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Utilization of the method of linear matrix equations to solve a quasi-birth-death problem

Part of: Special processes Operations research and management science Markov processes

Published online by Cambridge University Press:  14 July 2016

Richard M. Feldman ,
Bryan L. Deuermeyer  and
Ciriaco Valdez-Flores
Richard M. Feldman
Affiliation:
Texas A&M University
Bryan L. Deuermeyer
Affiliation:
Texas A&M University
Ciriaco Valdez-Flores*
Affiliation:
Texas A&M University
*
∗ Postal address for all authors: Industrial Engineering Department, Texas A&M University, College Station, TX 77843, USA.
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Abstract

The steady-state analysis of a quasi-birth-death process is possible by matrix geometric procedures in which the root to a quadratic matrix equation is found. A recent method that can be used for analyzing quasi-birth–death processes involves expanding the state space and using a linear matrix equation instead of the quadratic form. One of the difficulties of using the linear matrix equation approach regards the boundary conditions and obtaining the norming equation. In this paper, we present a method for calculating the boundary values and use the operator-machine interference problem as a vehicle to compare the two approaches for solving quasi-birth-death processes.

Keywords

QUEUESQUASI-BIRTH-DEATH PROCESSMATRIX GEOMETRICLCI-COMPLETENESS

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60K30: Applications (congestion, allocation, storage, traffic, etc.) 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 3 , September 1993 , pp. 639 - 649
Copyright
Copyright © Applied Probability Trust 1993 

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References

[1] Beuerman, S. L. and Coyle, E. J. (1989) State space expansions and the limiting behavior of quasi-birth-death processes. Adv. Appl. Prob. 21, 284–314.Google Scholar
[2] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore, MD.Google Scholar
[3] Zhang, J. and Coyle, E. J. (1989) Transient analysis of quasi-birth-death processes. Stoch. Models 5, 459–496.Google Scholar