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Tails in generalized Jackson networks with subexponential service-time distributions

Part of: Special processes Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

François Baccelli ,
Serguei Foss  and
Marc Lelarge
François Baccelli*
Affiliation:
INRIA-ENS
Serguei Foss*
Affiliation:
Heriot-Watt University and Institute of Mathematics, Novosibirsk
Marc Lelarge*
Affiliation:
INRIA-ENS
*
∗Postal address: ENS-DI, 45 rue d'Ulm, 75005 Paris, France.
∗∗∗Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: s.foss@ma.hw.ac.uk
∗Postal address: ENS-DI, 45 rue d'Ulm, 75005 Paris, France.
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Abstract

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We give the exact asymptotics of the tail of the stationary maximal dater in generalized Jackson networks with subexponential service times. This maximal dater, which is an analogue of the workload in an isolated queue, gives the time taken to clear all customers present at some time t when stopping all arrivals that take place later than t. We use the property that a large deviation of the maximal dater is caused by a single large service time at a single station at some time in the distant past of t, in conjunction with fluid limits of generalized Jackson networks, to derive the relevant asymptotics in closed form.

Keywords

Generalized Jackson networksubexponential random variableheavy tailintegrated tailVeraverbeke's theoremfluid limit

MSC classification

Primary: 60K25: Queueing theory
Secondary: 62G20: Asymptotic properties
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 513 - 530
Copyright
© Applied Probability Trust 2005 

Footnotes

***

Supported by INTAS grant 265 and by EPSRC grant R58765/01.

References

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