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Quasi-Birth-and-Death Processes, Lattice Path Counting, and Hypergeometric Functions

Part of: Markov processes Lattices Special processes Combinatorial probability Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  14 July 2016

Johan S. H. van Leeuwaarden ,
Mark S. Squillante  and
Erik M. M. Winands
Johan S. H. van Leeuwaarden*
Affiliation:
Eindhoven University of Technology and EURANDOM
Mark S. Squillante*
Affiliation:
IBM Thomas J. Watson Research Center
Erik M. M. Winands*
Affiliation:
Eindhoven University of Technology
*
∗Postal address: EURANDOM, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
∗∗Postal address: Mathematical Sciences Department, IBM Thomas J. Watson Research Center, PO Box 218, Yorktown Heights, NY 10598, USA. Email address: [email protected]
∗∗∗Postal address: Department of Mathematics and Computer Science and Department of Technology Management, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
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Abstract

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In this paper we consider a class of quasi-birth-and-death processes for which explicit solutions can be obtained for the rate matrix R and the associated matrix G. The probabilistic interpretations of these matrices allow us to describe their elements in terms of paths on the two-dimensional lattice. Then determining explicit expressions for the matrices becomes equivalent to solving a lattice path counting problem, the solution of which is derived using path decomposition, Bernoulli excursions, and hypergeometric functions. A few applications are provided, including classical models for which we obtain some new results.

Keywords

Quasi-birth-and-death processmatrix-analytic methodsrate matrixlattice path countinghypergeometric function

MSC classification

Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J05: Discrete-time Markov processes on general state spaces 60C05: Combinatorial probability 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 06B99: None of the above, but in this section 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 60K25: Queueing theory 60J22: Computational methods in Markov chains
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 507 - 520
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. US Government Printing Office, Washington, D.C. Google Scholar
[2] Adan, I. J. B. F. and Weiss, G. (2006). Analysis of a simple Markovian re-entrant line with infinite supply of work under the LBFS policy. Queueing systems 54, 169–183.Google Scholar
[3] Adan, I. J. and, B. F. van der Wal, J. (1998). Combining make to order and make to stock. OR Spektrum 20, 73–81.CrossRefGoogle Scholar
[4] Cobham, A. (1954). Priority assignment in waiting line problems. Operat. Res. 2, 70–76.Google Scholar
[5] Cohen, J. W. (1987). A two-queue, one-server model with priority for the longer queue. Queueing systems 2, 261–283.CrossRefGoogle Scholar
[6] Dunford, N. and Schwartz, J. T. (1958). Linear Operators. Part I: General Theory. John Wiley, New York.Google Scholar
[7] Dunford, N. and Schwartz, J. T. (1963). Linear Operators. Part II: Spectral Theory. Self Adjoint Operators in Hilbert Space. John Wiley, New York.Google Scholar
[8] Dunford, N. and Schwartz, J. T. (1971). Linear Operators. Part III: Spectral Operators. John Wiley, New York.Google Scholar
[9] Flatto, L. (1989). The longer queue model. Prob. Eng. Inf. Sci. 3, 537–559.CrossRefGoogle Scholar
[10] Jaiswal, N. K. (1968). Priority Queues (Math. Sci. Eng. 50). Academic Press, London.Google Scholar
[11] Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
[12] van Leeuwaarden, J. S. H. and Winands, E. M. M. (2006). Quasi-birth-and-death processes with an explicit rate matrix. Stoch. Models 22, 77–98.CrossRefGoogle Scholar
[13] Liu, D. and Zhao, Y. Q. (1997). Determination of explicit solutions for a general class of Markov processes. In Matrix-Analytic Methods in Stochastic Models (Lecture Notes Pure Appl. Math. 183), Dekker, New York, pp. 343–357.Google Scholar
[14] Nelson, R. D. and Squillante, M. S. (1996). Stochastic analysis of affinity scheduling and load balancing in parallel processing systems. In Proc. 5th INFORMS Computer Science Technical Section Conference on Computer Science and Operations Research: Recent Advances in the Interface (January 1996).Google Scholar
[15] Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins Press, Baltimore, MD.Google Scholar
[16] Neuts, M. F. (1989). Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker, New York.Google Scholar
[17] Ramaswami, V. and Latouche, G. (1986). A general class of Markov processes with explicit matrix-geometric solutions. OR Spektrum 8, 209–218.Google Scholar
[18] Schrage, L. E. (1967). The queue M/G/1 with feedback to lower priority queues. Management Sci. 13, 466–474.Google Scholar
[19] Squillante, M. S. (1998). Matrix-analytic methods in stochastic parallel-server scheduling models. In Advances in Matrix Analytic Methods for Stochastic Models, Notable, New Jersey, pp. 311–340.Google Scholar
[20] Squillante, M. S. (2005). Stochastic analysis of resource allocation in parallel processing systems. In Computer System Performance Modeling in Perspective: A Tribute to the Work of Prof. K. C. Sevcik, Imperial College Press, London, pp. 227–256.Google Scholar
[21] Squillante, M. S. and Nelson, R. D. (1991). Analysis of task migration in shared-memory multiprocessors. In Proc. ACM SIGMETRICS (May 1991), ACM Press, New York, pp. 143–155.Google Scholar
[22] Takács, L. (1991). A Bernoulli excursion and its various applications. Adv. App. Prob. 23, 557–585.Google Scholar
[23] Tweedie, R. L. (1982). Operator-geometric stationary distributions for Markov chains, with application to queueing models. Adv. App. Prob. 14, 368–391.CrossRefGoogle Scholar
[24] Zheng, Y. and Zipkin, P. H. (1990). A queueing model to analyze the value of centralized inventory information. Operat. Res. 38, 296–307.CrossRefGoogle Scholar