Abstract
We consider a process 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.
Highlights
Fluid flow processes can be seen as a class of applied probability models which in many ways is parallel to queues
The use of Markov-modulation for modeling bursty traffic has led into more general Markov additive processes which are the key tool we use for studying fluid flow nodels, by representing them as reflected versions of finite Markov additive processes with the additive component having the simplest possible structure of a p,re linear drift
The purpose of this paper is to present a study of this aspect; in particular, we study the behavior within a busy cycle and bounds and approximations for the time-dependent state probabilities
Summary
Fluid flow processes can be seen as a class of applied probability models which in many ways is parallel to queues. Frown an application point of view, the historical origin is in both cases performance evaluation in telecommunication, with the difference being motivated in the change of technology: from switchboards in the days of Erlang to modern ATM (asynchronous transfer mode) devices Both class of models have fundamental relations to random walks and more general additive processes. Some basic matrices occurring in the steady-state solution are introduced; they are of basic importance in the present paper as well, since the computational evaluation of the busy period/transient behavior results turns out to require either just these matrices, or matrices of just the same form but defined via duality in terms of time reversion, sign reversion or change of parameters. The above definition of a busy period becomes trivial (Pi 0), so that instead one has to start the busy period at x > 0
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