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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: Operations research and management science Computer system organization 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: german.kanukov@biu.ac.il
Uri Yechiali
Affiliation:
Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel E-mail: uriy@post.tau.ac.il
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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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References

1.Altman, E., Jiménez, T., Núñez Queija, R., & Yechiali, U. (2004). Optimal routing among · /M/1 queues with partial information. Stochastic Models 20(2): 149–171.CrossRefGoogle Scholar
2.Armony, M., Perel, E., Perel, N., & Yechiali, U. (2019). Exact analysis for multi-server queueing systems with cross selling. Annals of Operations Research 274(1–2): 75–100.CrossRefGoogle Scholar
3.Artalejo, J.R., & Gómez-Corral, A. (2008). Retrial queueing systems: A computational approach. Berlin, Heidelberg: Springer-Verlag.CrossRefGoogle Scholar
4.Bini, D.A., Latouche, G., & Meini, B. (2005). Numerical methods for structured Markov chains. Oxford University Press.CrossRefGoogle Scholar
5.Hanukov, G., Avinadav, T., Chernonog, T., Spiegel, U., & Yechiali, U. (2017). A queueing system with decomposed service and inventoried preliminary services. Applied Mathematical Modeling 47: 276–293.CrossRefGoogle Scholar
6.Hanukov, G., Avinadav, T., Chernonog, T., Spiegel, U., & Yechiali, U. (2018). Improving efficiency in service systems by performing and storing “preliminary services”. International Journal of Production Economics 197: 174–185.CrossRefGoogle Scholar
7.Hanukov, G., Avinadav, T., Chernonog, T., & Yechiali, U. (2019). Performance improvement of a service system via stocking perishable preliminary services. European Journal of Operational Research 274(3): 1000–1011.CrossRefGoogle Scholar
8.Haviv, M. (2013). A Course in Queueing Theory.CrossRefGoogle Scholar
9.Jolles, A., Perel, E., & Yechiali, U. (2018). Alternating server with non-zero switch-over times and opposite-queue threshold-based switching policy. Performance Evaluation 126: 22–38.CrossRefGoogle Scholar
10.Kapodistria, S., & Palmowski, Z. (2017). Matrix geometric approach for random walks: Stability condition and equilibrium distribution. Stochastic Models 33(4): 572–597.CrossRefGoogle Scholar
11.Latouche, G., & Ramaswami, V. (1999) Introduction to matrix analytic methods in stochastic modeling. Philadelphia, PA: ASA-SIAM Series on Statistics and Applied Probability, SIAM.CrossRefGoogle Scholar
12.Levy, Y., & Yechiali, U. (1976). An M/M/s queue with servers’ vacations. INFOR: Information Systems and Operational Research 14(2): 153–163.Google Scholar
13.Neuts, M. (1981). Matrix-geometric solutions in stochastic models: An algorithmic approach. Baltimore, MD: Johns Hopkins University Press.Google Scholar
14.Paz, N., & Yechiali, U. (2014). An M/M/1 queue in random environment with disasters. Asia-Pacific Journal of Operational Research 31(3): 1450016.CrossRefGoogle Scholar
15.Perel, E., & Yechiali, U. (2008). Queues where customers of one queue act as servers of the other queue. Queueing Systems 60(3–4): 271–288.CrossRefGoogle Scholar
16.Perel, N., & Yechiali, U. (2013). The Israeli queue with priorities. Stochastic Models 29(3): 353–379.CrossRefGoogle Scholar
17.Perel, N., & Yechiali, U. (2014). The Israeli queue with infinite number of groups. Probability in the Engineering and Informational Sciences 28(1): 1–19.CrossRefGoogle Scholar
18.Perel, E., & Yechiali, U. (2017). Two-queue polling systems with switching policy based on the queue that is not being served. Stochastic Models 33(3): 430–450.CrossRefGoogle Scholar
19.Phung-Duc, T. (2017). Exact solutions for M/M/c/setup queues. Telecommunication Systems 64(2): 309–324.CrossRefGoogle Scholar
20.Van Leeuwaarden, J.S.H., & Winands, E.M.M. (2006). Quasi-birth-and-death processes with an explicit rate matrix. Stochastic Models 22(1): 77–98.CrossRefGoogle Scholar
21.Van Houdt, B., & van Leeuwaarden, J.S.H. (2011). Triangular M/G/1-type and tree-like quasi-birth-death Markov chains. INFORMS Journal on Computing 23(1): 165–171.CrossRefGoogle Scholar
22.Van Leeuwaarden, J.S.H., Squillante, M.S., & Winands, E.M. (2009). Quasi-birth-and-death processes, lattice path counting, and hypergeometric functions. Journal of Applied Probability 46(2): 507–520.CrossRefGoogle Scholar