The Formal Theory of Birth-and-Death Processes, Lattice Path Combinatorics, and Continued Fractions

Philippe Flajolet 1 Fabrice Guillemin
1 ALGO - Algorithms
Inria Paris-Rocquencourt
Abstract : Classic works of Karlin-McGregor and Jones-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 trajectories 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.
Type de document :
Rapport
[Research Report] RR-3667, INRIA. 1999
Liste complète des métadonnées

https://hal.inria.fr/inria-00073005
Contributeur : Rapport de Recherche Inria <>
Soumis le : mercredi 24 mai 2006 - 11:34:19
Dernière modification le : vendredi 25 mai 2018 - 12:02:02
Document(s) archivé(s) le : jeudi 24 mars 2011 - 12:24:22

Fichiers

Identifiants

  • HAL Id : inria-00073005, version 1

Collections

Citation

Philippe Flajolet, Fabrice Guillemin. The Formal Theory of Birth-and-Death Processes, Lattice Path Combinatorics, and Continued Fractions. [Research Report] RR-3667, INRIA. 1999. 〈inria-00073005〉

Partager

Métriques

Consultations de la notice

145

Téléchargements de fichiers

160