Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T19:55:07.119Z Has data issue: false hasContentIssue false
  • English
  • Français

Low density traffic streams

Published online by Cambridge University Press:  01 July 2016

Mark Brown
Mark Brown*
Affiliation:
Cornell University
Get access Rights & Permissions [Opens in a new window]

Abstract

Low density traffic refers to the study of macroscopic properties of a traffic stream when vehicles travel independently of one another. It is usually assumed that each vehicle travels at a constant velocity, the velocity varying from vehicle to vehicle. We allow very general vehicular motions and study various aspects of the traffic streams. For example, it is shown that if π [a,b] is the expected time for a vehicle to travel from a to b under the stochastic process governing the motion of vehicles, then a non-homogeneous Poisson spatial process with mean measure π is invariant.

Keywords

LOW DENSITY TRAFFICMACROSCOPIC EQUILIBRIUMPOISSON PROCESSESINVARIANT POISSON PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 4 , Issue 1 , April 1972 , pp. 177 - 192
Copyright
Copyright © Applied Probability Trust 1972 

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. (1963) The Poisson tendency in traffic distribution. Ann. Math. Statist. 34, 308–311.CrossRefGoogle Scholar
[2] Brill, E. A. (1969) Point processes and tandem queues; their application to traffic flow. Stanford University Technical Report.Google Scholar
[3] Brown, M. (1969) An invariance property of Poisson processes. J. Appl. Prob. 6, 453–458.CrossRefGoogle Scholar
[4] Brown, M. (1969) Some results on a traffic model of Rényi. J. Appl. Prob. 6, 293–300.CrossRefGoogle Scholar
[5] Brown, M. (1970) A property of Poisson processes and its application to macroscopic equilibrium of particle systems. Ann. Math. Statist. 41, 1935–1941.CrossRefGoogle Scholar
[6] Doob, J. L. (1953) Stochastic Processes. John Wiley, New York.Google Scholar
[7] Feller, W. (1966) An Introduction to Probability Theory and its Applications, Volume II. John Wiley, New York.Google Scholar
[8] Goldman, J. R. (1968) Stochastic point processes: Limit theorems. Ann. Math. Statist. 39, 771–779.Google Scholar
[9] Haight, F. A. (1963) Mathematical Theories of Traffic Flow. Academic Press, New York and London.Google Scholar
[10] Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York and London.Google Scholar
[11] Loève, M. (1963) Probability Theory. D. Van Nostrand, Princeton, N. J.Google Scholar
[12] Newell, G. F. (1966) Equilibrium probability distributions for low density highway traffic. J. Appl. Prob. 3, 247–260.CrossRefGoogle Scholar
[13] Rényi, A. (1964) On two mathematical models of the traffic on a divided highway. J. Appl. Prob. 1, 311–320.CrossRefGoogle Scholar
[14] Royden, H. L. (1963) Real Analysis. The Macmillan Co., New York.Google Scholar
[15] Spitzer, F. (1969) Random processes defined through the interaction of an infinite particle system. Springer Lecture Notes, 89, 201–223.CrossRefGoogle Scholar
[16] Spitzer, F. (1969) Uniform motion with elastic collision of an infinite particle system. J. Math. Mech. 18, 973–990.Google Scholar
[17] Stone, C. (1968) On a theorem of Dobrushnin. Ann. Math. Statist. 39, 1391–1402.CrossRefGoogle Scholar
[18] Stravastava, R. S. (1969) A note on a mathematical model of traffic flow on a divided highway. Transportation Research 3, 135–143.CrossRefGoogle Scholar
[19] Suzuki, T. (1967) A filtered Poisson process on road traffic flow. Mem. Defence Acad. VI, 501–510.Google Scholar
[20] Thedéen, T. (1964) A note on the Poisson tendency in traffic distribution. Ann. Math. Statist. 35, 1823–1824.CrossRefGoogle Scholar
[21] Thedéen, T. (1969) On road traffic with the free overtaking. J. Appl. Prob. 3, 524–549.CrossRefGoogle Scholar
[22] Weiss, G. and Herman, R. (1962) Statistical properties of low density traffic. Quart. Appl. Math. XX, No. 2.Google Scholar