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

Nam Kyoo Boots () and Perwez Shahabuddin
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Nam Kyoo Boots: Vrije Universiteit Amsterdam
Perwez Shahabuddin: Columbia University

No 01-012/4, Tinbergen Institute Discussion Papers from Tinbergen Institute

Abstract: This paper deals with estimating small tail probabilities of thesteady-state waiting time in a GI/GI/1 queue withheavy-tailed (subexponential) service times. The problem ofestimating infinite horizon ruin probabilities in insurancerisk processes with heavy-tailed claims can be transformed into thesame framework. It is well-known that naivesimulation is ineffective for estimating small probabilities andspecial fast simulation techniques like importancesampling, multilevel splitting, etc., have to be used. Though thereexists a vast amount of literature on the rare eventsimulation of queuing systems and networks with light-taileddistributions, previous fast simulation techniques forqueues with subexponential service times have been confined to theM/GI/1 queue. The general approach is to use thePollaczek-Khintchine transformation to convert the problem into thatof estimating the tail distribution of a geometricsum of independent subexponential random variables. However, no suchuseful transformation exists when one goesfrom Poisson arrivals to general interarrival-time distributions. Wedescribe and evaluate an approach that is based ondirectly simulating the random walk associated with the waiting-timeprocess of the GI/GI/1 queue, using a change ofmeasure called delayed subexponential twisting -an importancesampling idea recently developed and found useful inthe context of M/GI/1 heavy-tailed simulations.

Keywords: importance sampling; rare event simulation; subexponential distributions; insurance risk; GI/GI/1 queues (search for similar items in EconPapers)
Date: 2001-02-06
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