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A unified approach to fast teller queues and ATM

Part of: Special processes Limit theorems Operations research and management science Computer system organization

Published online by Cambridge University Press:  01 July 2016

B. Beck ,
A. R. Dabrowski  and
D. R. McDonald
B. Beck*
Affiliation:
Université Catholique de Louvain
A. R. Dabrowski*
Affiliation:
University of Ottawa
D. R. McDonald*
Affiliation:
University of Ottawa
*
∗ Postal address: Institut de Statistique, Université Catholique de Louvain, Voie du Roman Pays, 20, B-1348 Louvain-La-Neuve, Belgium.
∗∗ Postal address: Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5.
∗∗ Postal address: Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5.
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Abstract

This paper examines a problem of importance to the telecommunications industry. In the design of modern ATM switches, it is necessary to use simulation to estimate the probability that a queue within the switch exceeds a given large value. Since these are extremely small probabilities, importance sampling methods must be used. Here we obtain a change of measure for a broad class of models with direct applicability to ATM switches.

We consider a model with A independent sources of cells where each source is modeled by a Markov renewal point process with batch arrivals. We do not assume the sources are necessarily identically distributed, nor that batch sizes are independent of the state of the Markov process. These arrivals join a queue served by multiple independent servers, each with service times also modeled as a Markov renewal process. We only discuss a time-slotted system. The queue is viewed as the additive component of a Markov additive chain subject to the constraint that the additive component remains non-negative. We apply the theory in McDonald (1999) to obtain the asymptotics of the tail of the distribution of the queue size in steady state plus the asymptotics of the mean time between large deviations of the queue size.

Keywords

Rare eventschange of measureh-transformasynchronous transfer modehotspot simulation

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60F10: Large deviations 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 68M20: Performance evaluation; queueing; scheduling 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 3 , September 1999 , pp. 758 - 787
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Research of the last two authors supported in part by NSERC grant.

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