Queueing for ergodic arrivals and services

TL;DR

This paper revisits Loynes' 1962 results on queue stability with ergodic arrivals and services, providing examples where the limit distribution exhibits heavier-than-exponential tails despite stability.

Contribution

It demonstrates cases with bounded ergodic arrivals and constant service rate where the queue's limit distribution has heavier tails than exponential, extending classical stability results.

Findings

Limit distribution can have heavier-than-exponential tails.

Stability does not imply exponential tail decay.

Examples with bounded ergodic arrivals and constant service rate.

Abstract

In this paper we revisit the results of Loynes (1962) on stability of queues for ergodic arrivals and services, and show examples when the arrivals are bounded and ergodic, the service rate is constant, and under stability the limit distribution has larger than exponential tail.