Computing queue-length distributions for power-law queues

The interest sparked by observations of long-range dependent traffic in real networks has lead to a revival of interest in non-standard queueing systems. One such queueing system is the M/G/1 queue where the service-time distribution has infinite variance. The known results for such systems are asymptotic in nature, typically providing the asymptotic form for the tail of the workload distribution, simulation being required to learn about the rest of the distribution. Simulation however performs very poorly for such systems due to the large impact of rare events. We provide a method for numerically evaluating the entire distribution for the number of customers in the M/G/1 queue with power-law tail service-time. The method is computationally efficient and shown to be accurate through careful simulations. It can be directly extended to other queueing systems and more generally to many problems where the inversion of probability generating functions complicated by power-laws is at issue. Through the use of examples we study the limitations of simulation and show that information on the tail of the queue-length distribution is not always sufficient to answer significant performance questions. We also derive the asymptotic form of the number of customers in the system in the case of a service-time distribution with a regularly varying tail (e.g. infinite variance) and thus illustrate the techniques required to apply the method in other contexts