# zbMATH — the first resource for mathematics

Second-order fluid flow models: Reflected Brownian motion in a random environment. (English) Zbl 0821.60087
Summary: This paper considers a stochastic fluid model of a buffer content process $$\{X(t), t \geq 0\}$$ that depends on a finite-state, continuous-time Markov process $$\{Z(t), t \geq 0\}$$ as follows: During the time-intervals when $$Z(t)$$ is in state $$i$$, $$X(t)$$ is a Brownian motion with drift $$\mu_ i$$, variance parameter $$\sigma^ 2_ i$$ and a reflecting boundary at zero. This paper studies the steady-state analysis of the bivariate process $$\{(X(t), Z(t)), t \geq 0\}$$ 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.

##### MSC:
 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Full Text: