in many real world queueing systems , the server may be unavailable for a random period of time when there is no customer in the waiting line at a service completion instant .
this random period of server absence is often called server vacation ; see doshi   and tian and zhang .
the classical vacation scheme with bernoulli - schedule ( bs ) discipline was introduced and studied by keilson and servi .
various aspects of bernoulli - schedule vacation models for single server queueing systems have been studied by servi   and ramaswamy and servi .
servi and finn   introduced a class of semivacation policies known as working vacation ( wv ) wherein a customer is served at a slower rate rather than keeping completely inactive during a vacation . at a service completion epoch during a regular busy period if the queue length is empty , the server may take multiple working vacations ( mwv ) .
the analysis of gi / m/1 queue with mwv is carried out by baba   using the matrix - analytic method .
analysis of finite buffer gi / m/1 queue with mwv can be found in banik et al . .
in order to utilize the server effectively , vacation interruption ( vi ) has become an important aspect , where the server interrupts the vacation and resumes regular service if at least one customer is present in the queue at a service completion epoch during a vacation . li and tian   studied a markovian queue with vacation interruption .
the gi / m/1 queue with bernoulli - schedule vacation interruption ( bs - vi ) has been analyzed by tao et al . .
using the matrix - analytic method , they have obtained the steady state distributions for the queue length , waiting time , and sojourn times .
for some controllable queueing systems with vacations , it is usually assumed that the server is available or unavailable completely depending upon the number of customers present in the system .
whenever the system is empty , the server goes on vacation . in the instant at which the server returns back from a vacation and finds at least n customers in the system
this type of control policy is also called n - policy for the queueing systems with vacations .
a brief analysis on finite buffer gi / m/1 queue with n - policy has been given by ke and wang   and ke .
investigated a quorum queueing system with a random setup time under n - policy and with bs vacations . an infinite buffer markovian queue with mwv and n - policy
state dependent queues are dependent on the queue size , arrivals , and their service times .
this is applicable in many areas like cellular manufacturing cells , routers , and switches that regulate the transmission of information packages having finite buffer capacity . in
many of these applications , the arrival and service rates depend on the state of the queue .
these queues have applications on arrivals and service rates which yield less waiting time in the system .
chao and rahman   have analyzed an n - policy g / m(n)/1/k queue with state dependent vacations .
an efficient algorithm for the state dependent services and state dependent mwv for gi / m(n)/1/n has been presented by goswami et al . .
recently , a computational algorithm for the steady state probabilities in a gi / m(n)/1/n / swv - vi queue has been presented by laxmi et al . .
the present paper is an extension of the work of [ 10 , 16 ] wherein our aim is to include n - policy and bs - vi in a finite buffer gi / m(n)/1 queue .
the analysis of such state dependent service models has not been carried out so far to the best of our knowledge .
further , the inclusion of n - policy with bs - vi utilizes the server more and decreases the waiting lines effectively in order to economize operating cost and energy consumption .
motivated by the above observations , this paper aims to contribute to the theory of bs - vi models with n - policy .
the service times during service period , vacation period , and vacation times are exponentially distributed .
we provide a recursive method using the supplementary variable technique and treating the remaining interarrival time as the supplementary variable , to develop the steady state system length distributions at prearrival and arbitrary epochs .
some performance measures such as blocking probability , the expected queue length , and the expected waiting time have been evaluated .
section 5 contains some numerical results to show the effectiveness of the model parameters followed by conclusions in section 6 .
in this paper , we consider a gi / m(n)/1/k queue with working vacations and bs - vi under n - policy , where k is the finite buffer space .
we assume that the interarrival times of successive arrivals are independent and identically distributed ( i.i.d . )
random variables with cumulative distribution function a(x ) , probability density function a(x ) , x  0 , and laplace - stieltjes transform ( l .- s .
t ) a( ) with mean interarrival time 1/ = a(0 ) , where h(0 ) is the first derivative of h( ) evaluated at  = 0 .
whenever the system becomes empty , the server takes a wv . during a wv ,
a customer is served at a rate generally lower than the regular service rate . at a service completion epoch during wv , if there are at least n customers present in the queue , the server interrupts the vacation with probability q=1-q and switches to regular service and otherwise continues the vacation with probability q. the service times during regular busy period and during wv period are exponentially distributed with rates n and n , 1  n  k , respectively , if there are n customers present in the system before the beginning of a service .
let  ,  , and  be the mean service rates during a regular busy period , during wv period and mean vacation rate respectively .
they are given by  = n=1n / k ,  = n=1n / k , and  = n=1n / k .
the traffic intensity is given by  = /. let us define the state of the system at time t as ns(t ) denoting the number of customers present in the system including the one in service , u(t ) is the remaining interarrival time for the next arrival , and 
 ( 1)(t)=0,if  the  server  is  in  wv  period,1,if  the  server  is  in  regular  busy  period . 

