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

Part of:Nontrigonometric harmonic analysisSpecial processesLatticesCombinatorial probabilityMarkov processes

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: j.s.h.v.leeuwaarden@tue.nl
∗∗ Postal address: Mathematical Sciences Department, IBM Thomas J. Watson Research Center, PO Box 218, Yorktown Heights, NY 10598, USA. Email address: mss@watson.ibm.com
∗∗∗ 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: e.m.m.winands@tue.nl
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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 spaces60J05: Discrete-time Markov processes on general state spaces60C05: Combinatorial probability33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)06B99: None of the above, but in this section42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)60K25: Queueing theory60J22: Computational methods in Markov chains

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009, pp. 507 - 520
Copyright
Copyright © Applied Probability Trust 2009 

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