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Fluid limit of generalized Jackson queueing networks with stationary and ergodic arrivals and service times

Part of: Limit theorems Special processes

Published online by Cambridge University Press:  14 July 2016

Marc Lelarge
Marc Lelarge*
Affiliation:
INRIA-ENS
*
∗Postal address: ENS-DI, 45 rue d'Ulm, 75005 Paris, France. Email address: [email protected]
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Abstract

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We use a sample-path technique to derive asymptotics of generalized Jackson queueing networks in the fluid scale; that is, when space and time are scaled by the same factor n. The analysis only presupposes the existence of long-run averages and is based on some monotonicity and concavity arguments for the fluid processes. The results provide a functional strong law of large numbers for stochastic Jackson queueing networks, since they apply to their sample paths with probability 1. The fluid processes are shown to be piecewise linear and an explicit formulation of the different drifts is computed. A few applications of this fluid limit are given. In particular, a new computation of the constant that appears in the stability condition for such networks is given. In a certain context of a rare event, the fluid limit of the network is also derived explicitly.

Keywords

Generalized Jackson networksample-path techniquefixed pointfluid limit

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60F17: Functional limit theorems; invariance principles
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 491 - 512
Copyright
© Applied Probability Trust 2005 

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