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The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions

Part of: Markov processes Stochastic processes Combinatorics

Published online by Cambridge University Press:  19 February 2016

Philippe Flajolet  and
Fabrice Guillemin
Philippe Flajolet*
Affiliation:
INRIA
Fabrice Guillemin*
Affiliation:
France Telecom
*
∗ Postal address: Algorithms Project, INRIA, F-78150 Rocquencourt, France. Email address: [email protected]
∗∗ Postal address: France Telecom, BD.CNET, DAC/ATM, 2, Avenue Pierre Marzin, 22300 Lannion, France. Email address: [email protected]
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Abstract

Classic works of Karlin and McGregor and Jones and Magnus have established a general correspondence between continuous-time birth-and-death processes and continued fractions of the Stieltjes-Jacobi type together with their associated orthogonal polynomials. This fundamental correspondence is revisited here in the light of the basic relation between weighted lattice paths and continued fractions otherwise known from combinatorial theory. Given that sample paths of the embedded Markov chain of a birth-and-death process are lattice paths, Laplace transforms of a number of transient characteristics can be obtained systematically in terms of a fundamental continued fraction and its family of convergent polynomials. Applications include the analysis of evolutions in a strip, upcrossing and downcrossing times under flooring and ceiling conditions, as well as time, area, or number of transitions while a geometric condition is satisfied.

Keywords

Lattice path combinatoricscontinued fractionsorthogonal polynomialsbirth-and-death processfirst passage timeexcursionstransient characteristics

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 05A15: Exact enumeration problems, generating functions
Secondary: 60G17: Sample path properties
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000 , pp. 750 - 778
Copyright
Copyright © Applied Probability Trust 2000 

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