Continuous feedback fluid queues

We investigate a fluid buffer which is modulated by a stochastic background process, while the momentary behavior of the background process depends on the current buffer level in a continuous way. Loosely speaking the feedback is such that the background process behaves `as a continuous-time Markov chain' with generator Q(y) at times when the buffer level is y, where the entries of Q(y) are continuous functions of y. Moreover, the fluid-flow rates for the buffer may also depend continuously on the current buffer level. First we define the feedback behavior precisely. Then we deduce the Kolmogorov forward equations for the joint background/buffer-process under some regularity assumptions. After presenting the differential equations and boundary conditions for the stationary distributions, we find an explicit solution when the background process has two states