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On the dynamics and performance of stochastic fluid systems

Part of: Special processes Stochastic 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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References

Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory. Springer, Berlin.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Kella, O. (1997). Stochastic storage networks: stationarity and the feedforward case. J. Appl. Prob. 34, 498–507.Google Scholar
Kella, O. (1997). Personal communication.Google Scholar
Kella, O., and Whitt, W. (1996). Stability and structural properties of stochastic storage networks. J. Appl. Prob. 33, 1169–1180.CrossRefGoogle Scholar
Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press.Google Scholar
Konstantopoulos, T., Zazanis, M., and de Veciana, G. (1997). Conservation laws and reflection mappings with an application to multiclass mean value analysis for stochastic fluid queues. Stoch. Proc. Appl. 65, 139–146.Google Scholar
Matthes, K., Kerstan, J., and Mecke, J. (1977). Infinitely Divisible Point Processes. John Wiley, New York.Google Scholar
Miyazawa, M. (1994). Palm calculus for a process with a stationary random measure and its applications to fluid queues. Queueing Systems 17, 183–211.Google Scholar
Miyazawa, M. (1994). Rate conservation laws: a survey. Queueing Systems 15, 1–58.Google Scholar
Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, Berlin.Google Scholar
Takács, L. (1967). Combinatorial Methods in the Theory of Stochastic Processes. John Wiley, New York.Google Scholar