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Queues in transportation systems, I: a Markovian system

Published online by Cambridge University Press:  14 July 2016

Michael A. Crane
Michael A. Crane*
Affiliation:
Control Analysis Corporation, Palo Alto, California
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Abstract

We study a transportation system consisting of S vehicles of unit capacity and N passenger terminals. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to a terminal j, 1 ≦ j ≦ N, with probability Pij. Customers at each terminal are served as vehicles become available. Each vehicle is dispatched from a terminal when loaded, whereupon it travels to the destination of its passenger, according to a stochastic travel time. It is shown under mild conditions that the system is unstable, due to random fluctuations of independent customer arrival processes. We obtain limit theorems, in certain special cases, for the customer queue size processes. Where a steady-state limit exists, this limit is expressed in terms of the corresponding limit in a related GI/G/S queue. In other cases, functional central limit theorems are obtained for appropriately normalized random functions.

Keywords

TRANSPORTATION QUEUESMARKOV CHAINSASSEMBLY-LIKE QUEUESDOUBLE-ENDED QUEUESTAXI-STAND QUEUESWEAK CONVERGENCEUNSTABLE QUEUESHEAVY TRAFFICFUNCTIONAL CENTRAL LIMIT THEOREMSNORMAL APPROXIMATIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 3 , September 1973 , pp. 630 - 643
Copyright
Copyright © Applied Probability Trust 1973 

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Footnotes

Research partially supported by Office of Naval Research Contract N00014-67-A-0112-0031 and N.S.F. Grant GP-20223.

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