Continuous feedback fluid queues

Werner Scheinhardt

University of Twente, the Netherlands


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.

[Werner Scheinhardt]
[University of Twente, the Netherlands]