Ergodicity of single server queues with preemptive priority

A. V. Mistryukov

Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University, 1-52 Leninskiye Gory, GSP-1, Moscow 119991, Russian Federation

Abstract: Well known results on ergodicity of queues with preemptive priority were obtained under the assumption that jobs arrive according to Poisson process. However, this assumption does not always hold true in practice. In this paper, the author finds sufficient ergodicity conditions for queues with two priority classes with single server, where interarrival times of high priority jobs have either Erlang or hyperexponential distribution and interarrival times of low priority jobs and service times of jobs of both classes have arbitrary continuous distributions. To formulate desired conditions, the authors use Lindley's recursion for waiting times of each priority class queue. Using Lyapunov–Foster criteria, the authors obtain sufficient conditions for a given recursion to be Harris-ergodic Markov chain.

Keywords: preemptive priority, ergodicity, Lyapunov–Foster criteria, hyperexponential arrivals, Erlang arrivals.

Received: 28.04.2020

DOI: 10.14357/08696527200303