Queueing Theory 2

1Stability Analysis of Queueing Systems based on Synchronization of the Input and Majorizing Output Flows

Larisa AFANASEVA

Department of Probability, Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, Russia

This chapter is focused on the stability conditions for a multiserver queueing system with heterogeneous servers and a regenerative input flow X. The main idea is constructing an auxiliary service process Y, which is also a regenerative flow and determination of the common points of regeneration for both processes X and Y. Then the traffic rate is defined in terms of the mean of the increments of these processes on a common regeneration period. It allows us to use well-known results from the renewal theory to find the instability and stability conditions. The possibilities of the proposed approach are demonstrated by examples. We also present the applications to transport system capacity analysis.

1.1. Introduction

This chapter deals with the study of stability conditions for a heterogeneous multiserver queueing system with regenerative input flow (Afanaseva 2019).

We consider queueing systems with regenerative input flow for three reasons. First, a process describing the performance of the system under some natural conditions turns out to be a classical regenerative process (Asmussen 2003; Thorisson 2000) and the renewal theory gives very effective tools for asymptotic analysis of the system. Second, the class of regenerative flows is rather wide and ...