Continued fraction analysis of the duration of an excursion in an M/M/∞ system

We show in this paper how the Laplace transform theta(star) of the duration a of an excursion by the occupation process {Lambda(t)} of an M/M/infinity system above a given threshold can be obtained by means of continued fraction analysis. The representation of theta(star) by a continued fraction is established and the [m-1/m] Pade 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 theta(star) 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 {Lambda(t)}. The continued fraction representation enables us to further characterize the distribution of the random variable theta.