the joint probabilities denoted by i,0(x , t ) and i,1(x , t ) are defined as 
 ( 2)n , jx , tdx  = limtpnstn , xutx+dx,t = j ,  x0 ,  jnk ,  j=0,1 . 

the system length distributions at prearrival epoch are obtained by developing differential - difference equations at steady state .
treating the remaining interarrival time as supplementary variable , we write the equations as 
 ( 3)0,0(1)(x)=11,1(x)+11,0(x),n,01x=nn,0x+axn1,00+n+1n+1,0x ,  1nn1,n,01x=nn,0x+axn1,00+qn+1n+1,0(x ) ,  nnk1,k,01x=kk,0x+axk1,00+k,00,1,11x=11,1x+22,1x,n,11x=nn,1x+n+1n+1,1x+a(x)n1,1(0 ) ,  2nn1,n,11x=nn,1x+n+1n+1,1x+qn+1n+1,0(x)+nn,0x+axn1,10 ,  nnk1,k,11x=kk,1x+kk,0x+axk1,10+k,10 , 
 where n = n + n , n , j(0 ) , j = 0,1 , are the respective rates of arrivals ; that is , an arrival is about to occur .
let us define the laplace transforms of n , j(x ) as n , j( ) = 0en , j(x)dx , re  0 .
hence , n , j  n , j(0 ) are the joint probabilities that there are n customers in the system and the server is in state j , j = 0,1 . multiplying the above set of equations by e and integrating with respect to x from 0 to  yield 
 ( 4)0,0()=11,1()+11,0()0,0(0),(5)nn,0=n+1n+1,0+an1,00n,00 ,  1nn1,(6)nn,0=qn+1n+1,0+an1,00n,00 ,  nnk1,(7)kk,0=ak1,00+k,00k,00,(8)(1)1,1()=22,1()1,1(0),(9)nn,1=n+1n+1,1+an1,10n,10 ,  2nn1,(10)nn,1=n+1n+1,1+qn+1n+1,0+nn,0+an1,10n,10 ,  nnk1,(11)kk,1=kk,0+ak,10+k1,10k,10 . 
 further , adding ( 4 ) to ( 11 ) and taking limit as   0 , we obtain the following result : 
 ( 12)n=0kn,00+n=1kn,10=. 
 the left hand side denotes the mean number of entrances into the system per unit time and is equal to the mean arrival rate . solving the set of equations from ( 5 ) to ( 11 ) in a backward recursion we get the following expressions . substituting  = n in ( 7 ) to ( 6 ) and  = n in ( 5 ) , we get 
 ( 13)k1,00=1akakk,00,n1,00=n,00qn+1n+1,0nan ,  n = k1,  ,n,n1,0(0)=n,00n+1n+1,0nan ,  n = n1,  ,1 , 
 where , for   n and   n , the unknowns n,0( ) are obtained from ( 7 ) to ( 5 ) as 
 ( 14)k,0=aakkakk,00,n,0=qn+1n+1,0+an1,00n,00n ,  n = k1,  ,n,n,0=n+1n+1,0+an1,00n,00n ,  n = n1,  ,2 . 
 substituting  = n in ( 11 ) to ( 9 ) we get 
 ( 15)k1,10=1akakk,10kakk,0k,n1,10=n,10ann+1n+1,1nanqn+1n+1,1nannn,0nan ,  n = k1,  ,n,n1,1(0)=n,10ann+1n+1,1nan ,  n = n1,  ,2 . 
 for   n , n,1( )
are given by the following : 
 ( 16)k,1  = kk,0+ak1,10+k,10k,10k,n,1=nn,0+qn+1n+1,0+n+1n+1,1n+a()n1,1(0)n,1(0)(n ) ,  n = k1,  ,n,n,1=n+1n+1,1+an1,10n,10n , 
n = n1,  ,2,1,1()=22,1()1,101. 
 differentiating ( 5 ) to ( 7 ) and setting  = k in ( 7 ) ,  = n in ( 6 ) , and  = n in ( 5 ) , we obtain , respectively , 
 ( 17)k,0k=a1kk1,00+k,00,(18)n,0n=qn+1n+1,01n+a1nn1,00 ,  n = k1,  ,n,(19)n,0n=n+1n+1,01n+a1nn1,00 . 
 similarly , differentiating ( 11 ) to ( 8) and setting  = n  ( 1  n  k ) , the expressions are given by 
 ( 20)k,1k=kk,01(k)+a1kk1,10+k,10,n,1n=nn,01n+qn+1n+1,01n  +  n+1n+1,01n+a1nn1,10 ,  n = k1,  ,n,n,1n=n+1n+1,01n+a1nn1,10 ,  n = n1,  ,2,1,11=22,111 . 
 using the above expressions one can evaluate n , j(0 ) , ( j 
n  k , j = 0,1 , be the arbitrary and joint probabilities that there are n customers in the system and the server is in state j , j = 0,1 , and let n , j denote the prearrival epoch probabilities . applying bayes ' theory and the result ( 12 ) , we have 
 ( 21)n , j=n , j0 ,  jnk ,  j=0,1 . 
 to obtain the arbitrary epoch probabilities
, we develop relations between prearrival and arbitrary epoch probabilities . setting  = 0 in ( 7 ) to ( 5 ) and ( 11 ) to ( 9 ) and using ( 21 ) , we obtain 
 ( 22)k,0=kk1,0,n,0=nn1,0+qn+1n+1n+1n,0j = n+1k1  + j = n+1k1qj+1j+1j+1s = n+1jqsss,0 ,  n = k1 ,  k2,  ,n,n,0=nn1,0+nnn1,0+qn+1n+1n+1n,0j = n+1k1  + j = n+1k1qj+1j+1j+1s = n+1jqsss,0 ,  n = n1, 
,1,k,1=kkkk1,0+k1,1,n,1=nn1,1+nnn1,0j = nk1qj+1j+1j+1  j = n+1k1s = njqsnsss,0 ,  n = k1 ,  k2,  ,n,n,1=nn1,1+nnn1,0j = nk1qj+1j+1j+1  j = n+1k1s = njqsisss,0 ,  n = n1 ,  n2,  ,2,1,1=1nnn1,0j = nk1qj+1j+1j+1  j = n+1k1s = njqsisss,0. 
 using the normalization condition , the only unknown 0,0 is obtained as 
 ( 23)0,0=1n=1kn,0+n,1 .
some models available in the literature are deduced as special cases of our model by taking specific values of the parameters n , q , n , n , and n . 
 case 1 .
if q = n = 1 , our model reduces to finite buffer gi / m(n)/1 queue with mwv . in this case
our results are found to match with the results available in goswami et al . . 
 
