Queueing Theory 2

7A Stability Analysis Method of Regenerative Queueing Systems

Evsey MOROZOV^1,2 and Bart STEYAERT³

¹Institute of Applied Mathematical Research, Russian Academy of Sciences, Petrozavodsk, Russia

²Petrozavodsk State University, Russia

³Department of TELIN, Ghent University, Belgium

This chapter is devoted to a stability analysis of regenerative queueing systems based on a renewal analysis technique. This approach allows us to obtain simple proofs for stability conditions for the stochastic processes that are considered. In order to illustrate the method, we specifically focus on the stability conditions of the well-known classical continuous-time single-server GI/G/1 queueing system, and the GI/G/m multiserver system. For the latter system, we consider the zero-delayed case, as well as the delayed case, meaning that the system initially is non-empty. Some applications of the regenerative method with respect to the stability analysis of other, more complicated, queueing models are briefly discussed as well.

7.1. Introduction

The stability analysis of stochastic models for telecommunication systems is one of the topics intensively studied nowadays. This analysis is, as a rule, a difficult problem requiring a refined mathematical technique, especially when we endeavor beyond the framework of Markovian models. Stability analysis allows us to delimit the region of the system parameters where the stationary regime of the processes describing the dynamics of the associated queuing system ...