Journal of Applied Mathematics and Stochastic Analysis, 13:4 (2000), 411-414. ON FINITE CAPACITY QUEUEING SYSTEMS WITH A GENERAL VACATION POLICY

We consider a Poisson arrival queueing system with finite capacity and a 
general vacation policy as described in Loris-Teghem [Queueing Systems 3 
(1988), 41-52]. From our previous results regarding the stationary queue 
length distributions immediately after a departure and at an arbitrary 
epoch, we derive a relation between both distributions which extends a 
result given in Frey and Takahashi [Operations Research Letters 21 
(1997), 95-100] for the particular case of an exhaustive service multiple 
vacation policy.


Introduction
For the M/GI/1 queueing system with finite capacity and exhaustive service multiple vacation policy studied previously by Courtois [1] and Lee [3], Frey and Takahashi  [2] analyze the stationary queue length distribution at an arbitrary epoch by express- ing it in terms of the stationary queue length distribution at service completion epochs, for which they provide a system of recurrence equations.
According to them, they would be first to consider the departure epoch imbedded Markov chain for vacation models with finite capacity.We would like to emphasize that in Loris-Teghem [5], dealing with a finite capacity queueing system with a general vacation policy, we derived the stationary queue length distribution imme- diately after a departure and at an arbitrary epoch, by relating each of both distribu- tions to the corresponding distribution in the model without vacations.
In the present note, we use our previous results to express the stationary queue length distribution at an arbitrary epoch in terms of the stationary queue length dis- tribution immediately after a departure, thus extending to the general vacation policy a result obtained in Frey and Takahashi [2] for the exhaustive service multiple vaca- tion policy.

The Model
We consider a queueing system with a Poisson arrival process (with rate ) and a finite capacity (L).We assume that at time t 0, a departure occurs with e custom- ers left in the system (where e is a fixed non-negative integer, with < L), and an in- active phase then begins for the server, who will become active again at time 7-1 The active phase initiated at r 1 will end the first time a departure occurs with custom- ers left in the system.Thus the server is alternatively in the inactive and active states.
Let 0 t o < t I <... < t n <... (0 < 7" <... < T n <...) be the epochs at which the server "enters" the inactive (active) state and xn, n > 1, the number of customers pre- sent in the system at time r n + 0. Let u n v n t n 1 (n >_ 1).We make the follow- ing further assumptions: the epochs t n are the times at which the number of customers in the system decreases to .(For > 0, service is non-exhaustive); the random variables us, n > 1, are i.i.d, with + 1 < x n < L a.s. and with finite expectation; the service times are i.i.d random variables-with finite expectation E(S)independent of the arrival process and of the sequence {(us, xn) n > 1}; service is non-preemptive and customers are served in an order independent of their service times.

Queue Length Distributions
We first consider the Markov chain {ln, n > 1}, where n denotes the number of cus- tomers in the system immediately after the nth departure.As proved in Loris- Teghem [5], the stationary distribution of {ln, n > 1} denoted by {Tr L,e(j),j e,...,L-1} is related to the corresponding distribution in the M/GI/1/L model without vacations-denoted by {rL(j),j_ O,...,L-1}-according to the following formula: J 7rL, e(J) (aL,) -lzL E Pr[x > i]Tr L-e(j-i) (j , ..., L-1) (1) where" x is distributed as the xn, n > 1; a L is the expected duration of an active phase; a m is the expected duration of the busy period in the M/GI/1/m model without vacations.
We now consider the stationary distribution of the queue length at an arbitrary pL ...,L}.As proved in Loris-Teghem [5] it is epoch denoted by { v,e(J),J , related to the corresponding distribution in the M/GI/1/L model without vacations- denoted by {pL(j), j 0,...,L}-according to the following formula: Le(j)_("cLe)-l(1 +'c e) EPv[x>i]P L-s(j-i) (j-s,.. L 1) (2) P, , L and another formula for j-L, which will not be used here, where c L is the expect- ed duration of a cycle.
Using (1) for j-and taking into account that E(S)-amrm(O) for m _> 1 (see [4]), we obtain (auL,)-1 7r L e(c)(E(S))-1 so that pL s(j) r L s(j) E(S) + d L r L (j ,...,L-1) L pu, e(L)-1- E(S) + d These relations extend to the general vacation policy considered here, relations (13) and ( 14) obtained in Frey and Takahashi [2] for the exhaustive service multiple qn vacation policy.In this particular case, e-0 and u n is distributed as i Un, where the random variables un, (n >_ 1, _> 1) are i.i.d, and q, is the smallest integer qn verifying the condition that the system is non-empty at time (t n 1-t-E i= Un, i)" Thus d L d L E(U)(1 bo)- where E(U)-E(un, i) and b 0 is the probability that no arrival occurs in a time