Control policy of a hysteretic bulk queueing system

K e y w o r d s B a t c h arrival, Bulk service, Hysteresis, Embedded Markov chain, Semiregenerative process, Control policy. 1. I N T R O D U C T I O N The goal of this paper is to design an optimal control policy for a batch arrival, bulk service queueing system under N-policy. In this system, customers arrive to the queueing facility in groups of random size and are served in groups of a fixed size. The server checks the queue at every service completion instant. Given two fixed thresholds r and N, (N > r), if more than r customers are in line, then the server picks a group of r customers and processes them in a single batch. If less than r customers are in line, then the server idles and waits for the queue to reach the value N. Once the level N is reached (or exceeded, because arrivals are in batches), then a group is picked and the r customers are served altogether. Using Kendall 's notation, the model considered here is an MX/Gr/1 under N-policy. The optimal control of bulk service systems was started by the authors in [1] where an M / G r / 1 under N-policy was considered. In this paper, the results of [1] are generalized to the case where 0895-7177/05/$ see front mat te r (~ 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2003.07.017 Typeset by ~4j~4S-TEX 572 L. TADJ AND J.-C. K~. the arrival process is no longer orderly Poisson but compound Poisson. The literature on the optimal control of N-policy queueing systems is rich and varied. Various scenarios have been considered by the researchers, see [1] for a survey. Nevertheless, to the best of the authors' knowledge, except in [1], the optimal control of a bulk service system has not been dealt with. We analyze the model by the method of embedded Markov chain and semiregenerative techniques. We derive all the key elements required to build a prescriptive model for the queueing system: system-size steady-state probabilities of the discrete and continuous time parameter processes, along with some important system characteristics such as mean system size, mean busy period, mean idle period, and mean busy cycle. The prescriptive model of this system consists in designing an optimal control policy. We seek optimal values for the threshold levels r and N. What should these values be in order to minimize the system's expected cost per unit of time? The model is described and analyzed in Section 2. Section 3 discusses the control problem. Our interest is in the numerical results of Section 4, in which the model is studied in detail and some important operating characteristics are revealed. Conclusions are given in Section 5. 2. M O D E L D E S C R I P T I O N A N D S T E A D Y S T A T E RESULTS Consider the MX/Gr/1 queueing system. The arrival process is compound Poisson with rate A. Sizes of successive arriving batches are Xo, X1,. . . , where X0 = i and X1,X2,.. . are lid, distributed as a(z) = E[zX1]. The first and second moments, a = E[X1] and a2 = E[X~], respectively, are assumed to be finite. The process Sn = Xo + X1 + ... + Xn is a delayed renewal process. The service time as of n th batch of customers has a general distribution B(t) with LaplaceStieltjes transform B*(s) = E[e-8a~], and finite first and second moments, b --E[a~] and b2 = E[a2], respectively. The waiting room is unlimited. The service discipline is FIFO. The process of interest is the number Q(t) of customers in the system at time t. Let T~ denote the n th service completion instant. Then, Q~ = Q(T~+) is the embedded process and represents the number of customers in the system at a service completion epoch. The service discipline is as follows. If at a service completion there are more than r customers in the queue, then the server takes a batch of customers of size r to process in a single batch. On the other hand, if at a service completion there are less than r customers in the queue, then the server starts an idle period and when the queue size reaches (or, most probably, exceeds) for the first time the value N, (N > r), then he takes a batch of customers of size r to process in a single batch. Analysis of this bulk queueing system requires knowledge of the first excess level and other related processes. Let the index ~ be, such that u = inf{k : Sk >_ N}. If ~'i denotes the arrival epoch of group Xi, then the instant T. is the first passage time after Tn of the queueing process to reach or exceed level N. The level S~ is called the level of the first excess. It was studied in a work of Abolnikov and Dshalalow [2], who show that its probability generating function L(~ ) (z) = Ei[zSv], satisfies L(~)(z)= N,,-,g-i-l { a(z)~la(x) } z :a x)l ' where the operator ~ is defined, for k > 0, by :D~ (.) = I lim Ok k. (')" i < N , (1)