Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-09T06:55:08.177Z Has data issue: false hasContentIssue false
  • English
  • Français

Point processes arising in vehicular traffic flow

Published online by Cambridge University Press:  14 July 2016

Edward A. Brill
Edward A. Brill*
Affiliation:
Naval Postgraduate School, Monterey, California
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we investigate the properties of stationary point processes motivated by the following traffic model. Suppose there is a dichotomy of slow and fast points (cars) on a road with limited overtaking. It is assumed that fast points are delayed behind (or are clustered at) a slow point in accordance with the principles of a GI/G/s queue, the order of service being irrelevant. Thus each slow point represents a service station, with the input into each station consisting of a fixed (but random) displacement of the output of the previous queueing station. It is found that tractable results for stationary point processes occur for the cases M/M/s (s = 1, 2, ···, ∞) and M/G/∞. In particular, it is found that for these cases the steady state point processes are compound Poisson and that for the M/M/1 case the successive headways form a two state Markov renewal process. In addition it is shown that the input, output, and queue size processes in a steady state M/G/∞ queue are independent at any fixed time; this is a result I have been unable to find in the literature.

Type
Short Communications
Information
Journal of Applied Probability , Volume 8 , Issue 4 , December 1971 , pp. 809 - 814
Copyright
Copyright © Applied Probability Trust 1971 

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] Breiman, L. (1968) Point and trajectory processes in one-way traffic flow. System Development Corporation Document TM-3858/006/00.Google Scholar
[2] Brill, E. A. (1971) A model of a traffic jam behind a bottleneck. Submitted for publication.Google Scholar
[3] Brown, M. (1969) Some results on a traffic model of Rényi. J. Appl. Prob. 6, 293–301.Google Scholar
[4] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[5] Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York and London.Google Scholar
[6] Miller, A. J. (1961) A queueing model for road traffic flow, J. R. Statist. Soc. B 23, 64–75.Google Scholar
[7] Newell, G. F. (1966) Equilibrium probability distributions for low density highway traffic. J. Appl. Prob. 3, 247–260.Google Scholar
[8] Reich, E. (1957) Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768–773.Google Scholar
[9] Renyi, A. (1964) On two mathematical models of the traffic on a divided highway. J. Appl. Prob. 1, 311–320.Google Scholar
[10] Ross, S. (1969) Applied Probability Models with Optimization Applications. Holden-Day, San Francisco.Google Scholar
[11] Tanner, J. C. (1961) Delays on a two-lane road. J. R. Statist. Soc. B 23, 38–63.Google Scholar