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We consider a finite-capacity M/G/l/K queueing system with a mixture of the single vacationl close-down, and set-up times; arising out of the switched virtual channel connection (SVCC) mechanism for an IP over ATM network. Applying the supplementary variable approach, we derive the steady-state queue length and waiting time distributions, as we11 as the set-up occurring rate. Taking the limit of our results as the capacity (K) tends to infinity, we obtain the results for an infinite-capacity queueing systeml covering the previous work by Hassan and Atiquzzaman as a special case. %. Inkroduction Teletraffic engineers frequently encounter queueing situations with vacation time and exhaustive service discipline. By vacation t ime, we mean that the server becomes unavailable for occasional intervals of time, and by exhaustive we mean that customers are served continuously until there is no customer in the system. A classical example for this situation is the time division multiple access (TD MA) scheme where the vacation time corresponds to a constant slotted time period in the TDMA system. Under the exhaustive service discipline, the well known stochastic decomposition formula will be useful for infinite capacity queues; see Doshi [I] and Miyazawa [5 ] . However, if we would like to evaluate the loss probability we have to treat a finite capacity system rather than infinite capacity systems. Assuming a Poisson input and a finite capacity queue, Lee [4] provided a numerical algorithm for this system via the embedded Markov chain technique; see also Frey and Takahashi [2] for a more simplified analysis for the M/G/l /K queue with vacation time and exhaustive service discipline. Very recently in the IP over ATM networks, we can see a more complicated queueing situation where the close-down time as well as the set-up time are further neededThe close-down time here corresponds to an inactivity timer in the switched virtual channel connections (SVCC) environment (see Hassan and Atiquzzaman [3]). Also, the set-up time cannot be negligible for the ATM server. An approximate queueing result (using the M/G/ 1 queue) has been reported in Hassan and Atiquzzaman [3]. Their approximation is based 011 i.i.d. service time of each packet to obtain the first and second moments of the waiting time. However, under the existance of SET-UP time, this i.i*d. service time does not hold. The main goal of this paper is 1) to present a new finite capacity queueing system with vacation> set-up, and close-down times for the SVCC operation in IP over ATM networks? generalizing the queueing system of Hassan and Atiquzzaman; 2) and to provide an exact result for the generalized system. The approach taken here is the supplementary variable © 1998 The Operations Research Society of Japan A Queue with Vacation, Set-up, and Close-down Times 69 technique. It should be noted that our approach enables one to treat the infinite capacity system by taking the limit of our result as the capacity (K) tends to infinity. We can derive an exact formula to the Hassan and Atiquzzaman's system [3] for a special case (the infinite queue without vacation). This paper is organized as follows. We describe the model and introduce notations in Section 2. Section 3 derives a set of Laplace-Stieltjes transform equations through the supplementary variable approach. Section 4 is then devoted to the numerical algorithm procedure for the steady-state probabilities. The procedure is expressed step by step to show our algorithm. Using these results, we express performance measures such as the mean waiting time, set-up rate, and loss probability in Section 5 and Section 6 presents the numerical result S for the performance measures. Section 7 shows that the limiting behavior of the finite capacity system is consistent with that of the infinite capacity system analyzed by the imbedded Markov chain approach. 2. The model We consider an M/G/l/K system with single vacation, set-up and close-down time, where K represents the maximum number of customers allowed in the system. The system consists of the waiting room and the server. We assume a customer arrives at the system according to a Poisson process with intensity A independent of the system state. These arriving customers are served under the First-In-First-Out (FIFO) discipline, where the customers will be served in order of their arrivals to the system. Since we consider the system with a finite capacity, a customer will be lost if it finds the waiting room all occupied. We assume the exhaustive service discipline and the server continues to serve the customers in the queue until all of them are served. The service times form a sequence of i.i.d. random variables with distribution function (DF) B(x) and the Laplace-Stieltjes transform (LST) Â¥*(S $ e-"B(dx). We call the period where the server is working a busy period. If the server finds that there is no customer in the queue a t a service completion, the system goes into a close-down period