This paper considers a stochastic fluid model of a buffer content process {X(t),t≥0} that depends upon an external environment process {Z(t),t≥0} as follows: whenever the environment is in state z the X process changes state at rate η(z). The X process is restricted to stay in [0,3], where B••. The aim is to study the steady state distribution of the bivariate process {(X(t),z(t)),t≥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.
