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Fluid models for single buffer systems. (English) Zbl 0871.60079
Dshalalow, Jewgeni H. (ed.), Frontiers in queueing: models and applications in science and engineering. Boca Raton, FL: CRC Press. Probability and Stochastics Series. 321-338 (1997).
Summary: This chapter considers a stochastic fluid model of a buffer content process \(\{X(t),\) \(t\geq 0\}\) that depends upon an external environment process \(\{Z(t),\) \(t\geq 0\}\) as follows: whenever the environment is in state \(z\) the \(X\) process changes state at rate \(\eta(z)\). The \(X\) process is restricted to stay in \([0,B)\), where \(B\leq \infty\). The aim is to study the steady state distribution of the bivariate process \(\{(X(t),Z(t))\), \(t\geq 0\}\). Three main cases are considered: the environment is (i) a continuous time Markov chain (CTMC), (ii) a CTMC and white noise, and (iii) an Ornstein-Uhlenbeck process. Spectral representations are obtained for the steady state distributions. Finally, an extension to state dependent drift is considered, where the rate of change of the \(X\) process depends on both the \(X\) and \(Z\) processes. The chapter ends with some interesting open problems in this area.
For the entire collection see [Zbl 0857.00015].

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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