### Abstract

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.

Original language | English |
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Place of Publication | Amsterdam |

Publisher | Tinbergen Instituut |

Publication status | Published - 2001 |

### Publication series

Name | Discussion paper TI |
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No. | 01-012/4 |

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## Cite this

Boots, N. K., & Shahabuddin, P. (2001).

*Simulating Tail Probabilities in GI/GI.1 Queues and Insurance Risk Processes with Subexponentail Distributions*. (Discussion paper TI; No. 01-012/4). Amsterdam: Tinbergen Instituut.