A system is provided for maintenance of a manufactured product composed of a plurality of parts. A modeling engine may receive failure data for the plurality of parts in which the failure data indicates individual instances of failure of at least two of a multiple quantity of a part of the plurality of parts. Each of the at least two of the multiple quantity may be located at a respective distinct location in the manufactured product. The modeling engine may generate a superimposed failure model (SFM) for the part, including determining a lifetime distribution of the part based at least partially on application of a lifetime distribution model to the SFM. A maintenance engine coupled to the modeling engine may perform a maintenance activity including determining a maintenance interval determined for the part according to the lifetime distribution of the part.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A system for maintenance of a manufactured product composed of a plurality of parts including a multiple quantity of a part of the plurality of parts, the system including at least: a data collection engine configured to receive failure data for the plurality of parts in which the failure data indicates individual instances of failure of at least two of the multiple quantity of the part, each of the at least two of the multiple quantity being located at a respective distinct location in the manufactured product; a modeling engine configured to generate a superimposed failure model (SFM) for the part based at least partially on the failure data, including the modeling engine being configured to determine a lifetime distribution of the part based at least partially on application of a lifetime distribution model to the SFM; and a maintenance engine coupled to the modeling engine and configured to perform a maintenance activity for the part, including the maintenance engine being configured to determine a maintenance interval for the part according to the lifetime distribution of the part. 2. The system of claim 1, wherein the modeling engine being configured to generate the SFM includes being configured to: identify an observed failure time for each of the individual instances of failure of the at least two of the multiple quantity of the part; and generate the SFM according to the following: S  ( t ) · [ ∑ m = 1 i - 1  ( S  ( t + ∑ k = 1 m  T i - k ) S  ( ∑ k = 1 m  T i - k ) · P ( X > ∑ k = 1 m  T i - k   T 1 , …  , T i - 1 ) ] n - 1 wherein t represents a given time, n represents the multiple quantity of the part, i represents an index for a sequence of observed failure times, x represents a given variable lifetime, S represents a survival function, Ti represents an observed time interval between at least two observed failure times, and F(x) represents a distribution function of the part that indicates a probability that a lifetime of the part is less than x, and wherein the modeling engine being configured to generate the SFM includes being configured to calculate P (X>Σk=1mTi-k|T1, . . . , Ti-1 according to: F(T1+T2+ . . . +Ti-m-1)−S(Ti-m-1)P(X fails at Σi=1i−m−2Tl|T1, . . . , Ti-1)−S(Ti-m-1+Ti-m-2)P(X fails at Σi=1i−m−3Tl|T1, . . . , Ti-1)− . . . −S(T2+T3+ . . . +Ti-m-1)F(T1) 3. The system of claim 2, wherein the modeling engine being configured to generate the SFM includes being configured to calculate T1 according to: Ti=min(X, R1, . . . , Rn-1), wherein X represents a random lifetime of the part, and R represents a corresponding residual lifetime of the part at each respective distinct location. 4. The system of claim 2, wherein S  ( t + x ) s  ( x ) represents the lifetime distribution model, and application of the lifetime distribution model to the SFM includes application of an exponential distribution model according to the following: S  ( t + x ) S  ( x ) =  - λ  ( t + x )  - λ   x wherein λ represents a constant failure rate at any observed failure time. 5. The system of claim 2, wherein S  ( t + x ) s  ( x ) represents the lifetime distribution model, and application of the lifetime distribution model to the SFM includes application of a Weibull distribution model according to the following: S  ( t + x ) S  ( x ) =  - [ ( t + x ) / α ] β  - ( x / α ) β wherein α and β represent respectively a scale parameter and a shape parameter. 6. The system of claim 1, wherein the modeling engine is further configured to at least: generate a scale parameter or shape parameter utilizing the lifetime distribution model; and determine upper and lower bounds of the lifetime distribution of the part based at least partially on the scale parameter or shape parameter, the maintenance interval being based at least partially on the upper and lower bounds of the lifetime distribution of the part. 7. The system of claim 1, wherein the maintenance engine is further configured to automatically schedule the part for removal or replacement based at least partially on the maintenance interval determined for the part; and communicate the schedule to a user. 8. A method for maintenance of a manufactured product composed of a plurality of parts including a multiple quantity of a part of the plurality of parts, the method comprising: receiving failure data for the plurality of parts in which the failure data indicates individual instances of failure of at least two of the multiple quantity of the part, each of the at least two of the multiple quantity being located at a respective distinct location in the manufactured product; generating a superimposed failure model (SFM) for the multiple quantity of the part based at least partially on the failure data, including determining a lifetime distribution of the part based at least partially on application of a lifetime distribution model to the SFM; and performing a maintenance activity for the part, including determining a maintenance interval for the part according to the lifetime distribution of the part. 9. The method of claim 8, wherein generating the SFM includes: identifying an observed failure time for each of the individual instances of failure of the at least two of the multiple quantity of the part; and generating the SFM includes generating the SFM according to the following: S  ( t ) · [ ∑ m = 1 i - 1  ( S  ( t + ∑ k = 1 m  T i - k ) S  ( ∑ k = 1 m  T i - k ) · P ( X > ∑ k = 1 m  T i - k   T 1 , …  , T i - 1 ) ] n - 1 wherein t represents a given time, n represents the multiple quantity of the part, i represents an index for a sequence of observed failure times, x represents a given variable lifetime, S represents a survival function, T, represents an observed time interval between at least two observed failure times, and F(x) represents a distribution function of the part that indicates a probability that a lifetime of the part is less than x, and wherein generating the SFM includes calculating P(X>Σk=1mTi-k|T1, . . . , Ti-1 according to: F(T1+T2+ . . . +Ti-m-1)−S(Ti-m-1)P(X fails at Σl=1i−m−2Tl|T1, . . . , Ti-1)−S(Ti-m-1+Ti-m-2)P(X fails at Σl=1i−m−3Tl|T1, . . . , Ti-1)− . . . −S(T2+T3+ . . . +Ti-m-1)F(T1) 10. The method of claim 9, wherein generating the SFM includes being configured to calculate Ti according to: Ti=min(X, R1, . . . , Rn-1), wherein X represents a random lifetime of the part, and R represents a corresponding residual lifetime of the part at each respective distinct location. 