Delay analysis of a discrete-time single-server queue with an occasional extra server
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Delay analysis of a discrete-time single-server queue with an occasional extra server Freek Verdonck1
· Herwig Bruneel1
· Sabine Wittevrongel1
Accepted: 13 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this work we look at the delay analysis of a customer in a discrete-time queueing system with one permanent server and one occasional extra server. The arrival process is assumed to be general independent, the buffer size infinite and the service times deterministically equal to one slot. The system is assumed to be in one of two different states; during the UP-state 2 servers are available and during the DOWN-state 1 server is available. State changes can only occur at slot boundaries and mark the beginnings and ends of UP-periods and DOWN-periods. The lengths of the UP-periods, expressed in their number of slots, are assumed to follow a geometric distribution, while the lengths of the DOWN-periods follow a general distribution with rational probability generating function. We provide a method to compute the tail characteristics of the delay of an arbitrary customer based on the theory of the dominant singularity. We illustrate the developed method with several numerical examples. Keywords Queueing theory · Discrete-time · Multiserver · Delay · Tail
1 Introduction This paper focusses on a discrete-time queueing system with two servers, where one server is permanently available and one server is subject to random interruptions. The buffer size is assumed to be infinite. The time horizon is divided into slots of equal lengths and the service times are deterministic and equal to one slot. The interruption process divides the system into two states: the UP-state with two servers available and the DOWN-state with one server available. State changes can only occur at slot boundaries, and these mark the beginnings and ends of UP-periods and DOWN-periods. The lengths of the UP-periods are according to a geometric distribution, while for the lengths of the DOWN-periods we allow a general distribution with rational probability generating function (pgf). This introduces correlation in the number of servers available from slot to slot. In the earlier paper by Bruneel and Wittevrongel (2017), the system content of such a system was described. The current paper is an extension to that work, we now focus on
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Freek Verdonck [email protected] Department of Telecommunications and Information Processing (TELIN), SMACS Research Group, Ghent University (UGent), Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium
123
Annals of Operations Research
the delay of an arbitrary customer. In our conference paper (Verdonck et al. 2019), the delay analysis was performed when the period lengths of the DOWN-periods are distributed according to a (mixture of) geometrical distribution(s). In the current paper we develop a different method to obtain the delay of a customer when the lengths of the DOWN-periods follow a distribution with rational pgf. We obtain the tail characteristics of th
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