Analysis of a discrete-time two-class randomly alternating service model with Bernoulli arrivals
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Analysis of a discrete-time two-class randomly alternating service model with Bernoulli arrivals Arnaud Devos1
· Joris Walraevens1 · Dieter Fiems1 · Herwig Bruneel1
Received: 28 November 2019 / Revised: 29 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We analyze a discrete-time two-class queueing system with a single server which is alternately available for only one customer class. The server is each time allocated to a customer class for a geometrically distributed amount of time. Service times of the customers are deterministically equal to 1 time slot each. During each time slot, both classes can have at most one arrival. The bivariate process of the number of customers of both classes can be considered as a two-dimensional nearest-neighbor random walk. The generating function of this random walk has to be obtained from a functional equation. This type of functional equation is known to be difficult to solve. In this paper, we obtain closed-form expressions for the joint probability distribution for the number of customers of both classes, in steady state. Keywords Two-class queueing model · Processor sharing · Singularity analysis · Analytic continuation · Nearest-neighbor random walk Mathematics Subject Classification 68M20 · 90B22
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Arnaud Devos [email protected] Joris Walraevens [email protected] Dieter Fiems [email protected] Herwig Bruneel [email protected]
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Department of Telecommunications and Information Processing, Ghent University, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium
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Queueing Systems
1 Introduction Multi-class queueing systems are characterized by the presence of different classes of customers who are requesting service from the same server(s). Well-known examples of such queuing systems are priority and polling systems [16,17]. Analysis of multiclass queueing systems is often more difficult than the analysis of single queues. The main reason is that in order to describe the system completely and to calculate most performance measures, the joint probability distribution of the number of customers of the different classes in the queueing systems must be found. In the present paper, we study a two-class queueing model where the server serves customers of either class alternately according to a Bernoulli process. Figure 1 depicts the queueing model that we consider. We assume that the server has no information about the queue contents. As a consequence, when a class with an empty queue is chosen to be served in a particular time slot, no one gets service. This non-workconserving system is therefore different, but still closely connected, to a discrete-time generalized processor sharing (GPS) queueing model [18]. It has to be noticed that, in contrast to the joint distribution, the marginal distributions of the number of customers present in both queues in this model can be obtained in a relatively easy way, since, from the perspective of one queue, this is a single-sever single-class queueing system with server in
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