The rate of convergence to stationarity for M/G/1 models with admission controls via coupling

TL;DR

This paper investigates the convergence rates to stationarity in two restricted M/G/1 queue models with admission controls, using coupling methods to derive bounds and conditions for geometric ergodicity.

Contribution

It provides new bounds and conditions for the rate of convergence to stationarity in restricted M/G/1 queues, including uniform bounds for certain subclasses and criteria based on service time distributions.

Findings

Uniform bounds for geometric ergodicity in Model 1 subclasses

No uniform bound exists for all Model 1 workload processes

Geometric ergodicity in Model 2 depends on the finiteness of the service time distribution's moment-generating function

Abstract

We study the workload processes of two restricted M/G/1 queueing systems: in Model 1 any service requirement that would exceed a certain capacity threshold is truncated; in Model 2 new arrivals do not enter the system if they have to wait more than a fixed threshold time in line. For Model 1 we obtain several results concerning the rate of convergence to equilibrium. In particular we derive uniform bounds for geometric ergodicity with respect to certain subclasses. However, we prove that for the class of all Model 1 workload processes there is no uniform bound. For Model 2 we prove that geometric ergodicity follows from the finiteness of the moment-generating function of the service time distribution and derive bounds for the convergence rates in special cases. The proofs use the coupling method.