if q = n = 1 , our model reduces to finite buffer gi / m(n)/1 queue with mwv . in this case
our results are found to match with the results available in goswami et al . . 

if n  0 , the model becomes gi / m(n)/1/k with n - policy ; our results are in accordance with chao and rahman . 
 
if n  0 , the model becomes gi / m(n)/1/k with n - policy ; our results are in accordance with chao and rahman . 

if n , n =  , n   , n  0 , q = 1 , the model reduces to gi / m/1/k queue and the results match with the results available in ke and wang . 
 
if n , n =  , n   , n  0 , q = 1 , the model reduces to gi / m/1/k queue and the results match with the results available in ke and wang .
in this section , some operating characteristics such as the average number of customers in the queue ( lq ) ( system ( ls ) ) , the average waiting time of a customer in the queue ( wq ) ( system ( ws ) ) , and the blocking probability of the server ( ploss ) are evaluated .
they are , respectively , given by 
 ( 24)lq=n=1kn1n,0+n=1kn1n,1,ls=n=1knn,0+n=1knn,1,ploss=k,0+k,1,wq = lq^ ,  ws = ls^ , 
 where ^=(1-ploss ) is the effective arrival rate .
in this subsection , we formulate an expected cost model , in which mean service rate during vacation  is the decision variable .
let us define the following :  c cost per unit time during regular busy period ,  c cost per unit time during working vacation period ,  clq cost per unit time for a customer waiting in the queue ,  cploss cost per unit time when a customer is lost due to blocking.the total expected cost function per unit time is given by 
 ( 25)minimize : f=c+c+clqlq+cplossploss . 

