3 , Step  301  indicates the beginning of a new NPI OM measurement period. In this example, there are multiple NPI OMs taking place simultaneously, as indicated by steps  303 - 1 ,  303 - 2 ,  303 - 3 , . . .  303 -n (the value of n with regard to  FIG. 3  corresponds to the number of NPI OMs being measured by the system). At steps  305 - 1 ,  305 - 2 ,  305 - 3 , . . .  304 -n, a determination is made as to whether or not a severe performance degradation (SPD) alarm has been issued for any of the NPIs being measured. The process for issuing or not issuing an SPD alarm for each NPI can follow the process described with respect to  FIG. 2 . 
     As can be seen, if an SPD alarm has been issued for a particular NPI OM, this fact is conveyed to a summing process  307  (for example, if an SPD alarm has been issued for a particular NPI, a “1” can be forwarded to the summing process  307 , and if no SPD alarm has been issued for a particular NPI, a “0” can be forwarded to the summing process  307 ). 
     At step  307 , the summing process determines how many of the NPI OMs are indicating an SPD condition as indicated by the issuance of an SPD alarm. At step  309 , a determination is made as to whether or not a Potential Outage Alarm threshold had been met. This threshold can be arbitrarily set by the network operator so that a certain number of simultaneous SPD alarms during the same NPI measurement period must occur before an alarm condition is considered to exist. For example, if the Potential Outage Alarm threshold is set at 3, and the if the sytems is set to issue an Potential Outage Alarm if the threshold is exceeded, then if 4 or more NPI OMs cause SPD alarms to be triggered at the same time, at step  313  a potential outage alarm is issued so that investigations and/or corrective actions can be instituted, and then at step  311  the summing process  307  is reset. If the potential outage alarm threshold is not exceeded (i.e., if the number of SPD alarms issued for a particular NPI measurement period is 3 or less, at step  311  the summing process  307  is reset (e.g., the sum is returned to zero to await the next NPI measurement period data). 
     To summarize the operations performed by the processes of  FIGS. 2 and 3 , the process detects potential outages by detecting simultaneous large abrupt and sudden change in, for example, the call success rates from multiple NPIs. The false alarm probability is reduced while maximizing the probability of outage detection. Each NPI has a severe and abrupt performance degradation dynamic threshold which adapts to changes in the most recent mean and variance values of the NPI success rate. The severe and abrupt NPI degradation decision rule compares the most recent measurement of NPI call success rate against this dynamic threshold to determine if there is a statistically significant large and sudden change from the most recent mean value. If the NPI call success rate has dropped below the threshold, a severe performance degradation alarm is issued for that NPI. A Potential Outage Alarm is set whenever a pre-determined number of severe performance degradation alarms are raised in the same NPI measurement period. 
     An additional embodiment of the invention is described with reference to  FIG. 4 . An objective of this embodiment is NPI Slow &amp; Persistent Degradation Detection Based on NPI Monitoring. Slow &amp; persistent degradation of NPIs is detected to enable early detection of problems like memory leak and thus help reduce outage events. The process uses the uniformly most powerful (UMP) hypothesis testing to identify the slow &amp; persistent NPIs degradation events. The UMP test is the test with highest detection probability under the constraint that the false alarm probability does not exceed a given value. In an example using call success rate as the NPI of interest, a slow &amp; persistent NPI degradation decision rule compares a test statistic, which is a function of M most recent measurements of NPI call success rate, against a fixed threshold to determine if there is a statistically significant NPI degradation trend. If the test statistic at a time instant i drops below the threshold, the slow performance degradation alarm is issued for that NPI. If the slow performance degradation alarm is currently set and the test statistic at time instant i for that NPI is greater than the threshold, the slow performance degradation alarm is cleared for that NPI. 
     Referring to  FIG. 4 , at step  401 , the most recent NPI value is read and stored by the system. At step  403 , a moving average is obtained for the NPI data immediately preceding the current NPI value; however, in contrast to the embodiment of  FIG. 2 , in this embodiment, a larger window size is utilized, for example, the window could be of a size that would cover several days, weeks or months of NPI OMs. Although the exact number of NPI OMs that would be considered “large” is a decision left up to the network operator, it is contemplated that a “larger window” would comprise more than 10 NPI measurement periods beyond the time period t. The moving average can be obtained, for example, by low-pass filtering the data to smooth out the NPIs daily cycle fluctuation. 
     At step  407 , the slope of the data trend line resulting from the smoothed data is determined by, for example using the linear least square fit for some or all of the data trend line. At step  409 , the determined slope is compared with a predetermined Slope and Persistent Degradation threshold. At step  411 , if the determined slope meets or exceeds the predetermined Slope and Persistent Degradation threshold, an Long Term Performance Anomaly alarm is triggered so that investigation and/or corrective measures can be taken. If at step  411  it is determined that the determined slope does not meet or exceed the threshold, the process continues back to step  401  to perform the same steps for the next current NPI value. 
     For any of the alarm conditions described with respect to  FIGS. 2-4 , the resetting of the alarms can be triggered by using similar processes that essentially flip the process around so that the threshold levels to be met are indicative of a recovery in performance (abatement) rather than a problem with performance. 
     What follows are examples of specific algorithms and elements that can be used to perform the processes described in  FIGS. 2-4 .
     1. Summary   

     An NPI OMs dynamic thresholding approach is described that automatically detects an onset/abatement of network performance anomaly events utilizing Network Performance Indicators Operation Measurement (NPI OMs) as input data. Network performance anomaly events considered in this document are the following: 
     1. Severe and abrupt degradation of the NPI OM, 
     2. Potential network outage events, 
     3. Slow and persistent degradation of the NPI OM. 
     The following are the NPI performance anomaly onset and abatement events defined in this algorithm:
     A1i) Severe and abrupt performance degration event detected for the ith NPI   A2i) Slow and persistent performance degradation event detected for the ith NPI   A3) Network outage event detected (detection of simultaneous severe and abrupt performance degration from the NPIs in some non-empty set K   R1i) Severe and abrupt performance anomaly event recovery detected for the ith NPI   R2i) Slow and persistent performance anomaly event recovery detected fro the ith NPI   R3) Network outage event recovery detected (detection of recovery from performance degradation for every NPIs in non-empty set K   R4i) Recovery to long term average performance event detected for the ith NPI   

     
       
         
               
             
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 NPI log, set/clear alarm behaviors based on network states 
               
             
          
           
               
                 Condi- 
                 Set/Clear 
                   
               
               
                 tions 
                 Alarm 
                 Information Log 
               
               
                   
               
               
                 A1 i  or 
                 Set network 
                 If A1 i , then generate severe and abrupt perf. 
               
               
                 A2 i   
                 performance 
                 degrad. detection Info Log for the i th  NPI. 
               
               
                   
                 anomaly alarm 
                 If A2 i , then generate slow and persistent perf. 
               
               
                   
                 for the i th   
                 degrad. detection Info Log for the i th  NPI. 
               
