Approximations for the performance evaluation of a discrete-time two-class queue with an alternating service discipline

• Published in: Annals of operations research 2022-03, Vol.310 (2), p.477-503
• Main Authors: Devos, Arnaud, Walraevens, Joris, Fiems, Dieter, Bruneel, Herwig
• Format: Article
• Language: English
• Subjects: Analysis; Approximation; Approximation theory; Business and Management; Combinatorics; Customer service; Customer services; Discrete time systems; Functional equations; Markov analysis; Mathematical analysis; Methods; Operations research; Operations Research/Decision Theory; Performance evaluation; Probability; Probability distribution; Queuing theory; S.I.: QueueStochMod2019; Stochastic models; Stochastic processes; Theory of Computation
• ISSN: 0254-5330; 1572-9338
• DOI: 10.1007/s10479-020-03776-5

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.