our objective is to determine the optimal mean service rate during vacation  to minimize the cost function f( ) .
we employ the qfsm to solve the above optimization problem , as the computation of derivatives of the above expected cost function is a nontrivial task .
c cost per unit time during regular busy period , c cost per unit time during working vacation period , clq cost per unit time for a customer waiting in the queue , cploss cost per unit time when a customer is lost due to blocking . given a 3-point pattern , we can fit a quadratic function through corresponding functional values that has a unique minimum , x , for the given objective function f(x ) .
quadratic fit uses this approximation to improve the current 3-point pattern by replacing one of its points with approximate optimum x. the unique optimum x of the quadratic function agreeing with f(x ) at 3-point operation ( x , x , x ) occurs at 
 ( 26)xq12fxlxm2xh2+fxmxm2xh2+fxhxlxm1  xh2xl2+fxhxl2xm2  fxlxmxh+fxmxhxlxm2xh2  xm2xh2+fxhxlxm1 .
to validate the results obtained earlier , some numerical computations have been done and some of them are presented in the form of tables and graphs . the parameters of the system are taken as k = 10 , n = 5 , the traffic intensity  = 0.5 , n = ln(n + 0.4 ) , n = ln(n + 0.2 ) , and n = ln(n + 0.3 ) with mean values  = 1.617224 ,  = 1.566202 , and  = 1.592235 , respectively , unless otherwise mentioned separately in the respective graphs and tables .
the various cost parameters are taken as c = 20 , c = 18 , clq = 30 , and cploss = 10 . 

it can be observed that as q increases , the system characteristics increase and model with vi ( q = 0 ) performs better than the model without vi ( q = 1 ) as expected in practice . 

figure 1 depicts the effect of  on the expected queue length ( lq ) in models with and without vi for he2 interarrival time distribution with 1 = 0.381248 , 2 = 3.2 , 1 = 0.4 , and 2 = 0.6 .
further , in both models ( mwv and mwv - vi ) lq converges to the same value as  approaches . 
 figure 2 shows the impact of threshold value n on wq when interarrival time is exponentially distributed .
it is clear from the figure that wq increases with the increase of n. this is because as n increases more customers are required for the service start - up that results in increase of waiting time .
further , the average waiting time in case of queues without vi is higher as compared to queues with vi particularly for smaller n values .
a similar observation of above figure can be made from figure 3 where the effect of  on lq for two different threshold values of n is shown .
it is clear that as n increases , difference between two models for lq reduces .
moreover we can observe that mwv - vi ( q = 0 ) model gives us the better expected queue lengths .
the effect of  on the total expected cost function ( f( ) ) is shown in figure 4 , with constant service rates during regular busy period and during working vacation period and constant vacation rates .
the state independent rates are chosen as  = 2.8 ,  = 0.9 , and  = 1.5 . with the information of figure 4 , qfsm is applied by choosing the stopping tolerance  = 10 and the initial 3-point pattern as (  ,  ,  ) = ( 2.5,2.6,2.7 ) .
after six iterations , table 2 shows that the minimum expected operating cost per unit time converges to the solution f( ) = 124.2712 for  = 2.583980 . 

table 3 describes the sensitivity analysis of cost function and the queue lengths between mwv and mwv - vi models for different  and n values .
the minimum expected cost and the average queue lengths increase with n and  in both the models and the model with vi has lower queue lengths when the arrival rate is lower .
the above observations highlight the fact that the model with mwv - vi has better performance than the mwv model for lower threshold value n and for  < .
in this paper , we have carried out an analysis of a renewal input state dependent n - policy queue with bernoulli - schedule vacation interruption that has potential applications in production , manufacturing , traffic signals , telecommunication systems , and so forth .
the bernoulli - schedule parameter q enables combined study of the models with and without vi . using the supplementary variable technique we have developed a recursive method to obtain the steady state system length distributions at various epochs .
various performance measures are evaluated and a cost optimization problem is considered using quadratic fit search method . the method and analysis used in this paper can be applied to multiserver gi / m(n)/c and map / m(n)/c queues .
 we study a finite buffer n - policy gi / m(n)/1 queue with bernoulli - schedule vacation interruption . 
 the server works with a slower rate during vacation period . at a service completion epoch during working vacation , if there are at least n customers present in the queue , the server interrupts vacation and otherwise continues the vacation . using the supplementary variable technique and recursive method 
 , we obtain the steady state system length distributions at prearrival and arbitrary epochs . 
 some special cases of the model , various performance measures , and cost analysis are discussed . finally , parameter effect on the performance measures of the model 
 is presented through numerical computations . 
 
1. Introduction
2. Model Description
3. Special Cases
4. Performance Measures
5. Numerical Results
6. Conclusions