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Exact and asymptotic solutions to a PDE that arises in time-dependent queues

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Charles Knessl
Charles Knessl*
Affiliation:
University of Illinois
*
∗ Postal address: Department of Mathematics, Statistics and Computer Science (M/C 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7045, USA.
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Abstract

We consider a diffusing particle in one dimension that is subject to a time-dependent drift or potential field. A reflecting barrier constrains the particle's position to the half-line X ≥ 0. Such models arise naturally in the study of queues with time-dependent arrival rates, as well as in advection-diffusion problems of mathematical physics. We solve for the probability distribution of the particle as a function of space and time. Then we do a detailed study of the asymptotic properties of the solution, for various ranges of space and time. We also relate our asymptotic results to those obtained by probabilistic approaches, such as central limit theorems and large deviations. We consider drifts that are either piecewise constant or linear functions of time.

Keywords

Diffusion processesQueueing theory

MSC classification

Primary: 60J60: Diffusion processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 256 - 283
Copyright
Copyright © Applied Probability Trust 2000 

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