               
                   
                 NPI 
               
               
                 R1 i   
                   
                 Generate severe and abrupt perf. degrad. 
               
               
                   
                   
                 recovery detection Info Log for the i th  NPI. 
               
               
                 R2 i   
                   
                 Generate slow and persistent perf. degrad. 
               
               
                   
                   
                 recovery detection Info Log for the i th  NPI. 
               
               
                 R4 i   
                   
                 Generate recovery to long term average perf. 
               
               
                   
                   
                 detection Info Log for the i th  NPI. 
               
               
                 R1 i  and 
                 Clear network 
                 Generate NPI perf. anomaly recovery detection 
               
               
                 R2 i  and 
                 performance 
                 Info Log for the i th  NPI 
               
               
                 R4 i   
                 anomaly 
               
               
                   
                 alarm for 
               
               
                   
                 the i th  NPI 
               
               
                 A3 
                 Set potential 
                 Generate potential network outage event due to 
               
               
                   
                 network 
                 simultaneous severe and abrupt perf. degrad. 
               
               
                   
                 outage event 
                 Detected for NPIs in set J Info Log. 
               
               
                   
                 alarm 
               
               
                 R3 
                 Clear potential 
                 Generate recovery from potential network 
               
               
                   
                 network 
                 outage event Info Log. 
               
               
                   
                 outage event 
               
               
                   
                 alarm 
               
               
                   
               
             
          
         
       
         
         2. Introduction 
       
    
     G/U and CDMA Voice Core Network Performance Indicator (NPI) process provides measurement of various call success rates at the end of every k OM transfer periods. A low and persistent degradation of NPIs can indicate an issue such as memory leak. Early detection of the network performance anomaly problem could help avoid network outage events. Additionally, simultaneous large abrupt and sudden changes in the call success rates from multiple NPIs can indicate the onset of an outage event (the outage can be partial, i.e., losing &gt;10% of capacity, or total outage), therefore, it is of interest to utilize the NPI process to help avoid or reduce the outage downtime of the network by devising an algorithm to automatically process the NPIs data in order to detect occurrence of slow and persistent NPI OM degradation, severe and sudden degradation in NPI OM, and potential outage events and raise an appropriate log or alarm to alert the operator of the observed performance anomaly so that they can be investigated and dealt with in a timely manner.
     3. Detection of Severe and Abrupt NPI Degradation and Potential Outage Event   

     An algorithm for detecting severe and abrupt degradation of NPI and the detecting a potential outage event is summarized in the next two Sections. Section 4.1 presents an algorithm assuming floating point arithmetic is used whereas Section 4.2 present an algorithm when the calculation is to be performed using integer arithmetic.
         3.1 Summary of the Severe and Abrupt NPI Degradation/Outage
           Detection Algorithm (Floating Point Implementation)   
           0) Initialization: Define the ordering of the M NPI, e.g., m=1 for the Mobile Originated call success rate, m=2 or the Land Originated call success rate, etc. Then,   read in the first NPI call success rates measurement, u m,current , m=1, 2, . . . , M, and set       

     
       
         
           
             
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             values of the m-th NPI performance anomaly set alarm indicator; p m =0, m=1, 2, . . . , M 
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             values of the weighted mean of m-th NPI; w m, current =u m, current , w m, previous =u m, current , m=1, 2, . . . , M 
             values of the standard variation of m-th NPI; σ m, current =1/b, σ m, previous =1/b, m=1, 2, . . . , M 
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             values of the severe and abrupt NPI degradation abatement thresholds; Z m =100, m=1, 2, . . . , M. 
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             values of the abrupt change decision function value; d1 m =0, m=1, 2, . . . M 
             value of the outage decision function value; D=0 (i.e., no outage alarm at initialization.) 
             1) At the next NPI measurement time instant, read in the new NPI values, u m,current,  m=1, 2, . . . , M 
             2) For m=1, 2, . . . , M, set the abrupt change decision function value 
           
         
       
    
     
       
         
           
             
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             3) For m=1, 2, . . . M, if d1 m =1 and p m =0, issue an alarm for m-th NPI performance anomaly and set p m =1, g m =u m,previous −10 
             4) Compute the outage decision function value 
           
         
       
    
     
       
         
           
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                               1 
                             
                             M 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               1 
                               m 
                             
                           
                         
                       
                       ≥ 
                       
                         
                           T 
                           outage 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     possible 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     outage 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     in 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     this 
                                   
                                 
                               
                               
                                 
                                   
                                     NPI 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     measurement 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     interval 
                                   
                                 
                               
                             
                             ) 
                           
                           . 
                         
                       
                     
                   
                 
               
             
           
         
       
         
         
           
             5) If D=1 and P=0, issue an alarm for possible outage and set P=1 
             6) If D=0 and P=1, clear outage alarm and set P=0 
             7) For m=1, 2, . . . M, set w m,current =λu m,current +(1−λ) w m,previous    
             8) For m=1, 2, . . . M, set 
           
         
       
    
     
       
         
           
             
               σ 
               
                 m 
                 , 
                 current 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             
                               min 
                               ⁢ 
                               
                                 { 
                                 
                                   
                                     b 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       σ 
                                       
                                         m 
                                         , 
                                         prev 
                                       
                                     
                                   
                                   , 
                                   
                                     
                                       λ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         σ 
                                         
                                           m 
                                           , 
                                           
                                             prev 
                                             . 
                                           
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           1 
                                           - 
                                           λ 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 u 
                                                 
                                                   m 
                                                   , 
                                                   curr 
                                                 
                                               
                                               - 
                                               
                                                 u 
                                                 
                                                   m 
                                                   , 
                                                   
                                                     prev 
                                                     . 
                                                   
                                                 
                                               
                                             
                                             ) 
                                           
                                           2 
                                         
                                         2 
                                       
                                     
                                   
                                 
                                 } 
                               
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               1 
                               / 
                               b 
                             
                             , 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 if 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   σ 
                                   
                                     m 
                                     , 
                                     previous 
                                   
                                 
                               
                               &lt; 
                               
                                 1 
                                 / 
                                 b 
                               
                             
                           
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     if 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       σ 
                       
                         m 
                         , 
                         
                           prev 
                           . 
                         
