# Continuous feedback fluid queues

##
Werner Scheinhardt

### University of Twente, the Netherlands

### Résumé:

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
Markov process' 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
flow rates for the buffer may also depend continuously on the current
buffer level. Such models are interesting in the context of closed-loop
telecommunication networks, in which sources interact with network
buffers, but may also be deployed in the study of certain production
systems.

After defining the feedback behavior precisely, we deduce the Kolmogorov
forward equations for the joint background/buffer-process of interest
under some regularity assumptions. For the system with two states for
the background process we find an explicit solution for the stationary
distribution of the process. For the multiple-state case we present a
numerical method that enables us to find this distribution.