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

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

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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