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Heavy traffic analysis of a transportation network model

Part of: Operations research and management science Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

William P. Peterson  and
Lawrence M. Wein
William P. Peterson*
Affiliation:
Middlebury College
Lawrence M. Wein*
Affiliation:
Massachusetts Institute of Technology
*
∗Postal address: Department of Mathematics and Computer Science, Middlebury College, Middlebury, Vermont 05753, USA.
∗∗Postal address: Sloan School of Management, Massachusetts Institute of Technology, 50 Memorial Drive, Cambridge, MA 02142–1347, USA.
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Abstract

We study a model of a stochastic transportation system introduced by Crane. By adapting constructions of multidimensional reflected Brownian motion (RBM) that have since been developed for feedforward queueing networks, we generalize Crane's original functional central limit theorem results to a full vector setting, giving an explicit development for the case in which all terminals in the model experience heavy traffic conditions. We investigate product form conditions for the stationary distribution of our resulting RBM limit, and contrast our results for transportation networks with those for traditional queueing network models.

Keywords

TRANSPORTATION NETWORKSQUEUEINGHEAVY TRAFFICREFLECTED BROWNIAN MOTION

MSC classification

Primary: 90B22: Queues and service
Secondary: 60K25: Queueing theory 60F17: Functional limit theorems; invariance principles
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 3 , September 1996 , pp. 870 - 885
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

Research supported by Middlebury College under the Faculty Leave Program.

Research supported by National Science Foundation grant DDM-9057297.

References

Crane, M. A. (1971) Limit theorems for queues in transportation systems. Ph.D. dissertation. Department of Operations Research, Stanford University.Google Scholar
Crane, M. A. (1974) Queues in transportation systems II: an independently dispatched system. J Appl. Prob. 11, 145–158.Google Scholar
Dai, J. G. and Harrison, J. M. (1992) Reflected Brownian motion in an orthant: numerical methods for steady-state analysis. Ann. Appl. Prob. 2, 65–86.Google Scholar
Harrison, J. M. (1985) Brownian Motion and Stochastic Flow Systems. Wiley, New York.Google Scholar
Harrison, J. M. and Nguyen, V. (1993) Brownian models of multiclass queueing networks: current status and open problems. Queueing Systems Theory Appl. 13, 5–10.Google Scholar
Harrison, J. M. and Williams, R. J. (1987) Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Prob. 15, 115–137.Google Scholar
Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic, I and II. Adv. Appl. Prob. 2, 150–157, 355–364.Google Scholar
Peterson, W. P. (1991) A heavy traffic limit theorem for networks of queues with multiple customer types. Math. Operat. Res. 16, 90–118.Google Scholar
Reiman, M. I. (1984) Open queueing networks in heavy traffic. Math. Operat. Res. 9, 441–458.CrossRefGoogle Scholar