Numerical Transform Inversion for Autocorrelations of Waiting Times

Abstract

The generating function of the autocorrelations of successive waiting times in a stationary M/G/l or in a stationary GI/M/1 system can be expressed in terms of the probability generating function of the number of customers served in a busy period. The latter function is only implicitly determined as a solution to a functional equation. More explicit expressions have been obtained with the aid of Lagrange’s theorem on the reversion of power series, but they involve increasingly higher order derivatives of a function which comprises several Laplace-Stieltjes transforms. A recently discovered substitution method for contour integrals allows the numerical inversion of an implicitly determined generating function without the numerical solution of the functional equation for many complex values.