Quasi-birth-and-death processes, lattice path counting, and hypergeometric functions
Johan S. H. van Leeuwaarden, Mark S. Squillante, and Erik M. M. Winands
Source: J. Appl. Probab. Volume 46, Number 2 (2009), 507-520.
Abstract
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.
Primary Subjects: 60J25, 60J05, 60C05, 33C45
Secondary Subjects: 06B99, 42C10, 60K25, 60J22
Keywords: Quasi-birth-and-death process; matrix-analytic methods; rate matrix; lattice path counting; hypergeometric function
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jap/1245676103
Digital Object Identifier: doi:10.1239/jap/1245676103
Zentralblatt MATH identifier:
05578822
Mathematical Reviews number (MathSciNet):
MR2535829
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