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Transient behavior of regulated Brownian motion, II: Non-zero initial conditions

Published online by Cambridge University Press:  01 July 2016

Joseph Abate  and
Ward Whitt
Joseph Abate*
Affiliation:
AT & T Bell Laboratories
Ward Whitt*
Affiliation:
AT & T Bell Laboratories
*
∗Postal address: AT & T Bell Laboratories, LC2W-E06, 184 Liberty Corner Road, Warren, NJ 07060, USA.
∗∗ Postal address: AT & T Bell Laboratories, MH2C-178, Murray Hill, NJ 07974, USA.
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Abstract

This paper continues an investigation of the time-dependent behavior of regulated or reflecting Brownian motion (RBM). Part I focused on RBM starting at the origin; Part II focuses on RBM starting at a fixed positive state. The first two moments of RBM as functions of time are analyzed by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied in Part I. By appropriate normalization, the two monotone components can be converted into cumulative distribution functions that can be analyzed probabilistically, e.g., their moments can be calculated. Simple approximations are then developed by fitting convenient distributions to these moments. Overall, the analysis yields a better understanding of the way RBM and related stochastic flow systems approach steady state.

Keywords

TIME-DEPENDENT BEHAVIORRELAXATION TIMESDIFFUSION PROCESSESFIRST-PASSAGE TIMEINVERSE GAUSSIAN DISTRIBUTIONCOUPLINGFITTING DISTRIBUTIONS BY MATCHING MOMENTS
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 3 , September 1987 , pp. 599 - 631
Copyright
Copyright © Applied Probability Trust 1987 

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