Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T00:58:36.690Z Has data issue: false hasContentIssue false
  • English
  • Français

Queues in transportation systems, II: An independently-dispatched 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 consider a transportation system consisting of a linear network of N + 1 terminals served by S vehicles of fixed capacity. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to some terminal j, 0 ≦ j ≦ i − 1, and are served as empty units of vehicle capacity become available at i. The vehicle fleet is partitioned into N service groups, with vehicles in the ith group stopping at terminals i, i − 1,···,0. Travel times between terminals and idle times at terminals are stochastic and are independent of the customer arrival processes. Functional central limit theorems are proved for random functions induced by processes of interest, including customer queue size processes. The results are of most interest in cases where the system is unstable. This occurs whenever, at some terminal, the rate of customer arrivals is at least as great as the rate at which vehicle capacity is made available.

Keywords

TRANSPORTATION QUEUESTRANSPORTATION NETWORKNETWORK QUEUESAUTOMATED HIGHWAYWEAK CONVERGENCEUNSTABLE QUEUESHEAVY TRAFFICFUNCTIONAL CENTRAL LIMIT THEOREMSNORMAL APPROXIMATIONSPRIORITY QUEUES
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 1 , March 1974 , pp. 145 - 158
Copyright
Copyright © Applied Probability Trust 1974 

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

[1] Bailey, N. (1954) On queueing processes with bulk service J. R. Statist. Soc. B 16, 80–87.Google Scholar
[2] Billingsley, P. (1968) Convergence of Probability Measures. John Wiley, New York.Google Scholar
[3] Crane, M. (1971) Limit theorems for queues in transportation systems. Technical Report No. 16, Department of Operations Research, Stanford University, California, (Ph. D. dissertation).Google Scholar
[4] Crane, M. (1973) Queues in transportation systems, I: a Markovian system. J. Appl. Prob. 10, 630–643.Google Scholar
[5] Iglehart, D. (1968) Weak convergence of probability measures on product spaces with applications to sums of random vectors. Technical Report No. 109, Department of Operations Research, Stanford University, California.Google Scholar
[6] Iglehart, D. and Whitt, W. (1970a) Multiple channel queues in heavy traffic, I. Adv. Appl. Prob. 2, 150–177.CrossRefGoogle Scholar
[7] Iglehart, D. and Whitt, W. (1970b) Multiple channel queues in heavy traffic, II: sequences, networks, and batches. Adv. Appl. Prob. 2, 355–369.CrossRefGoogle Scholar
[8] Whitt, W. (1968) Weak convergence theorems for queues in heavy traffic. Technical Report No. 2, Department of Operations Research, Stanford University, California, (Ph. D. Dissertation, Cornell University).Google Scholar