Simulating GI/GI/1 queues and insurance risk processes with subexponential distributions

Abstract

This paper deals with estimating small tail probabilities of the steady-state waiting time in a GI/GI/1 queue with heavy-tailed (subexponential) service times. The problem of estimating infinite horizon ruin probabilities in insurance risk processes with heavy-tailed claims can be transformed into the same framework. It is well-known that naive simulation is ineffective for estimating small probabilities and special fast simulation techniques like importance sampling, multilevel splitting, etc., have to be used. Previous fast simulation techniques for queues with subexponential service times have been confined to M/GI/1 queueing systems. The general approach is to use the Pollaczek-Khintchine transformation to transform the problem into that of estimating the tail distribution of a geometric sum of independent subexponential random variables. However, no such useful transformation exists when one goes from Poisson arrivals to general interarrival-time distributions. We describe an approach that is based on directly simulating the random walk associated with the waiting-time process of the GI/GI/1 queue, using a change of measure called delayed subexponential twisting --- an importance sampling idea recently developed and found useful in the context of M/GI/1 heavy-tailed simulations.