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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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References

Abate, J. and Whitt, W. (1987b) Transient behavior of the M/M/1 queue: starting at the origin. Queueing Systems 2.CrossRefGoogle Scholar
Abate, J. and Whitt, W. (1988) Transient behavior of the M/M/1 queue via Laplace transforms. Adv. Appl. Prob. 20(1).Google Scholar
Abate, J. and Whitt, W. (1987) Transient behavior of regulated Brownian motion, I: starting at the origin. Adv. Appl. Prob. 19, 560–598.Google Scholar
Apostol, T. M. (1957) Mathematical Analysis. Addison-Wesley, Reading, Mass. Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, Vol. II, 2 edn. Wiley, New York.Google Scholar
Gafarían, A. V., Ancker, C. J. Jr. and Morisaku, T. (1978) Evaluation of commonly used rules for detecting “steady state” in computer simulations. Naval. Res. Log. Quart. 25, 511–529.CrossRefGoogle Scholar
Gaver, D. P. Jr. (1968) Diffusion approximations and models for certain congestion problems. J. Appl. Prob. 5, 607–623.Google Scholar
Harrison, J. M. (1985) Brownian Motion and Stochastic Flow Systems Wiley, New York.Google Scholar
Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic II: sequences, networks and batches. Adv. Appl. Prob. 2, 355–369.Google Scholar
Kamae, T., Krengel, U. and O&Brien, G. L. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
Keilson, J. (1979) Markov Chain Models-Rarity and Exponentiality Springer-Verlag, New York.CrossRefGoogle Scholar
Kelton, W. D. (1985) Transient exponential-Erlang queues and steady-state simulation. Commun. Assoc. Comput. Mach. 28, 741–749.Google Scholar
Kelton, W. D. and Law, A. M. (1985) The transient behavior of the M/M/s queue, with implications for steady-state simulation. Operat. Res. 33, 378–396.Google Scholar
Lindvall, T. (1983) On coupling of diffusion processes. J. Appl. Prob. 20, 82–93.Google Scholar
Prabhu, N. U. (1980) Stochastic Storage Processes Springer-Verlag, New York.CrossRefGoogle Scholar
Sonderman, D. (1980) Comparing semi-Markov processes. Math. Operat. Res. 5, 110–119.Google Scholar
Stone, C. (1963) Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7, 638–660.CrossRefGoogle Scholar
Van Doorn, E. (1980) Stochastic Monotonicity and Queueing Applications of Birth-Death Processes. Lecture Notes in Statistics 4, Springer-Verlag, New York.Google Scholar
Whitt, W. (1982) Approximating a point process by a renewal process, I: two basic methods. Operat. Res. 30, 125–147.CrossRefGoogle Scholar
Whitt, W. (1985) The renewal-process stationary-excess operator. J. Appl. Prob. 22, 156–167.Google Scholar