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Fluid Model Driven by an Ornstein-Uhlenbeck Process

Published online by Cambridge University Press:  27 July 2009

Vidyadhar Kulkarni  and
Tomasz Rolski
Vidyadhar Kulkarni
Affiliation:
Department of Operations Research, CB# 3180, Smith Building, University of North Carolina, Chapel Hill, North Carolina 27599–3180
Tomasz Rolski
Affiliation:
Mathematical Institute, The University of Wroclaw, pl. Grunwaldzki 2/4 50 384 Wroclaw, Poland
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Abstract

In this paper we consider a fluid model of a single buffer in a random external environment modeled by an Ornstein-Uhlenbeck process. Such a model appears as the limiting case of the multiplexing model of Anick, Mitra, and Sondhi [2] in a heavy traffic environment. We use the change of measure technique to derive an exponential upper bound on the tail probabilities of the steady-state buffer content. We also establish an asymptotic upper and lower exponential bounds on the tail probabilities.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 8 , Issue 3 , July 1994 , pp. 403 - 417
Copyright
Copyright © Cambridge University Press 1994

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References

1.Andrews, L.C. (1985). Special functions for engineers and applied mathematicians. New York: Macmillan.Google Scholar
2.Anick, D., Mitra, D., & Sondhi, M.M. (1982). Stochastic theory of a data-handling system with multiple sources. Bell System Technical journal 61: 1871–1894.CrossRefGoogle Scholar
3.Asmussen, S. (1993). Fundamentals of ruin probability theory. Book manuscript.Google Scholar
4.Borovkov, A.A. (1976). Stochastic processes in queueing theory. New York: Springer.CrossRefGoogle Scholar
5.Borovkov, A.A. (1984). Asymptotic methods in queueing theory. Chichester: Wiley.Google Scholar
6.Harrison, J.M. (1985). Brownian motion and stochastic flow systems. New York: Wiley.Google Scholar
7.Iglehart, D.L. (1965). Limit diffusion approximations for the many server queue and the repairman problem. Journal of Applied Probability 2: 429–441.CrossRefGoogle Scholar
8.Karlin, S. & Taylor, H. (1981). A second course in stochastic processes. New York: Academic Press.Google Scholar
9.Knessl, C. & Morrison, J.A. (1991). Heavy traffic analysis of a data-handling system with many sources. SIAM Journal of Applied Mathematics 51: 187–213.CrossRefGoogle Scholar
10.Ott, T. & Daley, D.J. (1993). On a fluid model for a packet switch. Manuscript.Google Scholar
11.Parthasarathy, K.R.introduction to probability and measure. New York: Springer-Verlag.Google Scholar
12.Petrovski, I.G. (1954). Lectures on partial differential equations. New York: Intersciences Publisher.Google Scholar
13.Simonian, A. (1991). Stationary analysis of a fluid queue with input rate varying as an Ornstein-Uhlenbeck process. SIAM Journal of Applied Mathematics 51: 823–842.Google Scholar
14.Simonian, A. & Virtamo, J. (1991). Transient and stationary distributions for fluid queues and input processes with a density. SIAM Journal of Applied Mathematics 51: 1731–1739.CrossRefGoogle Scholar
15.Stroock, D.W. (1987). Lecture on stochastic analysis: Diffusion theory. London Mathematical Society Student Texts 6, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
16.Stroock, D.W. & Varadhan, S.R.S. (1979). Multidimensional diffusion processes. Berlin: Springer.Google Scholar
17.Wentzell, A.D. (1981). A course in the theory of stochastic processes. New York: McGraw-Hill.Google Scholar