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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].

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