Journal of Applied Mathematics and Stochastic Analysis
Volume 7 (1994), Issue 3, Pages 269-299
doi: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 Søren Asmussen. 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)}t≥0 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.