Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T08:27:27.938Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abate, J. and Whitt, W. (1987). Transient behavior of regulated Brownian motion, I and II. Adv. Appl. Prob. 19, 560–631.Google Scholar
Abate, J. and Whitt, W. (1992). The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10, 5–88.Google Scholar
Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Collings, T. and Stoneman, C. (1976). The M/M/∞ queue with varying arrival and departure rates. Operat. Res. 24, 760–773.Google Scholar
Green, L. and Kolesar, P. (1989). Testing the validity of a queueing model of police patrol. Management Sci. 35, 127–148.Google Scholar
Keller, J. B. (1978). Rays, waves and asymptotics. Bull. Amer. Math. Soc. 84, 727–750.Google Scholar
Keller, J. B. (1982). Time-dependent queues. SIAM Rev. 24, 401–412.Google Scholar
Kolesar, P. (1984). Stalking the endangered CAT: A queueing analysis of congestion at automatic teller machines. Interfaces 14, 16–26.Google Scholar
Koopman, B. O. (1972). Air-terminal queues under time-dependent conditions. Operat. Res. 20, 1089–1114.Google Scholar
Landauer, E. G. and Becker, L. C. (1989). Reducing waiting time at security checkpoints. Interfaces 19, 57–65.Google Scholar
Mandelbaum, A. and Massey, W. A. (1995). Strong approximations for time-dependent queues. Math. Oper. Res. 20, 33–64.Google Scholar
Newell, G. F. (1968). Queues with time-dependent arrival rates I: The transition through saturation. J. Appl. Prob. 5, 436–451.Google Scholar
Newell, G. F. (1968). Queues with time-dependent arrival rates II: The maximum queue and the return to equilibrium. J. Appl. Prob. 5, 579–590.Google Scholar
Newell, G. F. (1968). Queues with time-dependent arrival rates III: A mild rush hour. J. Appl. Prob. 5, 591–606.Google Scholar
Yang, Y. and Knessl, C. (1997). Asymptotic analysis of the M/G/1 queue with a time-dependent arrival rate. Queueing Systems: Theory and Applications 26, 23–68.Google Scholar