On a random variable associated with excursions in an M/M/∞ system

Abstract

We show in this paper how the theory of continued fractions can be used to invert the Laplace transform of a transient characteristic associated with excursions in an M/M/∞ system with unit service rate and input intensity u. The characteristic under consideration is the area V swept under the occupation process of an M/M/∞ queue during an excursion period above a given threshold C. The Laplace transform V ^⋆ of this random variable has been established in earlier studies and can be expressed as a ratio of Tricomi functions. In this paper, we first establish the continued fraction representation of V ^⋆, which allows us to obtain an alternative expression of the Laplace transform in terms of Kummer functions. It then turns out that the continued fraction considered is the even part of a Stieltjes (S) fraction, which provides information on the location of the poles of V ^⋆. It appears that the Laplace transform has simple poles on the real negative axis. Taking benefit of the fact that the spectrum is compact and that the numerical values of the Laplace transform can easily be computed by means of the continued fraction, we finally use a classical Laplace transform inversion technique to numerically compute the survivor probability distribution function x➙ℙ{V > fx} of the random variable V, which exhibits an exponential decay only for very large values of the argument x when the ratio u/C is sufficiently smaller than one.