Waiting-time tail probabilities in queues with long-tail service-time distributions

We consider the standard GI/G/1 queue with unlimited waiting room and the first-in first-out service discipline. We investigate the steady-state waiting-time tail probabilities P(W > x) when the service-time distribution has a long-tail distribution, i.e., when the service-time distribution fails to have a finite moment generating function. We have developed algorithms for computing the waiting-time distribution by Laplace transform inversion when the Laplace transforms of the interarrival-time and service-time distributions are known. One algorithm, exploiting Pollaczek’s classical contour-integral representation of the Laplace transform, does not require that either of these transforms be rational. To facilitate such calculations, we introduce a convenient twoparameter family of long-tail distributions on the positive half line with explicit Laplace transforms. This family is a Pareto mixture of exponential (PME) distributions. These PME distributions have monotone densities and Pareto-like tails, i.e., are of order x − r for r > 1. We use this family of long-tail distributions to investigate the quality of approximations based on asymptotics for P(W > x) as x → ∞. We show that the asymptotic approximations with these long-tail service-time distributions can be remarkably inaccurate for typical x values of interest. We also derive multi-term asymptotic expansions for the waiting-time tail probabilities in the M/G/1 queue. Even three terms of this expansion can be remarkably inaccurate for typical x values of interest. Thus, we evidently must rely on numerical algorithms for determining the waiting-time tail probabilities in this case. When working with service-time data, we suggest using empirical Laplace transforms.