Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T22:08:21.186Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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 

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.)

Footnotes

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

References

[1] Billingsley, P. (1968) Convergence of Probability Measures. John Wiley and Sons, New York.Google Scholar
[2] Chung, K. (1960) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, Berlin, Gottingen, Heidelberg.10.1007/978-3-642-49686-8Google Scholar
[3] Chung, K. (1968) A Course in Probability Theory. Harcourt, Brace and World, New York.Google Scholar
[4] Crane, M. (1971) Limit theorems for queues in transportation systems. Technical Report No. 16, Department of Operations Research, Stanford University, Stanford, California, (Ph.D. Dissertation).Google Scholar
[5] Debbie, J. (1961) A double-ended queueing problem of Kendall. Operations Res. 9, 755–757.10.1287/opre.9.5.755Google Scholar
[6] Harrison, J. M. (1970) Queueing models for assembly-like systems. Technical Report No. 20, Department of Operations Research, Stanford University, Stanford, California (Ph.D. dissertation).Google Scholar
[7] Iglehart, D. and Whitt, W. (1970a) Multiple channel queues in heavy traffic. I. Adv. Appl. Prob. 2, 150–177.Google Scholar
[8] Iglehart, D. and Whitt, W. (1970b) Multiple channel queues in heavy traffic. II: sequences, networks, and batches. Adv. Appl. Prob. 2, 355–369.10.2307/1426324Google Scholar
[9] Kashyap, B. (1965) A double-ended queueing system with limited waiting space. Proc. Nat. Inst. Sci. India A 31, 559–570.Google Scholar
[10] Kashyap, B. (1966) The double-ended queue with bulk service and limited waiting space. Operations Res. 14, 822–834.10.1287/opre.14.5.822Google Scholar
[11] Kemeny, J. and Snell, J. (1960) Finite Markov Chains. Van Nostrand, Princeton.Google Scholar
[12] Kendall, D. (1951) Some problems in the theory of queues. J. R. Statist. Soc. B 13, 151–185.Google Scholar
[13] Sasieni, M. (1961) Double queues and impatient customers with an application to inventory theory. Operations Res. 9, 771–781.10.1287/opre.9.6.771Google Scholar