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On the Transient Behavior of Ehrenfest and Engset Processes

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  04 January 2016

Mathieu Feuillet  and
Philippe Robert
Mathieu Feuillet*
Affiliation:
INRIA
Philippe Robert*
Affiliation:
INRIA
*
∗ Postal address: INRIA Paris-Rocquencourt, Domaine de Voluceau, 78153 Le Chesnay, France.
∗ Postal address: INRIA Paris-Rocquencourt, Domaine de Voluceau, 78153 Le Chesnay, France.
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Abstract

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Two classical stochastic processes are considered, the Ehrenfest process, introduced in 1907 in the kinetic theory of gases to describe the heat exchange between two bodies, and the Engset process, one of the early (1918) stochastic models of communication networks. In this paper we investigate the asymptotic behavior of the distributions of hitting times of these two processes when the number of particles/sources goes to infinity. Results concerning the hitting times of boundaries in particular are obtained. We rely on martingale methods; a key ingredient is an important family of simple nonnegative martingales, an analogue, for the Ehrenfest process, of the exponential martingales used in the study of random walks or of Brownian motion.

Keywords

Ehrenfest processEngset formulaexponential martingalespace-time harmonic functionhitting time

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 2 , June 2012 , pp. 562 - 582
Copyright
© Applied Probability Trust 

References

Bingham, N. H. (1991). Fluctuation theory for the Ehrenfest urn. Adv. Appl. Prob. 23, 598–611.CrossRefGoogle Scholar
Diaconis, P., Graham, R. L. and Morrison, J. A. (1990). Asymptotic analysis of a random walk on a hypercube with many dimensions. Random Structures Algorithms 1, 51–72.CrossRefGoogle Scholar
Di Crescenzo, A. (1998). First-passage-time densities and avoiding probabilities for birth-and-death processes with symmetric sample paths. J. Appl. Prob. 35, 383–394.Google Scholar
Engset, T. O. (1998). On the calculation of switches in an automatic telephone system. Telektronikk 94, 99–142.Google Scholar
Feinsilver, P. J. (1978). Special Functions, Probability Semigroups, and Hamiltonian Flows (Lecture Notes Math. 696). Springer, Berlin.Google Scholar
Feuillet, M. and Robert, P. (2012). The time scales of a stochastic network with failures. Preprint. In preparation.Google Scholar
Flajolet, P. and Huillet, T. (2008). Analytic combinatorics of the Mabinogion urn. In Proc. 5th Colloquium on Mathematics and Computer Science (Discrete Math. Theoret. Comput. Sci. Proc. AI), ed. Rösler, U., Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp. 549–571.Google Scholar
Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge University Press.Google Scholar
Flegg, M. B., Pollett, P. K. and Gramotnev, D. K. (2008). Ehrenfest model for condensation and evaporation processes in degrading aggregates with multiple bonds. Phys. Rev. E 78, 031117, 9 pp.CrossRefGoogle ScholarPubMed
Fricker, C., Robert, P. and Tibi, D. (1999). On the rates of convergence of Erlang's model. J. Appl. Prob. 36, 1167–1184.CrossRefGoogle Scholar
Karlin, S. and McGregor, J. (1957). The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366–400.CrossRefGoogle Scholar
Karlin, S. and McGregor, J. (1959). Coincidence properties of birth and death processes. Pacific J. Math. 9, 1109–1140.Google Scholar
Karlin, S. and McGregor, J. (1965). Ehrenfest urn models. J. Appl. Prob. 2, 352–376.Google Scholar
Kennedy, D. P. (1976). Some martingales related to cumulative sum tests and single-server queues. Stoch. Process. Appl. 4, 261–269.Google Scholar
Lamperti, J. and Snell, J. L. (1963). Martin boundaries for certain Markov chains. J. Math. Soc. Japan 15, 113–128.CrossRefGoogle Scholar
Palacios, J. L. (1993). Fluctuation theory for the Ehrenfest urn via electric networks. Adv. Appl. Prob. 25, 472–476.Google Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion (Fundamental Principles Math. Sci. 293), 3rd edn. Springer, Berlin.CrossRefGoogle Scholar
Robert, P. (2003). Stochastic Networks and Queues (Appl. Math. 52). Springer, Berlin.Google Scholar
Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 2. Cambridge University Press.Google Scholar
Schoutens, W. (2000). Stochastic Processes and Orthogonal Polynomials (Lecture Notes Statist. 146). Springer, New York.Google Scholar
Schoutens, W. and Teugels, J. L. (1998). Lévy processes, polynomials and martingales. Commun. Statist. Stoch. Models 14, 335–349.CrossRefGoogle Scholar
Simatos, F. and Tibi, D. (2010). Spatial homogenization in a stochastic network with mobility. Ann. Appl. Prob. 20, 312–355.Google Scholar
Szegő, G. (1975). Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence, RI.Google Scholar
Whittaker, E. T. and Watson, G. N. (1996). A Course of Modern Analysis. Cambridge University Press.Google Scholar