                       
                     
                   
                   ≥ 
                   
                     
                       1 
                       / 
                       b 
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     where 
                     ⁢ 
                     
                        
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     min 
                     ⁢ 
                     
                       { 
                       
                         a 
                         , 
                         b 
                       
                       } 
                     
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           a 
                           , 
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               a 
                             
                             ≤ 
                             b 
                           
                           , 
                         
                       
                     
                     
                       
                         
                           b 
                           , 
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               b 
                             
                             &lt; 
                             a 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
         
         
           
             9) For m,=1, 2, . . . M, set the severe and abrupt NPI degradation onset/abatement thresholds 
           
         
       
    
     
       
         
           
             
               T 
               m 
             
             = 
             
               max 
               ⁢ 
               
                 { 
                 
                   0 
                   , 
                   
                     
                       w 
                       
                         m 
                         , 
                         previous 
                       
                     
                     - 
                     
                       max 
                       ⁢ 
                       
                         { 
                         
                           c 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             σ 
                             
                               m 
                               , 
                               previous 
                             
                           
                           ⁢ 
                           
                             
                               
                                 ( 
                                 
                                   λ 
                                   
                                     2 
                                     - 
                                     λ 
                                   
                                 
                                 ) 
                               
                               , 
                               10 
                             
                           
                         
                         } 
                       
                     
                   
                 
                 } 
               
             
           
         
       
       
         
           
             
               Z 
               m 
             
             = 
             
               max 
               ⁡ 
               
                 ( 
                 
                   
                     min 
                     ⁢ 
                     
                       { 
                       
                         100 
                         , 
                         
                           
                             w 
                             
                               m 
                               , 
                               previous 
                             
                           
                           + 
                           
                             min 
                             ⁢ 
                             
                               { 
                               
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   σ 
                                   
                                     m 
                                     , 
                                     previous 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       ( 
                                       
                                         λ 
                                         
                                           2 
                                           - 
                                           λ 
                                         
                                       
                                       ) 
                                     
                                     , 
                                     10 
                                   
                                 
                               
                               } 
                             
                           
                         
                       
                       } 
                     
                   
                   , 
                   
                     g 
                     m 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             where 
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 max 
                 ⁢ 
                 
                   { 
                   
                     a 
                     , 
                     b 
                   
                   } 
                 
               
               = 
               
                 { 
                 
                   
                     
                       
                         a 
                         , 
                         
                           
                             if 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             a 
                           
                           ≥ 
                           b 
                         
                         , 
                       
                     
                   
                   
                     
                       
                         b 
                         , 
                         
                           
                             if 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             b 
                           
                           &gt; 
                           a 
                         
                       
                     
                   
                 
               
             
           
         
       
         
         
           
             10) Set u m,previous =u m,current,  m=1, 2, . . . M 
             11) Set w m,previous =w m,current,  m=1, 2, . . . M 
             12) Set σ m,previous =σ m,current,  m=1, 2, . . . M 
             13) Repeat step 1). 
             3.2 Summary of a Severe &amp; Abrupt NPI Degradation/Outage
           Detection Algorithm (Integer Arithmetic Implementation)   
         
             0) Initialization: Define the ordering of the M NPI, e.g. m=1 for the Mobile Originated call success rate, m=2 or the Land Originated call success rate, etc. Then, 
             read in the first NPI call success rates measurement, u m,current , m=1, 2, . . . , M, and set 
           
         
       
    
     
       
         
           
             
               c 
               = 
               3 
             
             , 
             
               
 
             
             ⁢ 
             
               λ 
               = 
               80 
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   
                     b 
                     = 
                       
                     ⁢ 
                     
                       100 
                       + 
                       
                         
                           [ 
                           
                             
                               ( 
                               
                                 100 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     100 
                                     - 
                                     λ 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     200 
                                     + 
                                     λ 
                                   
                                   ) 
                                 
                               
                             
                             
                               100 
                               + 
                               λ 
                             
                           
                           ] 
                         
                         
                           1 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   
                     = 
                       
                     ⁢ 
                     156 
                   
                 
               
             
           
         
       
         
         
           
             values of the m-th NPI severe and abrupt NPI degradation secondary threshold; g m =0, m=1, 2, . . . , M 
             values of the m-th NPI performance anomaly set alarm indicator; p m =0, m=1, 2, . . . , M 
             values of the set outage alarm indicator; P=0 
             values of the m-th NPI call success rates; u m,previous =u m,current,  m=1, 2, . . . , M 
             values of the weighted mean of the m-th NPI; w m,current =u m,current,  w m,previous =u m,current , m=1, 2, . . . , M 
             values of the standard variation of m-th NPI; σ m,current =10000/b=64, σ m previous =10000/b=64, m=1, 2, . . . , M 
             values of the severe and abrupt NPI degradation thresholds; T m =0, m=1, 2, . . . , M. (This will suppress the outage alarm at the next decision instant after initialization.) 
             values of the severe and abrupt NPI degradation abatement thresholds; Z m =100, m=1, 2, . . . , M. 
             value of the outage detection threshold; T outage =3 
             values of the abrupt change decision function value; d1 m =0, m=1, 2, . . . , M 
             value of the outage decision function value; D=0 (i.e., no outage alarm at initialization.) 
             1) At the next NPI measurement time instant, read in the new NPI values, u m,current , m=1, 2, . . . , M 
             2) For m=1, 2, . . . , M, set the abrupt change decision function value 
           
         
       
    
     
       
         
           
             
               d 
               ⁢ 
               
                   
               
               ⁢ 
               
                 1 
                 m 
               
             
             = 
             
               { 
               
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             
                               m 
                               , 
                               current 
                             
                           
                         
                         ≥ 
                         
                           
                             Z 
                             m 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     there 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     is 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     no 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     sudden 
                                   
                                 
                               
                               
                                 
                                   
                                     change 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     of 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     the 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       m 
                                       th 
                                     
                                     ⁢ 
                                     NPI 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         m 
                       
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           T 
                           m 
                         
                       
                       ≤ 
                       
                         u 
                         
                           m 
                           , 
                           current 
                         
                       
                       &lt; 
                       
                         
                           Z 
                           m 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   there 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   is 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   no 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sudde 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   change 
                                 
                               
                             
                             
                               
                                 
                                   of 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   the 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     m 
                                     th 
                                   
                                   ⁢ 
                                   NPI 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   alarm 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   state 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       1 
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           u 
                           
                             
                               m 
                               , 
                               current 
                             
                             ⁢ 
                             
                                 
                             
                           
                         
                       
                       &lt; 
                       
                         
                           T 
                           m 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   there 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   is 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   a 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sudden 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   change 
                                 
                               
                             
                             
                               
                                 
                                   of 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   the 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     m 
                                     th 
                                   
                                   ⁢ 
                                   NPI 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
         
         
           
             3) For m=1, 2, . . . M, if d1 m =1 and p m =0, issue an alarm for m-th NPI performance anomaly and set p m =1, g m =u m,previous −10 
             4) Compute the outage decision function value 
           
         
       
    
     
       
         
           
             D 
             = 
             
               { 
               
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               M 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 1 
                                 m 
                               
                             
                           
                         
                         = 
                         
                           0 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     no 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     outage 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     in 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     this 
                                   
                                 
                               
                               
                                 
                                   
                                     NPI 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     measurement 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     interval 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                     
                   
                 
                 
                   
                     
                       D 
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                       &lt; 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           M 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             1 
                             m 
                           
                         
                       
                       &lt; 
                       
                         
                           T 
                           outage 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   no 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   change 
                                 
                               
                             
                             
                               
                                 
                                   in 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   outage 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   state 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       1 
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             M 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               1 
                               m 
                             
                           
                         
                       
                       ≥ 
                       
                         
                           
                             T 
                             outage 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   
                                     possible 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     outage 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     in 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     this 
                                   
