CONTINUED-FRACTION ANALYSIS OF THE DURATION OF AN EXCURSION IN AN M M/INFINITY SYSTEM/

We show in this paper how the Laplace transform theta(star) of the dur ation 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 co ntinued fraction is established and the [m-1/m] Pade approximants are computed by means of well known orthogonal polynomials, namely associa ted Charlier polynomials. It turns out that the continued fraction con sidered is an S fraction and as a consequence the Stieltjes transform of some spectral measure. Then, using classic asymptotic expansion pro perties 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.