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

This paper deals with estimating small tail probabilities of thesteady-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. 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 confined to the M/GI/1 queue. The general approach is to use the Pollaczek-Khintchine 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

Tail behaviour of the area under a random process, with applications to queueing systems, insurance and percolations

The areas under workload process and under queuing process in a single server queue over the busy period have many applications not only in queuing theory but also in risk theory or percolation theory. We focus here on the tail behaviour of distribution of these two integrals. We present various open problems and conjectures, which are supported by partial results for some special cases

Queueing Systems with Heavy Tails

VI+227hlm.;24c

Fast simulation of a queue fed by a superposition of many (heavy-tailed) sources

We consider a queue fed by a large number, say $n$, of on-off sources with generally distributed on- and off-times. The queueing resources are scaled by $n$: the buffer is $Bequiv nb$ and link rate is $Cequiv nc$. The model is versatile: it allows us to model both long range dependent traffic (by using heavy-tailed distributed on-periods) and short range dependent traffic (by using light-tailed on-periods). A crucial performance metric in this model is the steady-state buffer overflow probability. This overflow probability decays exponentially in the number of sources $n$. Therefore, if the number of sources grows large, naive simulation is too time-consuming, and we have touse fast simulation techniques instead. Due to the exponential decay (in $n$), importance sampling with an exponential change of measure essentially goes through, irrespective of the on-times being heavy-tailed or light-tailed. An asymptotically optimal change of measure is found by using large deviations arguments. Notably, the change of measure is not constant during the simulation run, which is essentially different from many other studies (usually relying on large buffer asymptotics). We provide numerical examples to show that the resulting importance sampling procedure indeed improves considerably over naive simulation. We present some accelerations. Finally, we give short comments on the influence of the shape of the distributions on the loss probability, and we describe the limitations of our technique

Efficient Simulation and Conditional Functional Limit Theorems for Ruinous Heavy-tailed Random Walks

The contribution of this paper is to introduce change of measure based techniques for the rare-event analysis of heavy-tailed stochastic processes. Our changes-of-measure are parameterized by a family of distributions admitting a mixture form. We exploit our methodology to achieve two types of results. First, we construct Monte Carlo estimators that are strongly efficient (i.e. have bounded relative mean squared error as the event of interest becomes rare). These estimators are used to estimate both rare-event probabilities of interest and associated conditional expectations. We emphasize that our techniques allow us to control the expected termination time of the Monte Carlo algorithm even if the conditional expected stopping time (under the original distribution) given the event of interest is infinity -- a situation that sometimes occurs in heavy-tailed settings. Second, the mixture family serves as a good approximation (in total variation) of the conditional distribution of the whole process given the rare event of interest. The convenient form of the mixture family allows us to obtain, as a corollary, functional conditional central limit theorems that extend classical results in the literature. We illustrate our methodology in the context of the ruin probability $P(\sup_n S_n >b)$, where $S_n$ is a random walk with heavy-tailed increments that have negative drift. Our techniques are based on the use of Lyapunov inequalities for variance control and termination time. The conditional limit theorems combine the application of Lyapunov bounds with coupling arguments