New Representations for G/G/1 Waiting Times

TL;DR

This paper introduces explicit Laplace transform representations for G/G/1 waiting times using Wiener-Hopf factorization, enabling efficient numerical analysis of complex queue models.

Contribution

It provides a novel explicit representation and invertibility results for G/G/1 waiting time distributions, facilitating practical computations.

Findings

Explicit Laplace transform representations derived

Numerical methods demonstrated for various G/G/1 queues

Special tractable results for M/M/1 with gated arrivals

Abstract

We obtain an explicit representation for the Laplace transform of the waiting time for a wide class of distributions by solving the Wiener-Hopf factorization problem via the Hadamard product theorem. Under broad conditions it is shown that this representation is invertible by an infinite partial fraction expansion. Computational schema illustrated by a variety of examples demonstrate the feasibility of numerically solving a rich class of G/G/1 queue possessing either bounded service or arrival times. In particular very tractable computations are derived for the M/M/1 queue with gated arrivals.