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

Guven Mercankosk, Gopalan M. Nair

Research output: Chapter in Book/Conference paperChapter

Abstract

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
PublisherSpringer
Pages327-342
Number of pages16
Volume58
ISBN (Electronic)9783030111021
ISBN (Print)9783030111014
DOIs
Publication statusPublished - 3 Mar 2019

Publication series

NameDevelopments in Mathematics
Volume58
ISSN (Print)1389-2177

Fingerprint

M/M/1 Queue
Combinatorial Analysis
P-matrix
Count
Matrix-geometric Method
Nonlinear Matrix Equation
Generalized Hypergeometric Function
Lattice Paths
Reformulation
Markov chain
Closed-form
Theorem

Cite this

Mercankosk, G., & Nair, G. M. (2019). A combinatorial analysis of the M/M [ m ] /1 queue. In Developments in Mathematics (Vol. 58, pp. 327-342). (Developments in Mathematics; Vol. 58). Springer. https://doi.org/10.1007/978-3-030-11102-1_15
Mercankosk, Guven ; Nair, Gopalan M. / A combinatorial analysis of the M/M [ m ] /1 queue. Developments in Mathematics. Vol. 58 Springer, 2019. pp. 327-342 (Developments in Mathematics).
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Mercankosk, G & Nair, GM 2019, A combinatorial analysis of the M/M [ m ] /1 queue. in Developments in Mathematics. vol. 58, Developments in Mathematics, vol. 58, Springer, pp. 327-342. https://doi.org/10.1007/978-3-030-11102-1_15

A combinatorial analysis of the M/M [ m ] /1 queue. / Mercankosk, Guven; Nair, Gopalan M.

Developments in Mathematics. Vol. 58 Springer, 2019. p. 327-342 (Developments in Mathematics; Vol. 58).

Research output: Chapter in Book/Conference paperChapter

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Mercankosk G, Nair GM. A combinatorial analysis of the M/M [ m ] /1 queue. In Developments in Mathematics. Vol. 58. Springer. 2019. p. 327-342. (Developments in Mathematics). https://doi.org/10.1007/978-3-030-11102-1_15