                                 
                               
                               
                                 
                                   
                                     NPI 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     measurement 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     interval 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
             
           
         
       
         
         
           
             5) If D=1 and P=0, issue an alarm for possible outage and set P=1 
             6) If D=0 and P=1, clear outage alarm and set P=0 
             7) For m=1, 2, . . . M, set 
           
         
       
    
     
       
         
           
             
               
                 
                   
                     w 
                     
                       m 
                       , 
                       current 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       ⌊ 
                       
                         
                           λ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             
                               m 
                               , 
                               current 
                             
                           
                         
                         + 
                         
                           
                             ( 
                             
                               100 
                               - 
                               λ 
                             
                             ) 
                           
                           ⁢ 
                           
                             w 
                             
                               m 
                               , 
                               previous 
                             
                           
                         
                       
                       ⌋ 
                     
                     100 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           80 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             
                               m 
                               , 
                               current 
                             
                           
                         
                         + 
                         
                           20 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             w 
                             
                               m 
                               , 
                               previous 
                             
                           
                         
                       
                       ] 
                     
                     100 
                   
                 
               
             
           
         
       
         
         
           
             8) For m=1, 2, . . . M, set 
           
         
       
    
     
       
         
           
             
               
                 
                   
                     σ 
                     
                       m 
                       , 
                       current 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     { 
                     
                       
                         
                           min 
                           ⁢ 
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       b 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         σ 
                                         
                                           m 
                                           , 
                                           
                                             prev 
                                             . 
                                           
                                         
                                       
                                     
                                     , 
                                     
                                       + 
                                       
                                         
                                           100 
                                           ⁢ 
                                           
                                             ( 
                                             
                                               100 
                                               - 
                                               λ 
                                             
                                             ) 
                                           
                                         
                                         2 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           u 
                                           
                                             m 
                                             , 
                                             
                                               curr 
                                               . 
                                             
                                           
                                         
                                         - 
                                         
                                           u 
                                           
                                             m 
                                             , 
                                             
                                               prev 
                                               . 
                                             
                                           
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                             } 
                           
                         
                         
                           
                             10000 
                             / 
                             b 
                           
                           , 
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 σ 
                                 
                                   m 
                                   , 
                                   previous 
                                 
                               
                             
                             &lt; 
                             
                               10000 
                               / 
                               b 
                             
                           
                         
                       
                       , 
                       
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             σ 
                             
                               m 
                               , 
                               
                                 prev 
                                 . 
                               
                             
                           
                         
                         ≥ 
                         
                           10000 
                           / 
                           b 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 min 
                                 ⁢ 
                                 
                                   { 
                                   
                                     
                                       
                                         
                                           
                                             156 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               σ 
                                               
                                                 m 
                                                 , 
                                                 
                                                   prev 
                                                   . 
                                                 
                                               
                                             
                                           
                                           , 
                                           
                                             
                                               80 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 σ 
                                                 
                                                   m 
                                                   , 
                                                   
                                                     prev 
                                                     . 
                                                   
                                                 
                                               
                                             
                                             + 
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           1000 
                                           ⁢ 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   u 
                                                   
                                                     m 
                                                     , 
                                                     
                                                       curr 
                                                       . 
                                                     
                                                   
                                                 
                                                 - 
                                                 
                                                   u 
                                                   
                                                     m 
                                                     , 
                                                     
                                                       prev 
                                                       . 
                                                     
                                                   
                                                 
                                               
                                               ) 
                                             
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   } 
                                 
                               
                               100 
                             
                           
                         
                         
                           
                             
                               64 
                               , 
                               
                                 
                                   if 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     σ 
                                     
                                       m 
                                       , 
                                       previous 
                                     
                                   
                                 
                                 &lt; 
                                 64 
                               
                             
                           
                         
                       
                       , 
                       
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             σ 
                             
                               m 
                               , 
                               
                                 prev 
                                 . 
                               
                             
                           
                         
                         ≥ 
                         64 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             where 
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 min 
                 ⁢ 
                 
                   { 
                   
                     a 
                     , 
                     b 
                   
                   } 
                 
               
               = 
               
                 { 
                 
                   
                     
                       
                         a 
                         , 
                         
                           
                             if 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             a 
                           
                           ≤ 
                           b 
                         
                         , 
                       
                     
                   
                   
                     
                       
                         b 
                         , 
                         
                           
                             if 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             b 
                           
                           &lt; 
                           a 
                         
                       
                     
                   
                 
               
             
           
         
       
         
         
           
             9) For m=1, 2, . . . M, set the severe and abrupt NPI degradation onset/abatement thresholds 
           
         
       
    
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             T 
                             m 
                           
                           = 
                             
                           ⁢ 
                           
                             max 
                             ⁢ 
                             
                               { 
                               
                                 0 
                                 , 
                                 
                                   
                                     w 
                                     
                                       m 
                                       , 
                                       previous 
                                     
                                   
                                   - 
                                   
                                     max 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           
                                             c 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               σ 
                                               
                                                 m 
                                                 , 
                                                 previous 
                                               
                                             
                                             ⁢ 
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     10000 
                                                     ⁢ 
                                                     
                                                         
                                                     
                                                     ⁢ 
                                                     λ 
                                                   
                                                   
                                                     200 
                                                     - 
                                                     λ 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           10000 
                                         
                                         , 
                                         10 
                                       
                                       } 
                                     
                                   
                                 
                               
                               } 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             max 
                             ⁢ 
                             
                               { 
                               
                                 0 
                                 , 
                                 
                                   
                                     w 
                                     
                                       m 
                                       , 
                                       previous 
                                     
                                   
                                   - 
                                   
                                     max 
                                     ⁢ 
                                     
                                       { 
                                       
                                         
                                           
                                             245 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               σ 
                                               
                                                 m 
                                                 , 
                                                 previous 
                                               
                                             
                                           
                                           10000 
                                         
                                         , 
                                         10 
                                       
                                       } 
                                     
                                   
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             Z 
                             m 
                           
                           = 
                             
                           ⁢ 
                           
                             max 
                             ( 
                             
                               
                                 min 
                                 ⁢ 
                                 
                                   { 
                                   
                                     
                                       100 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         mw 
                                         
                                           m 
                                           , 
                                           previous 
                                         
                                       
                                     
                                     + 
                                     
                                       min 
                                       ⁢ 
                                       
                                         { 
                                         
                                           
                                             
                                               c 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 σ 
                                                 
                                                   m 
                                                   , 
                                                   previous 
                                                 
                                               
                                               ⁢ 
                                               
                                                 
                                                   ( 
                                                   
                                                     
                                                       10000 
                                                       ⁢ 
                                                       
                                                           
                                                       
                                                       ⁢ 
                                                       λ 
                                                     
                                                     
                                                       200 
                                                       - 
                                                       λ 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             10000 
                                           
                                           , 
                                           10 
                                         
                                         } 
                                       
                                     
                                   
