Calculation of the Gi/g/1 Waiting-time Distribution and Its Cumulants from Pollaczek’s Formulas

The steady-state waiting time in a stable GI/G/1 queue is equivalent to the maximum of a general random walk with negative drift. Thus, the distribution of the steady-state waiting time in the GI/G/1 queue is characterized by Spitzer’s (1956) formula. However, earlier, Pollaczek (1952) derived an equivalent contour-integral expression for the Laplace transform of the GI/G/1 steady-state waiting time. Since Spitzer’s formula is easier to understand probabilistically, it is better known today, but it is not so easy to apply directly except in special cases. In contrast, we show that it is easy to compute the GI/G/1 waiting-time distribution and its cumulants (and thus its moments) from Pollaczek’s formulas. For the waiting-time tail probabilities, we use numerical transform inversion, numerically integrating the Pollaczek contour integral to obtain the transform values. For the cumulants and the probability of having to wait, we directly integrate the Pollazcek contour integrals numerically. The resulting algorithm is evidently the first for a GI/G/1 queue in which neither the transform of the interarrival-time distribution nor the transform of the service-time transform distribution need be rational. The algorithm can even be applied to long-tail distributions, i.e., distributions with some infinite moments. To treat these distributions, we approximate them by suitable exponentially-damped versions of these distributions. Overall, the algorithm is remarkably simple compared to alternative algorithms requiring more structure. key words: Fe ́lix Pollaczek, queueing theory, computational probability, GI/G/1 queue, waiting time distribution, tail probabilities, random walk, Spitzer’s formula, numerical transform inversion, numerical integration, contour integrals.