The Matrix-Analytic Formalism

This third part of the book deals with retrial queues analyzed by matrixanalytic methods. We begin with the present chapter on some generalities of the matrix-analytic formalism for two reasons. First and foremost, this chapter summarizes basic results of the matrix-analytic methods, presents the main tools employed in Chapters 8 and 9, and fixes some notation. In addition, it helps the reader to acquire some feeling of the constant interplay between algebraic manipulations and reasoning on the probabilistic interpretation of the underlying expressions.

Although it is assumed that the reader will have some knowledge of the basic matrix algebra, we give in Subsection 7.1.1 a short glossary of the conventions we use in Part III of this book and of some notation used earlier. In the rest of Section 7.1, we introduce the QBD processes and the Markov chains of GI/M/1- and M/G/1-types. Then, we describe a point process, namely the batch Markovian arrival process, which generalizes and unifies several Markovian processes commonly used in queueing theory. We next examine the phase-type distribution and the semi-Markov service process. In Sections 7.2 and 7.3, we study in some detail concrete results on the QBD, GI/M/1 and M/G/1 structures. Our intention is only to facilitate the understanding of the material in later chapters, hence we omit proofs. Bibliographical notes in Section 7.4 are then useful for finding the corresponding sources.