Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-12T12:00:47.364Z Has data issue: false hasContentIssue false
  • English
  • Français

Excursions of birth and death processes, orthogonal polynomials, and continued fractions

Part of: Markov processes Convergence and divergence of infinite limiting processes

Published online by Cambridge University Press:  14 July 2016

Fabrice Guillemin  and
Didier Pinchon
Fabrice Guillemin*
Affiliation:
France Telecom
Didier Pinchon*
Affiliation:
Université Paul Sabatier
*
∗Postal address: France Telecom/CNET DAC/ARP, Technopole Anticipa, 2 Avenue Pierre Marzin, 22307 Lannion Cedex, France. Email address: [email protected].
∗∗Postal address: Laboratoire MIP, Université Paul Sabatier, 118 route de Narbonne, 31 062 Toulouse Cedex, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

Keywords

Continued fractionbirth and death processspectral measureorthogonal polynomial

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 40A15: Convergence and divergence of continued fractions
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 752 - 770
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M., and Stegun, I. (1972). Handbook of Mathematical Functions. Applied Mathematics Series 55. National Bureau of Standards, Washington DC.Google Scholar
Aldous, D. (1989). Probability approximations via the Poisson Clumping Heuristic. Springer, Berlin.CrossRefGoogle Scholar
Askey, R., and Ismail, M. E. H. (1984). Recurrence relations, continued fractions, and orthogonal polynomials. Memoirs of the American Mathematical Society, Vol. 49, Number 300.CrossRefGoogle Scholar
Asmussen, S. (1987). Applied Probability and Queues. Wiley, New York.Google Scholar
Bordes, G., and Roehner, B. (1983). Applications of Stieltjes theory for S-fractions to birth and death processes. Adv. Appl. Prob. 15, 507–530.CrossRefGoogle Scholar
Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.Google Scholar
Guillemin, F., and Pinchon, D. (1998). Continued fraction analysis of the duration of an excursion in an M/M/α system. J. Appl. Prob. 35, 165–183.CrossRefGoogle Scholar
Guillemin, F., and Simonian, A. (1995). Transient characteristics of an M/M/α system. Adv. Appl. Prob. 27, 862–888.CrossRefGoogle Scholar
Henrici, P. (1977). Applied and Computational Complex Analysis, Vol. 2. Wiley, New York.Google Scholar
Ismail, M. E. H., Letessier, J., Masson, D., and Valent, G. (1990). Birth and death processes and orthogonal polynomials. In Orthogonal polynomials, ed. Nevai, P. Kluwer, Dordrecht, pp. 229–255.CrossRefGoogle Scholar
Karlin, S., and McGregor, J. (1957). The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–401.CrossRefGoogle Scholar
Karlin, S., and McGregor, J. (1957). The differential equation of birth and death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.CrossRefGoogle Scholar
Karlin, S., and McGregor, J. (1958). Linear growth birth and death processes. J. Math. Mech. 7, 643–662.Google Scholar
Kleinrock, L. (1975). Queueing Systems, Vol. I: Theory. Wiley, New York.Google Scholar
Morrison, J., Shepp, J., and Van Wyck, C. (1987). A queueing analysis of hashing with lazy deletion. SIAM J. Computing 16, 1155–1164.CrossRefGoogle Scholar