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Low density traffic streams

Published online by Cambridge University Press:  01 July 2016

Mark Brown
Mark Brown*
Affiliation:
Cornell University
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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 

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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