A combinatorial analysis of the M/M [ m ] /1 queue

Guven Mercankosk, Gopalan M. Nair

Research output: Chapter in Book/Conference paperChapterpeer-review


Neuts’ Matrix Geometric Method makes use of the left-skip free characteristic of M/G/1-type Markov chains and determines the first passage distribution matrix G by solving a non-linear matrix equation. In this paper, we focus on the k-step first passage problem. In particular, we identify three associated matrices, namely the matrix G k , the conditional first passage probability matrix P k , and the first passage count matrix T k . The reformulation allows for combinatorial techniques. Specifically, we refer to an extension of Takács’ ballot theorem. We note that the matrix P k exhibits some ballot properties. In the case of the M/M [ m ] /1 queue, we establish the special structure of the count matrix T k using lattice path arguments. Furthermore, we obtain a closed-form expression for the G matrix, where the first passage probabilities are expressed in terms of generalized hypergeometric functions.

Original languageEnglish
Title of host publicationDevelopments in Mathematics
Number of pages16
ISBN (Electronic)9783030111021
ISBN (Print)9783030111014
Publication statusPublished - 3 Mar 2019

Publication series

NameDevelopments in Mathematics
ISSN (Print)1389-2177

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