                                   } 
                                 
                               
                               , 
                               
                                 g 
                                 m 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             max 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   min 
                                   ⁢ 
                                   
                                     { 
                                     
                                       100 
                                       , 
                                       
                                         
                                           w 
                                           
                                             m 
                                             , 
                                             previous 
                                           
                                         
                                         + 
                                         
                                           min 
                                           ⁢ 
                                           
                                             { 
                                             
                                               
                                                 
                                                   245 
                                                   ⁢ 
                                                   
                                                       
                                                   
                                                   ⁢ 
                                                   
                                                     σ 
                                                     
                                                       m 
                                                       , 
                                                       previous 
                                                     
                                                   
                                                 
                                                 10000 
                                               
                                               , 
                                               10 
                                             
                                             } 
                                           
                                         
                                       
                                     
                                     } 
                                   
                                 
                                 , 
                                 
                                   g 
                                   m 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     where 
                     ⁢ 
                     
                        
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         max 
                         ⁢ 
                         
                           { 
                           
                             a 
                             , 
                             b 
                           
                           } 
                         
                       
                       = 
                       
                         { 
                         
                           
                             
                               
                                 a 
                                 , 
                                 
                                   
                                     if 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     a 
                                   
                                   ≥ 
                                   b 
                                 
                                 , 
                               
                             
                           
                           
                             
                               
                                 b 
                                 , 
                                 
                                   
                                     if 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     b 
                                   
                                   &gt; 
                                   a 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
         
         
           
             10) Set u m,previous =u m,current , m=1, 2, . . . M 
             11) Set w m,previous =w m,current , m=1, 2, . . . M 
             12) Set σ m,previous =σ m,current , m=1, 2, . . . M 
             13) Repeat step 1). 
           
         
         4. Slow and Persistent NPI Degradation Detection Algorithm
       4.1 Summary of a Slow and Persistent NPI Degradation
           Detection Algorithm (Floating Point Implementation)   
           0) Initialization: Use the same ordering of the M NPI as in the severe &amp; abrupt NPI degradation/outage detection algorithm. Then initializes
           values of the NPI OM moving average window; W=100   values of the slow and persistent NPI degradation moving test window; N=50   values of the m-th NPI slow and persistent degradation test statistics; Q m =0, m=1, 2, . . . , M   values of the minimum negative slop threshold of −3% per week, b o =   
           
     
       
    
     
       
         
           
             - 
             
               3 
               
                 7 
                 × 
                 24 
               
             
           
         
       
         
         
           
             
               
                 values of the W present and past m-th NPI values; u mj =−1, j=0, 1, . . . , W, m=1, 2, . . . , M; u m,new =0, m=1, 2, . . . M 
                 values of the N m-th NPI average values; y mj =−1, j=0, 1, . . . , N, m=1, 2, . . . , M 
               
             
           
         
       
    
     
       
         
           
             D 
             = 
             
               
                 
                   
                     
                       N 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         N 
                         + 
                         1 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       N 
                       - 
                       1 
                     
                     ) 
                   
                 
                 12 
               
               ⁢ 
               520625 
             
           
         
       
       
         
           
             
               
                 
                   
                     S 
                     uu 
                     
                       ( 
                       m 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       D 
                       N 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                     ⁢ 
                     102.042 
                   
                   , 
                   
                     m 
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   M 
                 
               
             
           
         
       
         
         
           
             
               
                 values of the m-th NPI performance anomaly set alarm indicator; p m =0, m=1, 2, . . . , M 
                 values of the slow and persistent NPI degradation decision function value; d2 m =0, m=1, 2, . . . , M 
                 values of the slow and persistent NPI degradation detection threshold; T slow =−1.6772 
               
             
             1) At the next NPI measurement time instant, read in the new NPI values, u m,new, m=1, 2, . . . M 
             2) For m=1, 2, . . . M, update the set of W+1 present and past m-th NPI values
           for j=0: W−1, set u mj =u mj+1,  end   u m,W =u m,new      
         
             3) For m=1, 2, . . . M, if u m,0 ≠−1, update the set of N m-th NPI average values
           for j=1: N−1, set y mj =y mj+1,  end   
         
           
         
       
    
     
       
         
           
             
               Y 
               
                 m 
                 , 
                 N 
               
             
             = 
             
               
                 
                   
                     W 
                     
                       ym 
                       , 
                       N 
                     
                   
                   - 
                   
                     u 
                     
                       m 
                       , 
                       
                         0 
                         = 
                       
                     
                   
                 
                 = 
                 
                   u 
                   
                     m 
                     , 
                     W 
                   
                 
               
               W 
             
           
         
       
         
         
           
             4) For m=1, 2, . . . M, if y m,l ≠−1, update the m-th NPI slow and persistent degradation test statistics Q m : 
           
         
       
    
     
       
         
           
             
               A 
               m 
             
             = 
             
               
                 
                   
                     
                       
                         N 
                         ⁡ 
                         
                           ( 
                           
                             N 
                             + 
                             1 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             N 
                           
                           + 
                           1 
                         
                         ) 
                       
                     
                     6 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         y 
                         
                           m 
                           , 
                           j 
                         
                       
                     
                     ) 
                   
                 
                 - 
                 
                   
                     
                       N 
                       ⁡ 
                       
                         ( 
                         
                           N 
                           + 
                           1 
                         
                         ) 
                       
                     
                     2 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         jy 
                         
                           m 
                           , 
                           j 
                         
                       
                     
                     ) 
                   
                 
               
               D 
             
           
         
       
       
         
           
             
               B 
               m 
             
             = 
             
               
                 
                   N 
                   ⁡ 
                   
                     ( 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         jy 
                         
                           m 
                           , 
                           j 
                         
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                     
                 
                 - 
                 
                   
                     
                       N 
                       ⁡ 
                       
                         ( 
                         
                           N 
                           + 
                           1 
                         
                         ) 
                       
                     
                     2 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         y 
                         
                           m 
                           , 
                           j 
                         
                       
                     
                     ) 
                   
                 
               
               D 
             
           
         
       
       
         
           
             
               S 
               m 
             
             = 
             
               
                 
                   
                     ∑ 
                     
                       j 
                       = 
                       1 
                     
                     N 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           y 
                           
                             m 
                             , 
                             j 
                           
                         
                         - 
                         
                           A 
                           m 
                         
                         - 
                         
                           jB 
                           m 
                         
                       
                       ) 
                     
                     2 
                   
                 
                 
                   N 
                   - 
                   2 
                 
               
             
           
         
       
       
         
           
             
               Q 
               m 
             
             = 
             
               
                 
                   ( 
                   
                     
                       B 
                       m 
                     
                     - 
                     
                       b 
                       0 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   S 
                   uu 
                   
                     ( 
                     m 
                     ) 
                   
                 
               
               
                 S 
                 m 
               
             
           
         
       
         
         
           
             5) For m=1, 2, . . . M, if y m,l =−1, set the slow and persistent NPI degradation decision function value 
           
         
       
    
     
       
         
