Analysis of a queue with general service demands and correlated service capacities
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Analysis of a queue with general service demands and correlated service capacities Michiel De Muynck1
· Herwig Bruneel1 · Sabine Wittevrongel1
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We present the study of a non-classical discrete-time queueing model in which the customers each request a variable amount of service, called their “service demand”, from a server which is able to execute a variable amount of work, called its “service capacity”, during each time slot. We assume that the numbers of arrivals in consecutive time slots and the service demands of consecutive customers form two independent and identically distributed sequences. However, we allow the service capacities in consecutive time slots to be correlated according to a discrete-batch Markovian process. We study this model analytically and obtain expressions for the probability generating function of the steady-state system content and customer delay, as well as their moments and an approximation for their tail probabilities. The results are illustrated with several numerical examples. Keywords Discrete-time queueing theory · Service demands · Correlated service capacities · Discrete-batch Markovian service process
1 Introduction In classical queueing theory, queueing phenomena where customers require varying amounts of work from the server(s) are often modeled using the concept of “service time”, where the service time of a customer is the amount of time that a server needs to fully process that customer. It is then commonly assumed that the service times of the consecutive customers are independent from each other. However, in many queueing phenomena this assumption may not hold. Indeed, the service time of a customer is usually determined by two underlying quantities: the amount of work that the customer requires from the server, which we refer to as the “service demand” of
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Michiel De Muynck [email protected] Herwig Bruneel [email protected] Sabine Wittevrongel [email protected]
1
Department TELIN, Ghent University, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium
123
Annals of Operations Research
that customer, and the speed with which the server can process this work. The assumption that service times form an independent and identically distributed (i.i.d.) sequence generally breaks down into assuming that both these service demands and the service speeds are uncorrelated from customer to customer. The first of these two assumptions, i.e., that the service demands of subsequent customers are independent of each other, is valid for many queueing phenomena. For example, in most brick-and-mortar stores there is little correlation between the length of the shopping list of one customer entering that store and that of the next. However, the second assumption, i.e., that the speed with which subsequent customers are served is also uncorrelated, is often not valid. If the cashier at a register is working slowly, it is likely that the next person served at that register will also
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