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Journal of Applied Mathematics and Stochastic Analysis
Volume 7 (1994), Issue 3, Pages 269-299
http://dx.doi.org/10.1155/S1048953394000262

Busy period analysis, rare events and transient behavior in fluid flow models

Aalborg University, Institute of Electronic Systems, Fr. Bajersv. 7, Aalborg DK-9220, Denmark

Received 1 May 1993; Revised 1 January 1994

Copyright © 1994 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a process {(Jt,Vt)}t0 on E×[0,), such that {Jt} is a Markov process with finite state space E, and {Vt} has a linear drift ri on intervals where Jt=i and reflection at 0. Such a process arises as a fluid flow model of current interest in telecommunications engineering for the purpose of modeling ATM technology. We compute the mean of the busy period and related first passage times, show that the probability of buffer overflow within a busy cycle is approximately exponential, and give conditioned limit theorems for the busy cycle with implications for quick simulation. Further, various inequalities and approximations for transient behavior are given. Also explicit expressions for the Laplace transform of the busy period are found. Mathematically, the key tool is first passage probabilities and exponential change of measure for Markov additive processes.