Continuous feedback fluid queues.

ABSTRACT
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.
2000 Mathematics Subject Classification: 60K25
Keywords and Phrases: Fluid queue, feedback, forward equations