Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T03:50:41.686Z Has data issue: false hasContentIssue false
  • English
  • Français

On the convergence to stationarity of birth-death processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Pauline Coolen-Schrijner  and
Erik A. Van Doorn
Pauline Coolen-Schrijner*
Affiliation:
University of Durham
Erik A. Van Doorn*
Affiliation:
University of Twente
*
∗ Postal address: Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham, DH1 3LE, UK. Email address: [email protected]
∗∗ Postal address: Faculty of Mathematical Sciences, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

Taking up a recent proposal by Stadje and Parthasarathy in the setting of the many-server Poisson queue, we consider the integral ∫0∞[limu→∞E(X(u))-E(X(t))]dt as a measure of the speed of convergence towards stationarity of the process {X(t), t≥0}, and evaluate the integral explicitly in terms of the parameters of the process in the case that {X(t), t≥0} is an ergodic birth-death process on {0,1,….} starting in 0. We also discuss the discrete-time counterpart of this result, and examine some specific examples.

Keywords

Birth-death processspeed of convergence

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 3 , September 2001 , pp. 696 - 706
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

Blanc, J. P. C., and van Doorn, E. A. (1986). Relaxation times for queueing systems. In Mathematics and Computer Sci., eds de Bakker, J. W., Hazewinkel, M. and Lenstra, J. K. North-Holland, Amsterdam, pp. 139–162.Google Scholar
Coolen-Schrijner, P., and van Doorn, E. A. (2001). The deviation matrix of a continuous-time Markov chain. Submitted.Google Scholar
Feller, W. (1967). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Holewijn, P. J., and Hordijk, A. (1975). On the convergence of moments in stationary Markov chains. Stoch. Proc. Appl. 3, 55–64.CrossRefGoogle Scholar
Karlin, S., and McGregor, J. L. (1957). The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 589–646.CrossRefGoogle Scholar
Karlin, S., and McGregor, J. L. (1957). The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.CrossRefGoogle Scholar
Karlin, S., and McGregor, J. L. (1959). Random walks. Illinois J. Math. 3, 66–81.CrossRefGoogle Scholar
Keilson, J. (1971). Log-concavity and log-convexity in passage time densities of diffusion and birth–death processes. J. Appl. Prob. 8, 391–398.CrossRefGoogle Scholar
Kijima, M. (1997). Markov Processes for Stochastic Modeling. Chapman and Hall, London.CrossRefGoogle Scholar
Stadje, W., and Parthasarathy, P. R. (1999). On the convergence to stationarity of the many-server Poisson queue. J. Appl. Prob. 36, 546–557.CrossRefGoogle Scholar
Van Doorn, E. A. (1981). The transient state probabilities for a queueing model where potential customers are discouraged by queue length. J. Appl. Prob. 18, 499–506.CrossRefGoogle Scholar
Van Doorn, E. A. (1985). Conditions for exponential ergodicity and bounds for the decay parameter of a birth–death process. Adv. Appl. Prob. 17, 514–530.CrossRefGoogle Scholar
Whitt, W. (1992). Asymptotic formulas for Markov processes with applications to simulation. Operat. Res. 40, 279–291.CrossRefGoogle Scholar