A fluid model with upward jumps at the boundary

Abstract

We consider a single buffer fluid system in which the instantaneous rate of change of the fluid is determined by the current state of a background stochastic process called “environment”. When the fluid level hits zero, it instantaneously jumps to a predetermined positive level q. At the jump epoch the environment state can undergo an instantaneous transition. Between two consecutive jumps of the fluid level the environment process behaves like a continuous time Markov chain (CTMC) with finite state space. We develop methods to compute the limiting distribution of the bivariate process (buffer level, environment state). We also study a special case where the environment state does not change when the fluid level jumps. In this case we present a stochastic decomposition property which says that in steady state the buffer content is the sum of two independent random variables: one is uniform over [0,q], and the other is the steady-state buffer content in a standard fluid model without jumps.