Computation of Autocorrelations of Interdeparture Times by Numerical Transform Inversion

The generating functions of the autocorrelations of the interdeparture times in a stationary M/G/1 system and in a stationary GI/M/1 system involve the probability generating functions of the number of customers served in a busy period. The latter functions are only implicitly determined as solutions to some functional equations. Standard methods for the numerical inversion of generating functions require the values of these functions at many complex arguments. A recently discovered substitution method for contour integrals allows the numerical inversion of implicitly determined generating functions without the numerical solution of the functional equations.