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Continued Fraction Analysis of the Duration of an Excursion in an M/M/∞ System

Part of: Miscellaneous topics of analysis in the complex domain Markov processes

Published online by Cambridge University Press:  14 July 2016

Fabrice Guillemin  and
Didier Pinchon
Fabrice Guillemin*
Affiliation:
France Télécom
Didier Pinchon*
Affiliation:
Université Paul Sabatier
*
∗Postal address: France Télécom, CNET Lannion A, Technopole Anticipa, 2, Avenue Pierre Marzin, 22300 Lannion, France
∗∗Postal address: Laboratoire MIP, Université Paul Sabatier, 118 Route de Narbonne, 31 062 Toulouse Cedex, France
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Abstract

We show in this paper how the Laplace transform θ* of the duration θ of an excursion by the occupation process {Λt} of an M/M/∞ system above a given threshold can be obtained by means of continued fraction analysis. The representation of θ* by a continued fraction is established and the [m−1/m] Padé approximants are computed by means of well known orthogonal polynomials, namely associated Charlier polynomials. It turns out that the continued fraction considered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion properties of hypergeometric functions, the representation of the Laplace transform θ* by means of Kummer's function is obtained. This allows us to recover an earlier result obtained via complex analysis and the use of the strong Markov property satisfied by the occupation process {Λt}. The continued fraction representation enables us to further characterize the distribution of the random variable θ.

Keywords

M/M/∞ systemLaplace transformscontinued fractionsStieltjes transformsorthogonal polynomials

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 30E15: Asymptotic representations in the complex domain
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 1 , March 1998 , pp. 165 - 183
Copyright
Copyright © Applied Probability Trust 1998 

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References

Askey, R., and Ismail, M. (1984). Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Amer. Math. Soc. 300, 12.Google Scholar
Doshi, B. (1993). Deterministic rule based traffic descriptors for broadband ISDN: worst case behavior and connection acceptance control. In Proc. Globecom'93. Houston, TX.Google Scholar
Guillemin, F., and Simonian, A. (1995). Transient characteristics of an M/M/∞ system. Adv. Appl. Prob. 27, 862–888.CrossRefGoogle Scholar
Guillemin, F., Rubino, G., Sericola, B., and Simonian, A. (1997). Transient analysis of statistical multiplexing of data connections on an ATM link. In Proc. ITC'15. Washington, DC.Google Scholar
Guillemin, F., and Pinchon, D. (1996). On a random variable associated with excursions in an M/M/∞ queueing system. Submitted.Google Scholar
Henrici, P. (1977). Applied and Computational Complex Analysis. Vol. 2. Wiley, New York.Google Scholar
Ince, E.L. (1956). Ordinary Differential Equations. Dover, New York.Google Scholar
Ismail, M., 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
Ismail, M., Letessier, J., and Valent, G. (1988). Linear birth and death models and associated Laguerre and Meixner polynomials. J. Approx. Theory 55, 337–348.CrossRefGoogle Scholar
Ismail, M., and Kelker, D. (1979). Special functions, Stieltjes transforms and infinite divisibility. SIAM J. Math. Anal. 10, 884–901.CrossRefGoogle Scholar
Karlin, S., and McGregor, J. (1957). The differential equations of birth and death processes and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489–456.CrossRefGoogle Scholar
Karlin, S., and McGregor, J. (1958). Linear growth birth and death processes. J. Math. Mech. 7, 643–662.Google Scholar
Lebedev, N. (1965). Special Functions and their Applications. Prentice Hall, New York.CrossRefGoogle Scholar
Morrison, J., Shepp, L., and Van Wyk, C. (1987). A queueing analysis of hashing with lazy deletion. SIAM J. Comput. 16, 1155–1164.CrossRefGoogle Scholar
Slater, L.J. (1960). Confluent Hypergeometric Functions. Cambridge University Press, Cambridge.Google Scholar
Szegö, G. (1975). Orthogonal Polynomials. American Mathematical Society, Providence, RI.Google Scholar