Random walks in the quarter-plane: invariant measures and performance bounds

Abstract

This monograph focuses on random walks in the quarter-plane. Such random walks are frequently used to model queueing systems and the invariant measure of a random walk is of major importance in studying the performance of these systems. In special cases the invariant measure of a random walk can be expressed as a geometric product-form and performance measures can readily be obtained. In general, however, no tractable closed-form expressions are available for the invariant measure and exact performance measures are not readily obtained. The aim of this monograph is two-fold. On one hand we consider measures that are a sum of geometric terms. We characterize the random walks in the quarter-plane of which the invariant measure is of this form. This extends the class of random walks for which tractable closed-form results can be obtained. On the other hand we develop approximation schemes that provide analytical upper and lower bounds on performance for the case that no tractable closed-form expressions for the invariant measure are available.