Transient behavior of regulated Brownian motion, I: Starting at the origin

Abstract

A natural model for stochastic flow systems is regulated or reflecting Brownian motion (RBM), which is Brownian motion on the positive real line with constant negative drift and constant diffusion coefficient, modified by an impenetrable reflecting barrier at the origin. As a basis for understanding how stochastic flow systems approach steady state, this paper provides relatively simple descriptions of the moments of RBM as functions of time. In Part I attention is restricted to the case in which RBM starts at the origin; then the moment functions are increasing. After normalization by the steady-state limits, these moment c.d.f.&s (cumulative distribution functions) coincide with gamma mixtures of inverse Gaussian c.d.f.&s. The first moment c.d.f. thus coincides with the first-passage time to the origin starting in steady state with the exponential stationary distribution. From this probabilistic characterization, it follows that the kth-moment c.d.f is the k-fold convolution of the first-moment c.d.f. As a consequence, it is easy to see that the (k + 1)th moment approaches its steady-state limit more slowly than the kth moment. It is also easy to derive the asymptotic behavior as t →∞. The first two moment c.d.f.&s have completely monotone densities, supporting approximation by hyperexponential (H2) c.d.f.&s (mixtures of two exponentials). The H2 approximations provide easily comprehensible descriptions of the first two moment c.d.f.&s suitable for practical purposes. The two exponential components of the H2 approximation yield simple exponential approximations in different regimes. On the other hand, numerical comparisons show that the limit related to the relaxation time does not predict the approach to steady state especially well in regions of primary interest. In Part II (Abate and Whitt (1987a)), moments of RBM with non-zero initial conditions are treated by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied here.