Sufficient ergodicity conditions for priority queues

Known results in ergodicity of priority queues are based on the assumption that interarrival times in each queue have exponential distribution. The aim of this paper is to relax this assumption, namely, to establish sufficient conditions of ergodicity for queues with two priority classes $GI|GI|1$ under assumption that interarrival times only in high priority class queue have exponentional distribution. Queues with nonpreemptive and different kinds of preemptive priority are considered. 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 a Harris-ergodic Markov chain, meaning to have a unique invariant measure, to which its transition probabilities converge in total variation.