Approximations for the performance evaluation of a discrete-time two-class queue with an alternating service discipline
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Approximations for the performance evaluation of a discrete-time two-class queue with an alternating service discipline Arnaud Devos1
· Joris Walraevens1 · Dieter Fiems1 · Herwig Bruneel1
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider a discrete-time queueing system with two queues and one server. The server is allocated in each slot to the first queue with probability α and to the second queue with probability 1−α. The service times are equal to one time slot. The queues have exponentially bounded, but general, arrival distributions. The mathematical description of this system leads to a single functional equation for the joint probability generating function of the stationary system contents. As the joint stochastic process of the system contents is not amenable for exact analysis, we focus on an efficient approximation of the joint probability generating function. In particular, first we prove that the partial probability generating functions, present in the functional equation, have a unique dominant pole. Secondly, we use this information to approximate these partial probability generating functions by truncating an infinite sum. The remaining finite number of unknowns are estimated from a noise perturbed linear system. We illustrate our approach by various numerical examples and verify the accuracy by means of simulation. Keywords Queueing theory · Two-class queueing model · Joint probability generating function · Dominant singularities · Approximation 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]
1
Department of Telecommunications and Information Processing, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium
123
Annals of Operations Research
1 Introduction In many practical queueing problems, different types of customers (or packets) need the same source of service. The differentiation of customers in different customer types can be necessary because they may have different characteristics such as different arrival patterns, different Quality of Service requirements (QoS), etc. Once the customers are distributed in various traffic classes, a scheduling policy decides in which order the customer types have access to the server. Differentation of customers is often done by giving different priorities to customers. For example, in the two-class priority queueing system (Walraevens et al. 2003) customers are stored in two queues (the low and high priority queue) and a customer of the high priority queue is served when a new service opportunity occurs. The biggest drawback of this service policy, is that customers from the low priority queue can suffer from potential starvation in case there is a high load of high-priority customers. A possible solution to this starvation problem is a Generalized Processor Sharing scheduling (GPS) (Parekh and Gallager 1993). In GPS, customers arrive and are classified in clas
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