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

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

A spectral theory approach for extreme value analysis in a tandem of fluid queues

“…Asmussen and Bladt [3] propose a sample-path approach to study mean busy periods in Markov Modulated fluid queues, and derive a simple way of calculating mean busy periods in terms of steady-state quantities. In [1], Asmussen shows that the probability of buffer overflow within a busy cycle has an exponential tail, gives an explicit expression for the Laplace Transform of the busy period and, moreover, derives several inequalities and approximations for the transient behaviour. Boxma and Dumas [7] study the busy period of a fluid queue fed by N ON/OFF sources with exponential OFF periods and heavy tailed activity durations (more specifically, with regularly varying activity duration distributions).…”

“…Their analysis leads to matrix differential Ricatti equations for which there is a unique solution. Asmussen [1] investigates a more general setting than the one considered in this paper, which focuses on the streaming video setting. In our work we use an alternative matrix-theoretic analysis technique that is better suited for standard computational methods.…”

Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications

“…Recently, there has been some interest in deriving the Laplace transform of the busy period in fluid-flow models (Ahn and Ramaswami 2005;Bean et al 2005b); see also (Asmussen 1994) for an earlier contribution. It is our present aim to show how our general theory reproduces some of the most important busy-period results.…”

“…1. We remark that the killing technique is an alternative to other approaches that have been proposed for fluid-flow models (Ahn and Ramaswami 2005;Asmussen 1994;Bean et al 2005b). …”

Efficiency improvement techniques

“…A similar conclusion holds for Markov modulated queues, in which either the arrival epochs or the sum of the service times (or both) are Markov random walks. For additional details on importance sampling for random walks and queues, see Chang et al [4], Asmussen [1], Sadowsky [33], Glasserman and Kao [14], and Heidelberger [25]. For some discussion of how to compute the relevant likelihood ratios for general discrete-event simulations, see…”