with DF C(x) and LST C*(s). During the close-down period, if a customer arrives, the server immediately begins service for that customer without set-up period and another busy period begins. On the other hand, if no customer arrives until the end of a close-down period, the system goes into a single vacation period. At the end of a vacation period, if the server finds one or more customers waiting, the server begins to serve for them after a set-up period. We denote by S(x), S*(s) (V(x), V*(s)) the DF or the LST of the set-up period length (vacation period length). If the server finds no customers waiting, the system goes into an idle period and an arrival of a customer ends this period. Then the server begins to serve after a set-up period. It should be noted that our model reduces to that of Hassan and Atiquzzaman [3] if the capacity (K) is infinite and the vacation length is zero. We define B as the remaining service time. Similarly we define P,S' and C. We further need the following notations for our subsequent analysis. L : number of customers in the system including the one in the server. (L = 0 , ,K ) IIn,i(x)dx = Prob[L = i , x < B < X + dx, the system is busy] A Qi{x)dx Prob[L = i, X < V < X + dx, the system is in vacation] II,s,i(x)dx s Prob[L = ! " , X < 5'5 X + dx, the system is setting-up] Copyright © by ORSJ. Unauthorized reproduction of this article is prohibited. Y. Sakai & Y. Takahashi & Y. Takahashi & T. Hasegawa Prob[L = i , X < C < X + dx, the system is closing-down] 00 e-"nB,; (dx) I I G , i ( ~ ) 1 e-sxIIv,i(dx) 00 00 s x I I s , i (dx) e-sxIIc,i (dx) 3. Supplementary variable approach In this section, we obtain Laplace-Stieltjes transforms, II;,o(s), IIb(s) (0 < i < K), II'5,;(5) (1 < i < K), and I'I$ld-s) (1 < i < 1% which are the bases of analysis in the following section. Since the service, vacation, set-up, close-down times are generally distributed (having non-Markovian property), the queue length process does not form any Markov chain. Here, the supplementary variable technique is applied to make our system Markovian. To be more exact, the joint distribution of the queue length and the remaining service (vacation, set-up, or close-down) time forms a Markov process. Observing the system state at time t and t + At, we have following equations. where II* is the probability that the system is idle at an arbitrary time. Equating the rate at which a vacation period ends with no customers present to the rate at which the idle period following ends because of arrivals, we have During the close-down time, an arriving customer may end the period, thus Copyright © by ORSJ. Unauthorized reproduction of this article is prohibited. A Queue with Vacation, Set-up, and Close-down Times 71 Taking into account (1) and (2)) we obtain Laplace-Stieltjes transforms, 4. System state probability In this section we give IIc,o(0), II&,o(0), IIv,,(O) (0 5 i I(), IIki(0) (0 5 i 5 K), WO) ( l 5 i 5 K ) , Q,;(()) (l 5 i 5 I<), I IB , , (O) (1 5 i 5 K ) and q ( 0 ) (1 5 i 5 K ) . From these results we obtain the system state probability a t which close-down/vacation/setup/service/idle periods end and at an arbitrary point. In the following we express all probabilities in terms of llB1 (0). At last we have TiBl (0) from the normalization condition. 4.1 IIc,o(O) and II;,o(0) in terms of I I B , ~ ( O ) We express IIo,o(O) and I I S O ) in terms of IIB,~(O). Substituting s = A and s = 0 into (3) yields 4.2 IIv,;(O) (1 5 i 5 K) and 11$,,(0) (1 5 i 5 K ) in terms of IIB,l(0) We then express IIv,i(0) (1 5 i S I() and lIF,i(0) (1 i 5 K) in terms of IIB,~(O). Substituting s = A into (4) and (5), we have WO) = w O ) C * ( A ) V * ( A ) nv,i (0) = AII$,i_l (A), for 1 5 i 5 I< 1. Differentiating (4) and (5), and inserting s = A, we have -(n + l)~lV,t)(A) = n B , l ( o ) ~ * ( ~ ) v * ( n + l ) ( ~ ) , for n > O (17) -(n + ~ ) I I $ ) ( A ) = A I I ~ , ~ _ ~ *^"(A), f o r n > o , i > l . (18) Using (17) and (18)) we have &,;(A) = (-1)'"" Ai (i + l )! D ~ , ~ ( O ) C * ( A ) V * ( ~ + ~ ) ( A ) , f o r 0 5 i 5 A : 2 . (19) Copyright © by ORSJ. Unauthorized reproduction of this article is prohibited. 72 Y. Sakai & Y. Takahashi & Y. Takahashi & T. Hasegawa From (19), (15) and (16) we can express MO) (0 5 i S I{ 1) in terms of H p l (0). Indeed we have Next we substitute S = 0 into (4), (5) and ( 6 ) , yielding Since we have IIV,i(0) (1 <: z <: I{ 1) in terms of IIBj1(O) we can easily express Tl'y,(O) (0 < i 5 K 1) in terms of IIB,l(0) using (22) and (23). Together with these results and (24), we have nv,l<{O) in terms of nB,l(0) Differentiating (4), (5) and (6), and substituting S = 0 yield AII$,)(0) = II;,o(0) + n ~ , 1 (o)c*(A)v*(~)(o) (26) AII$(O) = n;,,(o) + AII$^O), for l < i 5 K 1 (27) -r~;,~<-(o) = (0). (28) From these results, we can express IIcIs(0) in terms of IIB,l (0) Hereby we complete our procedure. 4.3 MO) (1 5 i S A') and II;,(O) (1 5 i < K ) in terms of IIB,l(0) We are next in a position

A COMPOSITE QUEUE WITH VACATION/SET-UP/CLOSE-DOWN TIMES FOR SVCC IN IP OVER ATM NETWORKS