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Linear stochastic fluid networks

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Offer Kella  and
Ward Whitt
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
Ward Whitt*
Affiliation:
AT&T Labs
*
∗Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: mskella@olive.mscc.huji.ac.il
∗∗Postal address: Room A117, AT&T Labs, Shannon Laboratory, 180 Park Avenue, Florham Park, NJ 07932–0971, USA. Email address: wow@research.att.com
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Abstract

We introduce open stochastic fluid networks that can be regarded as continuous analogues or fluid limits of open networks of infinite-server queues. Random exogenous input may come to any of the queues. At each queue, a c.d.f.-valued stochastic process governs the proportion of the input processed by a given time after arrival. The routeing may be deterministic (a specified sequence of successive queue visits) or proportional, i.e. a stochastic transition matrix may govern the proportion of the output routed from one queue to another. This stochastic fluid network with deterministic c.d.f.s governing processing at the queues arises as the limit of normalized networks of infinite-server queues with batch arrival processes where the batch sizes grow. In this limit, one can think of each particle having an evolution through the network, depending on its time and place of arrival, but otherwise independent of all other particles. A key property associated with this independence is the linearity: the workload associated with a superposition of inputs, each possibly having its own pattern of flow through the network, is simply the sum of the component workloads. As with infinite-server queueing models, the tractability makes the linear stochastic fluid network a natural candidate for approximations.

Keywords

Fluid modelsstochastic fluid networksstorage networksLévy processesinfinite-server queuesqueueing networksshot noise

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B05: Inventory, storage, reservoirs 90B15: Network models, stochastic
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 244 - 260
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

This work was partially supported by Israel Science Foundation Grant No. 794/97.

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