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On the dynamics and performance of stochastic fluid systems

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  14 July 2016

Takis Konstantopoulos  and
Günter Last
Takis Konstantopoulos*
Affiliation:
University of Texas at Austin
Günter Last*
Affiliation:
Technische Universität Braunschweig
*
∗Postal address: Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78712, USA. Email address: takis@alea.ece.utexas.edu
∗∗Postal address: Institut für Mathematische Stochastik, Universität Karlsruhe, Englerstraße 2, 76128 Karlsruhe, Germany. Email address: g.last@math.uni-karlsruhe.de
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Abstract

A (generalized) stochastic fluid system Q is defined as the one-dimensional Skorokhod reflection of a finite variation process X (with possibly discontinuous paths). We write X as the (not necessarily minimal) difference of two positive measures, A, B, and prove an alternative ‘integral representation’ for Q. This representation forms the basis for deriving a ‘Little's law’ for an appropriately constructed stationary version of Q. For the special case where B is the Lebesgue measure, a distributional version of Little's law is derived. This is done both at the arrival and departure points of the system. The latter result necessitates the consideration of a ‘dual process’ to Q. Examples of models for X, including finite variation Lévy processes with countably many jumps on finite intervals, are given in order to illustrate the ideas and point out potential applications in performance evaluation.

Keywords

Random measuresstationary processesPalm probabilitiesqueueing theoryLittle's lawperformance evaluation

MSC classification

Primary: 60G17: Sample path properties 60G10: Stationary processes 60G57: Random measures
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 3 , September 2000 , pp. 652 - 667
Copyright
Copyright © Applied Probability Trust 2000 

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