The time-dependent expected reward and deviation matrix of a finite QBD process

Sarah Dendievel; Sophie Hautphenne; Guy Latouche; Peter G Taylor

Journal article

The time-dependent expected reward and deviation matrix of a finite QBD process

Sarah Dendievel, Sophie Hautphenne, Guy Latouche, Peter G Taylor

Linear Algebra and Its Applications | Elsevier | Published : 2019

DOI: 10.1016/j.laa.2019.02.002

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Abstract

Deriving the time-dependent expected reward function associated with a continuous-time Markov chain involves the computation of its transient deviation matrix. In this paper we focus on the special case of a finite quasi-birth-and-death (QBD) process, motivated by the desire to compute the expected revenue lost in a MAP/PH/1/C queue. We use two different approaches in this context. The first is based on the solution of a finite system of matrix difference equations; it provides an expression for the blocks of the expected reward vector, the deviation matrix, and the mean first passage time matrix. The second approach, based on some results in the perturbation theory of Markov chains, leads t..

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University of Melbourne Researchers

Sophie Hautphenne Author

Peter Taylor Author

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Grants

Awarded by Australian Research Council (ARC)

Awarded by Ministere de la Communaute francaise de Belgique through the Action de Recherche Concertee (ARC) grant

Funding Acknowledgements

Sophie Hautphenne and Peter Taylor would like to thank the Australian Research Council (ARC) for supporting this research through Discovery Early Career Researcher Award DE150101044 and Laureate Fellowship FL130100039, respectively. In addition, they both acknowledge the support of the ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS).Sarah Dendievel and Guy Latouche thank the Ministere de la Communaute francaise de Belgique for funding this research through the Action de Recherche Concertee (ARC) grant AUWB-08/13-ULB 5.Sarah Dendievel would also like to thank the Methusalem program of the Flemish Community of Belgium for supporting this work.