           
             
               d 
               ⁢ 
               
                   
               
               ⁢ 
               
                 2 
                 m 
               
             
             = 
             
               { 
               
                 
                   
                     
                       0 
                       , 
                       
                           
                       
                       ⁢ 
                       
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Q 
                             m 
                           
                         
                         ≥ 
                         
                           T 
                           slow 
                         
                       
                     
                   
                   
                     
                       ( 
                       
                         
                           
                             
                               there 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               is 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               no 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               slow 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               change 
                             
                           
                         
                         
                           
                             
                               of 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               the 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 m 
                                 th 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               NPI 
                             
                           
                         
                       
                       ) 
                     
                   
                 
                 
                   
                     
                       1 
                       , 
                       
                           
                       
                       ⁢ 
                       
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             Q 
                             m 
                           
                         
                         &lt; 
                         
                           T 
                           slow 
                         
                       
                     
                   
                   
                     
                       ( 
                       
                         
                           
                             
                               there 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               is 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               a 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               slow 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               change 
                             
                           
                         
                         
                           
                             
                               of 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               the 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 m 
                                 th 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               NPI 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
         
         
           
             6) For m=1, 2, m . . . M, if d2 m =1 and  Pm =0, issue an alarm for m-th NPI performance anomaly and set p m =1 
             7) Repeat step 1). 
             4.2 Summary of a Slow and Persistent NPI Degradation
           Detection Algorithm (Integer Arithmetic Implementation: Signed 32 Bits)   
         
             0) Initialization: Use the same ordering of the M NPI as in the severe &amp; abrupt NPI degradation/outage detection algorithm, then initializes
           values of the NPI OM moving average window; W=100   values of the slow and persistent NPI degradation moving test window; N=50   values of the m-th NPI slow and persistent degradation test statistics; Q m =0, m=1, 2, . . . , M   values of the W present and past m-th NPI values; u mj =−1, j=0, 1, . . . , W, m=1, 2, . . . , M; u m,new =0, m=1, 2, . . . M   values of the N m-th NPI average values; y mj =−1, j=0, 1, . . . N, m=1, 2, . . . , M   value of the integer scale factor; G=1000   values of the minimum negative slop threshold of −3% per week,   
         
           
         
       
    
     
       
         
           
             
               
                 
                   
                     b 
                     0 
                   
                   = 
                     
                   ⁢ 
                   
                     - 
                     
                       
                         3 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         G 
                       
                       
                         7 
                         × 
                         24 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     - 
                     18 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   D 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         
                           N 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             N 
                             + 
                             1 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           N 
                           - 
                           1 
                         
                         ) 
                       
                     
                     12 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   520625 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     s 
                     uu 
                     
                       ( 
                       m 
                       ) 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     G 
                     ⁢ 
                     
                       
                         D 
                         N 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                     ⁢ 
                     102042 
                   
                   , 
                 
               
             
           
         
       
       
         
           
             
               m 
               = 
               1 
             
             , 
             2 
             , 
             … 
             ⁢ 
             
                 
             
             , 
             M 
           
         
       
         
         
           
             
               
                 values of the m-th NPI performance anomaly set alarm indicator; p m =0, m=1, 2, . . . , M 
                 values of the slow and persistent NPI degradation decision function value; d2 m =0, m=1, 2, . . . , M 
                 values of the slow and persistent NPI degradation detection threshold; T slow =−1677 
               
             
             1) At the next NPI measurement time instant, read in the new NPI values, u m,new, m=1, 2, . . . M 
             2) For m=1, 2, . . . M, update the set of W+1 present and past m-th NPI values
           for j=0: W−1, set u mj =u mj+1,  end   u m,W =U m,new      
         
             3) For m=1, 2, . . . M, if u m,0 ≠−1, update the set of N m-th NPI average values 
           
         
       
    
     
       
         
           
             
               
                 for 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 j 
               
               = 
               
                 1 
                 : 
                 
                   N 
                   - 
                   1 
                 
               
             
             , 
             
               
                 set 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   y 
                   mj 
                 
               
               = 
               
                 
                   y 
                   
                     
                       mj 
                       + 
                       1 
                     
                     , 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 end 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     y 
                     m 
                   
                   ⁢ 
                   
                     , 
                     N 
                   
                   ⁢ 
                   
                     = 
                       
                     ⁢ 
                     
                       
                         
                           Wy 
                           
                             n 
                             , 
                             N 
                           
                         
                         - 
                         
                           100 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             
                               m 
                               , 
                               0 
                             
                           
                         
                         + 
                         
                           100 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             
                               m 
                               , 
                               W 
                             
                           
                         
                       
                       W 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       y 
                       
                         m 
                         , 
                         N 
                       
                     
                     + 
                     
                       
                         100 
                         W 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             u 
                             
                               m 
                               , 
                               W 
                             
                           
                           - 
                           
                             u 
                             
                               m 
                               , 
                               0 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
         
         
           
             4) For m=1, 2, . . . M, if y m,l ≠−1, update the m-th NPI slow and persistent degradation test statistics Q m : 
           
         
       
    
     
       
         
           
             
               A 
               m 
             
             = 
             
               
                 
                   2 
                   ⁢ 
                   
                     ( 
                     
                       
                         2 
                         ⁢ 
                         N 
                       
                       + 
                       1 
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           y 
                           
                             m 
                             , 
                             j 
                           
                         
                       
                       10 
                     
                     ) 
                   
                 
                 - 
                 
                   6 
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             y 
                             
                               m 
                               , 
                               j 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                       
                       10 
                     
                     ) 
                   
                 
               
               
                 N 
                 ⁡ 
                 
                   ( 
                   
                     N 
                     - 
                     1 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               B 
               m 
             
             = 
             
               
                 
                   12 
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           N 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           jy 
                           
                             m 
                             , 
                             j 
                           
                         
                       
                       10 
                     
                     ) 
                   
                 
                 - 
                 
                   6 
                   ⁢ 
                   
                     ( 
                     
                       N 
                       + 
                       1 
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             y 
                             
                               m 
                               , 
                               j 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                       
                       10 
                     
                     ) 
                   
                 
               
               
                 N 
                 ( 
                 
                   N 
                   + 
                   
                     1 
                     ⁢ 
                     _ 
                     ⁢ 
                     
                       ( 
                       
                         N 
                         - 
                         1 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               S 
               m 
             
             = 
             
               
                 
                   100 
                   ⁢ 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       N 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             y 
                             
                               m 
                               , 
                               j 
                             
                           
                           - 
                           
                             10 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               A 
                               m 
                             
                           
                           - 
                           
                             10 
                             ⁢ 
                             j 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               B 
                               m 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
                 
                   N 
                   - 
                   2 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Q 
                     m 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             100 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               B 
                               m 
                             
                           
                           - 
                           
                             b 
                             0 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         S 
                         uu 
                         
                           ( 
                           m 
                           ) 
                         
                       
                     
                     
                       S 
                       m 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             100 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               B 
                               m 
                             
                           
                           + 
                           28 
                         
                         ) 
                       
