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On the dynamics and performance of stochastic fluid systems

  • Takis Konstantopoulos (a1) and Günter Last (a2)


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.


Corresponding author

Postal address: Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78712, USA. Email address:
∗∗ Postal address: Institut für Mathematische Stochastik, Universität Karlsruhe, Englerstraße 2, 76128 Karlsruhe, Germany. Email address:


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[1] Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory. Springer, Berlin.
[2] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.
[3] Kella, O. (1997). Stochastic storage networks: stationarity and the feedforward case. J. Appl. Prob. 34, 498507.
[4] Kella, O. (1997). Personal communication.
[5] Kella, O., and Whitt, W. (1996). Stability and structural properties of stochastic storage networks. J. Appl. Prob. 33, 11691180.
[6] Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press.
[7] 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, 139146.
[8] Matthes, K., Kerstan, J., and Mecke, J. (1977). Infinitely Divisible Point Processes. John Wiley, New York.
[9] Miyazawa, M. (1994). Palm calculus for a process with a stationary random measure and its applications to fluid queues. Queueing Systems 17, 183211.
[10] Miyazawa, M. (1994). Rate conservation laws: a survey. Queueing Systems 15, 158.
[11] Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, Berlin.
[12] Takács, L. (1967). Combinatorial Methods in the Theory of Stochastic Processes. John Wiley, New York.


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