Queueing Theory 2

8Transient Analysis of Markovian Queueing Systems: a Survey with Focus on Closed-forms and Uniformization

Gerardo RUBINO¹

¹Inria, Rennes, France

Analyzing the transient behavior of a queueing system is much harder than studying its steady-state, the difference being basically that of moving from a linear system to a linear differential system. However, a huge amount of efforts has been made for the former problem from all kinds of points of view: trials to find closed-forms of the main state distributions, algorithms for numerical evaluations, approximations of different types, exploration of other transient metrics than the basic state distributions, etc.

In this chapter, we focus on the first two elements, the derivation of closed-forms for the main transient state distributions and the development of numerical techniques. This chapter is organized as a survey, and the main goal is to position and to underline the role of the uniformization technique, for both finding closed-forms and for developing efficient numerical evaluation procedures. In some cases, we extend the discussion to other related transient metrics that are relevant for applications.

8.1. Introduction

Queueing models are the most frequently used when analyzing the performance of a resource sharing system, and they are applied on a daily basis in many different areas. As for other dynamical systems, most of their analysis is done in steady-state and for several reasons (see the nice developments in the Introduction ...