                       ⁢ 
                       
                         S 
                         uu 
                         
                           ( 
                           m 
                           ) 
                         
                       
                     
                     
                       S 
                       m 
                     
                   
                 
               
             
           
         
       
         
         
           
             5) For m=1, 2, . . . M, if y m,l ≠−1, set the slow and persistent NPI degradation decision function value 
           
         
       
    
     
       
         
           
             
               d 
               ⁢ 
               
                   
               
               ⁢ 
               
                 2 
                 m 
               
             
             = 
             
               { 
               
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Q 
                           m 
                         
                       
                       ≥ 
                       
                         T 
                         slow 
                       
                     
                   
                   
                     
                       ( 
                       
                         
                           
                             
                               there 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               is 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               no 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               slow 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               change 
                             
                           
                         
                         
                           
                             
                               of 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
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                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 m 
                                 th 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               NPI 
                             
                           
                         
                       
                       ) 
                     
                   
                 
                 
                   
                     
                       1 
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           Q 
                           m 
                         
                       
                       &lt; 
                       
                         T 
                         slow 
                       
                     
                   
                   
                     
                       ( 
                       
                         
                           
                             
                               there 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               is 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               a 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               slow 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               change 
                             
                           
                         
                         
                           
                             
                               of 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               the 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 m 
                                 th 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               NPI 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
         
         
           
             6) For m=1, 2, m . . . M, if d2 m =1 and p m =0, issue an alarm for m-th NPI performance anomaly and set p m =1 
             7) Repeat step 1). 
           
         
         5. Recovery of NPI Long Term Average Performance Detection Algorithm 
       
    
     Once the system has entered the network performance anomaly state and the alarm has been set, in order to declare that the network performance anomaly event has abated and the system has entered the ‘normal’ state it is necessary to make sure that the system performance has reverted back to its most recent long term average performance. To achieve this recovery detection goal, a test statistic constructed from a 7-day moving average estimate of the mean and variance of each of the NPI OMs can be used. Suppose there are J samples of the OMs over the 7-day period, then the sample mean value of the m-th NPI at the time instant k is given by 
                     l     m   ,   k       =       ⁢       1   J     ⁢       ∑     i   =     k   -   J   +   1       k     ⁢           ⁢     u     m   ,   i                       =       ⁢           Jl     m   ,     k   -   1         -     u     k   -   J       +     u     m   ,   k         J     .                 
Let q m,k :=Σ i=k-J+1   k u m,i   2 , then the sample variance of the m-th NPI at the time instant k is given by
 
                     L     m   ,   k       =       ⁢       1     J   -   1       ⁢       ∑     i   =     k   -   J   +   1       k     ⁢           ⁢       (       u     m   ,   i       -     l     m   ,   k         )     2                     =       ⁢       1     J   -   1       ⁢       (         ∑     i   =     k   -   J   +   1       k     ⁢           ⁢     u     m   ,   i     2       -     Jl     m   ,   k     2       )     .                   =       ⁢       1     J   -   1       ⁢       (       q     m   ,     k   -   1         -     u     m   ,     k   -   J       2     +     u     m   ,   k     2     -     Jl     m   ,   k     2       )     .                   
With the above recursive relations for sample mean and sample variance, it is straight forward to construct a recovery to long term performance detection algorithm.
         5.1 Summary of a Recovery of NPI Long Term Average
           Performance Detection Algorithm (Floating Point Implementation)   
           0.a) Initialization Option 1 (Without using data from prior to the start time of the algorithm): Use the same ordering of the M NPI as in the severe &amp; abrupt NPI degradation/outage detection algorithm. Then initializes   values of the NPI OM moving average window is the number of samples during a 7-day period; J   values of the m-th NPI long term average performance recovery threshold; V m =0, m=1, 2, . . . , M   values of the J present and past m-th NPI; u mj =−1, j=0, 1, . . . , J 0, 1, . . . , J, m=1, 2, . . . , M; u m,new =0, m=1, 2, . . . M   values of the m-th NPI sample average; l m =−1, m=1, 2, . . . , M   values of the m-th NPI sum of square; q m =J 2 , m=1, 2, . . . , M   values of the m-th NPI sample variance; L M =0, m=1, 2, . . . , M   values of the m-th NPI long term average performance recovery decision function value; d3 m =0, m=1, 2, . . . , M       1) At the next NPI measurement time instant, read in the new NPI values, u m,new , m=1, 2, . . . M   2) For m=1, 2, . . . M, update the set of J+1 present and past m-th NPI values
       for j=0: J−1, set u mj =u mj+1,  end   u m,J =u m,new      
       3) For m=1, 2, . . . M, if u m,0 ≠−1, update the m-th NPI sample average values   

     
       
         
           
             
               l 
               m 
             
             = 
             
               
                 
                   Jl 
                   m 
                 
                 - 
                 
                   u 
                   
                     m 
                     , 
                     0 
                   
                 
                 + 
                 
                   u 
                   
                     m 
                     , 
                     J 
                   
                 
               
               J 
             
           
         
       
         
         4) For m=1, 2, . . . M, if u m,0 ≠−1, update the m-th NPI sample variance values 
       
    
     
       
         
           
             
               L 
               m 
             
             = 
             
               
                 1 
                 
                   J 
                   - 
                   1 
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     q 
                     m 
                   
                   - 
                   
                     u 
                     
                       m 
                       , 
                       0 
                     
                     2 
                   
                   + 
                   
                     u 
                     
                       m 
                       , 
                       J 
                     
                     2 
                   
                   - 
                   
                     Jl 
                     m 
                     2 
                   
                 
                 ) 
               
             
           
         
       
         
         5) For m=1, 2, . . . m, if u m,0 ≠−1, update the m-th NPI sample sum of squares values
 
 q   m = q   m − u   m,0   2 + u   m,J   2  
 
         6) For m=1, 2, . . . M, if l m ≠100, update the m-th NPI long term average performance recovery threshold V m :
 
 V   m =max{0, l   m −3√{square root over ( L   m )}}
 
For m=1, 2, . . . M, if l m ≠100, set the long term average performance recovery decision function value
 
       
    
     
       
         
           
             
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               ⁢ 
               
                   
               
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                 3 
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             = 
             
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         7) Repeat step 1).
       5.2 Summary of a Recovery of NPI Long Term Average
           Performance Detection Algorithm (Integer Arithmetic Implementation)   
           
     
         0) Initialization: Use the same ordering of the M NPI as in the severe &amp; abrupt NPI degradation/outage detection algorithm. Then initializes
       values of the NPI OM moving average window is the number of samples during a 7-day period; J   values of the m-th NPI long term average performance recovery threshold; V m =0, m=1, 2, . . . , M   values of the J present and past m-th NPI; u mj =−1, j=0, 1, . . . , J, m=1, 2, . . . , M; u m,new =0, m=1, 2, . . . M   values of the m-th NPI sample average; l m =100, m=1, 2, . . . , M   values of the m-th NPI sum of square; q m =0, m=1, 2, . . . , M   values of the m-th NPI sample variance; L m =0, m=1, 2, . . . , M   values of the m-th NPI long term average performance recovery decision function value; d3 m =0, m=1, 2, . . . , M   
     