11. The method of claim 9, wherein S  ( t + x ) s  ( x ) represents the lifetime distribution model, and application of the lifetime distribution model to the SFM includes application of an exponential distribution model according to the following: S  ( t + x ) S  ( x ) =  - λ  ( t + x )  - λ   x wherein λ represents a constant failure rate at any distinct failure time (t). 12. The method of claim 9, wherein S  ( t + x ) s  ( x ) represents the lifetime distribution model, and application of the lifetime distribution model to the SFM includes application of a Weibull distribution model according to the following: S  ( t + x ) S  ( x ) =  - [ ( t + x ) / α ] β  - ( x / α ) β wherein α and β represent respectively a scale parameter and a shape parameter. 13. The method of claim 8 further comprising: generating a scale parameter or shape parameter utilizing the lifetime distribution model; and determining upper and lower bounds of the lifetime distribution of the part based at least partially on the scale parameter or shape parameter, the maintenance interval being based at least partially on the upper and lower bounds of the lifetime distribution of the part. 14. The method of claim 8 further comprising automatically removing or replacing the part based at least partially on the maintenance interval determined for the part. 15. A computer-readable storage medium for maintenance of a manufactured product composed of a plurality of parts including a multiple quantity of a part of the plurality of parts, the computer-readable storage medium having computer-readable program code stored therein that, in response to execution by a processor, cause an apparatus to at least: receive failure data for the plurality of parts in which the failure data indicates individual instances of failure of at least two of the multiple quantity of the part, each of the at least two of the multiple quantity being located at a respective distinct location in the manufactured product; generate a superimposed failure model (SFM) for the multiple quantity of the part based at least partially on the failure data, including determining a lifetime distribution of the part based at least partially on application of a lifetime distribution model to the SFM; and perform a maintenance activity for the part, including determining a maintenance interval for the part according to the lifetime distribution of the part. 16. The computer-readable storage medium of claim 15, wherein the apparatus being caused to generate the SFM includes being caused to: identify an observed failure time for each of the individual instances of failure of the at least two of the multiple quantity of the part; and generate the SFM according to the following: S  ( t ) · [ ∑ m = 1 i - 1  ( S  ( t + ∑ k = 1 m  T i - k ) S  ( ∑ k = 1 m  T i - k ) · P ( X > ∑ k = 1 m  T i - k   T 1 , …  , T i - 1 ) ] n - 1 wherein t represents a given time, n represents the multiple quantity of the part, i represents an index for a sequence of observed failure times, x represents a given variable lifetime, S represents a survival function, Ti represents an observed time interval between at least two observed failure times, and F(x) represents a distribution function of the part that indicates a probability that a lifetime of the part is less than x, and wherein the apparatus being caused to generate the SFM includes being caused to calculate P(X>Σk=1mT1-k|T1, . . . , Ti-1 according to: F(T1+T2+ . . . +Ti-m-1)−S(Ti-m-1)P(X fails at Σl=1i−m−2Tl|T1, . . . , Ti-1)−S(Ti-m-1+Ti-m-2)P(X fails at Σi=1i−m−3Tl|T1, . . . , Ti-1)− . . . −S(T2+T3+ . . . +Ti-m-1)F(T1) 17. The computer-readable storage medium of claim 16, wherein the apparatus being caused to generate the SFM includes being caused to calculate T1 according to: Ti=min(X, R1, . . . , Rn-1), wherein X represents a random lifetime of the part, and R represents a corresponding residual lifetime of the part at each respective distinct location. 18. The computer-readable storage medium of claim 16, wherein S  ( t + x ) s  ( x ) represents the lifetime distribution model, and application of the lifetime distribution model to the SFM includes application of an exponential distribution model according to the following: S  ( t + x ) S  ( x ) =  - λ  ( t + x )  - λ   x wherein λ represents a constant failure rate at any distinct failure time (t). 19. The computer-readable storage medium of claim 16, wherein S  ( t + x ) s  ( x ) represents the lifetime distribution model, and application of the lifetime distribution model to the SFM includes application of a Weibull distribution model according to the following: S  ( t + x ) S  ( x ) =  - [ ( t + x ) / α ] β  - ( x / α ) β wherein α and β represent respectively a scale parameter and a shape parameter. 20. The computer-readable storage medium of claim 15, the apparatus being further caused to: generate a scale parameter or shape parameter utilizing the lifetime distribution model; and determine upper and lower bounds of the lifetime distribution of the part based at least partially on the scale parameter or shape parameter, the maintenance interval being based at least partially on the upper and lower bounds of the lifetime distribution of the part. 21. The computer-readable storage medium of claim 15, the apparatus being further caused to automatically schedule the part for removal or replacement based at least partially on the maintenance interval determined for the part; and communicate the schedule to a user.