Second-Order Fluid Flow Models: Reflected Brownian Motion in a Random Environment

This paper considers a stochastic fluid model of a buffer content process {X(t), t ≥ 0} that depends on a finite-state, continuous-time Markov process {Z(t), t ≥ 0} as follows: During the time-intervals when Z(t) is in state i, X(t) is a Brownian motion with drift μ_i, variance parameter σ_i²and a reflecting boundary at zero. This paper studies the steady-state analysis of the bivariate process {(X(t), Z(t)), t ≥ 0} in terms of the eigenvalues and eigenvectors of a nonlinear matrix system. Algorithms are developed to compute the steady-state distributions as well as moments. Numerical work is reported to show that the variance parameter has a dramatic effect on the buffer content process.