         1) At the next NPI measurement time instant, read in the new NPI values, u m,new , m=1, 2, . . . M 
         2) For m=1, 2, . . . M, update the set of J+1 present and past m-th NPI values
       for j=0: J−1, set u mj =u mj+1 , end   u m,J =u m,new      
     
         3) For m=1, 2, . . . M, if u m,0 ≠−1, update the m-th NPI sample average values 
       
    
     
       
         
           
             
               l 
               m 
             
             = 
             
               
                 
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                   m 
                 
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                   ⁢ 
                   
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                       , 
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                 + 
                 
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                   ⁢ 
                   
                     u 
                     
                       m 
                       , 
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               J 
             
           
         
       
         
         4) For m=1, 2, . . . M, if u m,0 ≠−1, update the m-th NPI sample variance values 
       
    
     
       
         
           
             Lm 
             = 
             
               
                 1 
                 
                   J 
                   - 
                   1 
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     q 
                     m 
                   
                   - 
                   
                     10000 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       u 
                       
                         m 
                         , 
                         0 
                       
                       2 
                     
                   
                   + 
                   
                     10000 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       u 
                       
                         m 
                         , 
                         J 
                       
                       2 
                     
                   
                   - 
                   
                     Jl 
                     m 
                     2 
                   
                 
                 ) 
               
             
           
         
       
         
         5) For m=1, 2, . . . M, if u m,0 ≠−1, update the m-th NPI sample sum of square values
 
 q   m = q   m −10000 u   m,0   2 +10000 u   m,J   2  
 
         6) For m=1, 2, . . . M, if l m ≠−100, update the m-th NPI long term average performance recovery threshold V m : 
       
    
     
       
         
           
             
               V 
               m 
             
             = 
             
               max 
               ⁢ 
               
                 { 
                 
                   0 
                   , 
                   
                     
                       
                         l 
                         m 
                       
                       - 
                       
                         3 
                         ⁢ 
                         
                           
                             L 
                             m 
                           
                         
                       
                     
                     100 
                   
                 
                 } 
               
             
           
         
       
         
         7) For m=1, 2, . . . M, if l m ≠−100, set the long term average performance recovery decision function value 
       
    
     
       
         
           
             
               d 
               ⁢ 
               
                   
               
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                 3 
                 m 
               
             
             = 
             
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                         ⁢ 
                         
                             
                         
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                         m 
                       
                     
                   
                   
                     
                       ( 
                       
                         
                           
                             
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                               ⁢ 
                               
                                   
                               
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                               ⁢ 
                               
                                   
                               
                               ⁢ 
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                               has 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               not 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               recovered 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               yet 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 m 
                                 th 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               NPI 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
         
         8) Repeat step 1). 
         6. Performance Anomaly Clear Alarm Algorithm
 
Use the same ordering of the M NPI as the severe &amp; abrupt NPI degradation/outage detection.
 
         1) At the next NPI measurement time instant after finish executing all the three performance anomaly detection algorithm, for m=1, 2, . . . M, read d1 m , d2 m , d3 m  and p m    
         2) If d1 m =0, d2 m =0, d3 m =0, and p m =1, clear alarm for NPI performance anomaly of the m-th NPI and set p m =0 
         3) Repeat 1). 
       
    
     As set forth above, a scheme for network performance anomaly detection has been disclosed based on 1) detecting severe and sudden change in the NPI OM 2) Detecting slow and persistent degradation of the NPI OM. Furthermore, utilizing multiple NPIs helps reduce the false alarm probability while maximizing the probability of outage detection. Each NPI has two network performance degradation thresholds which dynamically adapts to changes in the most recent mean and variance values of the NPI success rate in the severe and abrupt NPI degradation detection. The severe and abrupt performance degradation decision rule compares the most recent measurement of NPI call success rate against this dynamic threshold value in order to determine if there is a statistically significant large and sudden change from the most recent mean value. At any particular time instant at which the NPI call success rate dropped below the threshold, a network performance anomaly alarm is issued for that NPI. 
     The second dynamic threshold for network performance anomaly detection uses a low pass filter (i.e. long moving average window) to smooth out the normal NPIs daily fluctuation in order to discriminate real versus fictitious slow downward trend in the NPI OM performance. To ascertain that the network performance anomaly event has abated, three abatement dynamic thresholds are proposed. The first two thresholds concern the detection of the abatement event related to the severe and abrupt NPI degradation detection and the slow and persistent NPI detection algorithm. The last: abatement dynamic threshold is used to check whether the net-work performance has recovered to its long term average performance level. Once a network performance anomaly alarm has been set, it could only be cleared when the relevant NPI OM value exceeds all three abatement thresholds. 
     The above-described steps can be implemented using standard well-known programming techniques. The novelty of the above-described embodiment lies not in the specific programming techniques but in the use of the steps described to achieve the described results. Software programming code which embodies the present invention is typically stored in permanent storage. In a client/server environment, such software programming code may be stored with storage associated with a server. The software programming code may be embodied on any of a variety of known media for use with a data processing system, such as a diskette, or hard drive, or CD ROM. The code may be distributed on such media, or may be distributed to users from the memory or storage of one computer system over a network of some type to other computer systems for use by users of such other systems. The techniques and methods for embodying software program code on physical media and/or distributing software code via networks are well known and will not be further discussed herein. 
     It will be understood that each element of the illustrations, and combinations of elements in the illustrations, can be implemented by general and/or special purpose hardware-based systems that perform the specified functions or steps, or by combinations of general and/or special-purpose hardware and computer instructions. 
     These program instructions may be provided to a processor to produce a machine, such that the instructions that execute on the processor create means for implementing the functions specified in the illustrations. The computer program instructions may be executed by a processor to cause a series of operational steps to be performed by the processor to produce a computer-implemented process such that the instructions that execute on the processor provide steps for implementing the functions specified in the illustrations. Accordingly, the figures support combinations of means for performing the specified functions, combinations of steps for performing the specified functions, and program instruction means for performing the specified functions. 
     Although the present invention has been described with respect to a specific preferred embodiment thereof, various changes and modifications may be suggested to one skilled in the art and it is intended that the present invention encompass such changes and modifications as fall within the scope of the appended claims.
In order to provide an early and more accurate determination of network problems, current NPI OMs are compared with samples of recent historical NPI OMs so that changes in the NPI OM are detected based on current overall network conditions rather than on conditions that may have existed at statistically insignificant earlier operational periods. By constantly adjusting a performance threshold, against which the current NPI OM is compared, by using a smaller and very recent sampling of NPIs (in the case of sudden and abrupt performance-NPI degradation detection) or a larger and greater number of NPIs over a wider time period (in the case of slow and persistent NPI degradation detection) to establish the threshold, detection results are more accurate and meaningful.