Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-30T21:15:48.634Z Has data issue: false hasContentIssue false
  • English
  • Français

Further approaches to computing fundamental characteristics of birth-death processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Frank Ball  and
Valeri T. Stefanov
Frank Ball*
Affiliation:
University of Nottingham
Valeri T. Stefanov*
Affiliation:
The University of Western Australia
*
∗ Postal address: School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK. Email address: [email protected]
∗∗ Postal address: Department of Mathematics and Statistics, The University of Western Australia, Crawley 6009, WA, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

General and unifying approaches are discussed for computing fundamental characteristics of both continuous-time and discrete-time birth-death processes. In particular, an exponential family framework is used to derive explicit expressions, in terms of continued fractions, for joint generating functions of first-passage times and a whole collection of associated random quantities, and a random sum representation is used to obtain formulae for means, variances and covariances of stopped reward functions defined on a birth-death process.

Keywords

Birth-death processstopping timeexponential family

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60J22: Computational methods in Markov chains
Secondary: 92D30: Epidemiology
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 995 - 1005
Copyright
Copyright © by the Applied Probability Trust 2001 

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

Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, New York.Google Scholar
Artalejo, J. R., and Lopez-Herrero, M. J. (2001). Analysis of the busy period for the M/M/c queue: an algorithmic approach. J. Appl. Prob. 38, 209–222.Google Scholar
Barndorff-Nielsen, O. (1978). Information and Exponential Families. John Wiley, Chichester.Google Scholar
Bingham, N. H. (1991). Fluctuation theory for the Ehrenfest urn. Adv. Appl. Prob. 23, 598–611.CrossRefGoogle Scholar
Flajolet, F., and Guillemin, F. (2000). The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions. Adv. Appl. Prob. 32, 750–778.CrossRefGoogle Scholar
Guillemin, F., and Pinchon, D. (1998). Continued fraction analysis of the duration of an excursion in an M/M/∞ system. J. Appl. Prob. 35, 165–183.Google Scholar
Guillemin, F., and Pinchon, D. (1999). Excursions of birth–death processes, orthogonal polynomials, and continued fractions. J. Appl. Prob. 36, 752–770.Google Scholar
Hernández-Suárez, C. M., and Castillo-Chavez, C. (1999) A basic result on the integral for birth–death Markov processes. Math. Biosci. 161, 95–104.CrossRefGoogle ScholarPubMed
Jones, W. B., and Magnus, A. (1977). Application of Stieltjes fractions to birth–death processes. In Padé and Rational Approximation, eds Saff, E. B. and Varga, R. S., Academic Press, New York, pp. 173–179.CrossRefGoogle Scholar
Karlin, S., and McGregor, J. L. (1957a). The differential equation of birth and death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–546.Google Scholar
Karlin, S., and McGregor, J. L. (1957b). The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–401.Google Scholar
Karlin, S., and McGregor, J. L. (1958). Linear growth birth and death processes. J. Math. Mech. 7, 643–662.Google Scholar
Karlin, S., and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Karlin, S., and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Keilson, J. (1965). A review of transient behaviour in regular diffusion and birth–death processes. Part II. J. Appl. Prob. 2, 405–428.CrossRefGoogle Scholar
Murphy, J. A., and O'Donohoe, J. L. (1975). Some properties of continued fractions with applications in Markov processes. J. Inst. Math. Appl. 16, 57–71.CrossRefGoogle Scholar
Pakes, A. G. (1979). The age of a Markov process. Stoch. Process. Appl. 8, 277–303.Google Scholar
Stefanov, V. T. (1991). Noncurved exponential families associated with observations over finite state Markov chains. Scand. J. Statist. 18, 353–356.Google Scholar
Stefanov, V. T., and Wang, S. (2000). A note on integrals for birth–death processes. Math. Biosci. 168, 161–165.CrossRefGoogle ScholarPubMed
Waugh, W. A. O'N. (1972). Taboo extinction, sojourn times, and asymptotic growth for the Markovian birth and death process. J. Appl. Prob. 9, 486–506.Google Scholar