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EXPLICIT SOLUTIONS FOR CONTINUOUS-TIME QBD PROCESSES BY USING RELATIONS BETWEEN MATRIX GEOMETRIC ANALYSIS AND THE PROBABILITY GENERATING FUNCTIONS METHOD

Part of: Computer system organization Operations research and management science Special processes

Published online by Cambridge University Press:  03 January 2020

Gabi Hanukov Open the ORCID record for Gabi Hanukov [Opens in a new window]  and
Uri Yechiali
Gabi Hanukov
Affiliation:
Department of Management, Bar-Ilan University, Ramat Gan 5290002, Israel E-mail: [email protected]
Uri Yechiali
Affiliation:
Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel E-mail: [email protected]
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Abstract

Two main methods are used to solve continuous-time quasi birth-and-death processes: matrix geometric (MG) and probability generating functions (PGFs). MG requires a numerical solution (via successive substitutions) of a matrix quadratic equation A0 + RA1 + R2A2 = 0. PGFs involve a row vector $\vec{G}(z)$ of unknown generating functions satisfying $H(z)\vec{G}{(z)^\textrm{T}} = \vec{b}{(z)^\textrm{T}},$ where the row vector $\vec{b}(z)$ contains unknown “boundary” probabilities calculated as functions of roots of the matrix H(z). We show that: (a) H(z) and $\vec{b}(z)$ can be explicitly expressed in terms of the triple A0, A1, and A2; (b) when each matrix of the triple is lower (or upper) triangular, then (i) R can be explicitly expressed in terms of roots of $\det [H(z)]$; and (ii) the stability condition is readily extracted.

Keywords

continuous-time QBD processesprobability generating functionsmatrix geometriccalculation of the rate matrix R

MSC classification

Primary: 60K25: Queueing theory
Secondary: 68M20: Performance evaluation; queueing; scheduling 90B22: Queues and service
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 3 , July 2021 , pp. 565 - 580
Copyright
© Cambridge University Press 2020

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