Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-30T21:40:30.231Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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