Simulating Tail Probabilities in GI/GI.1 Queues and Insurance Risk Processes with Sub Exponential Distributions

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 interarrival times can have any distribution with a nite mean. The problem of estimating in nite 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 ine ective for estimating small probabilities and special fast simulation techniques like importance sampling, multilevel splitting, etc., have to be used. Though there exists a vast amount of literature on the rare event simulation of queuing systems and networks with light-tailed distributions, previous fast simulation techniques for queues with subexponential service times have been con ned to the M/GI/1 queue. The general approach is to use the PollaczekKhintchine transformation to convert 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 and evaluate 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. Some quantities other than those mentioned above can also be estimated via this approach.