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

Abstract

This chapter focuses on the first two elements, the derivation of closed-forms for the main transient state distributions and the development of numerical techniques. It 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. The chapter describes some elements of the transient behavior of queues when they are in their transient phase, limiting the focus on Markovian models, because they are by far the most frequently used in performance or dependability evaluation studies. It introduces some queueing models where closed-forms for the standard state distribution are not hard to obtain, and then, the most fundamental system, the M/M/1 model, together with its bounded version, the M/M/1/H queue. To complete the introduction of the duality concept, the chapter shows an example using a small Markov process that is not in the birth–death family.

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