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Equilibrium probability distributions for low density highway traffic

Published online by Cambridge University Press:  14 July 2016

G. F. Newell
G. F. Newell*
Affiliation:
University of California, Berkeley
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Abstract

If on a long homogeneous highway there is no interaction between cars, then, under a wide range of conditions, an initial distribution of cars will in the course of time tend toward that of a Poisson process with statistically independent velocities for the cars in any finite interval of highway. Here we will generalize this known property to obtain the following. Suppose cars do interact in such a way as to delay a car when it passes another, but the density of cars is so low that we can neglect simultaneous interactions between three or more cars. There will again be equilibrium distributions of cars to which general classes of initial distributions will converge. These equilibrium distributions are superpositions of two statistically independent processes, one a Poisson process of single free cars with statistically independent velocities, and the other a Poisson process of interacting pairs of cars with various velocities. In the limit of zero interaction, the density of pairs vanishes leaving only the Poisson process of single cars as a special case. To the same order of approximation, including the first order effects of interactions, the headway distribution between consecutive cars will still have exponential tail outside the range of interaction.

Type
Research Papers
Information
Journal of Applied Probability , Volume 3 , Issue 1 , June 1966 , pp. 247 - 260
Copyright
Copyright © Sheffield: Applied Probability Trust 

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References

[1] Weiss, G. and Herman, R. (1962) Statistical properties of low density traffic. Quart. Appl. Math. 20, 121–130.Google Scholar
[2] Breiman, Leo (1962) On some probability distributions occurring in traffic flow Bull Inst. Internat. Statist. 39, 155–161.Google Scholar
[3] Breiman, Leo (1963) The Poisson tendency in traffic distribution. Ann. Math. Statist. 34, 308–311.Google Scholar
[4] Rényi, A. (1964) On two mathematical models of the traffic on a divided highway. J. Appl. Prob. 1, 311–320.Google Scholar
[5] Thedeen, T. (1964) A note on the Poisson tendency in traffic distribution. Ann. Math. Statist. 35, 1823–1824.Google Scholar
[6] Riordan, J. (1951) Telephone traffic time averages. Bell System Tech. J. 30, 1129–1144.Google Scholar
[7] Beneş, V. E. (1957) Fluctuations of telephone traffic. Bell System Tech. J. 36, 965–973.Google Scholar
[8] Mirasol, N. M. (1963) The output of an M/G/8 queueing system is Poisson. Operations Res. 11, 282–284.CrossRefGoogle Scholar
[9] Newell, G. F. (1966) The M/G/8 queue. SIAM Journal 14, 86–88.Google Scholar
[10] Newell, G. F. (1955) Mathematical models of freely flowing highway traffic. Operations Res. 3, 176–186.Google Scholar
[11] Carleson, L. (1957) A mathematical model for highway traffic. Nordisk Mat. Tidskr. 5, 176–213.Google Scholar
[12] Bartlett, M. S. (1957) Some problems associated with random velocities. Publications de l'Institut de Statistique de l'Université de Paris 6, 261–270.Google Scholar
[13] Miller, A. J. (1961) Traffic flow treated as a stochastic process. Proc. symposium on the theory of traffic flow Detroit 1959, Herman, R., ed. Elsevier 165–174.Google Scholar
[14] Wardrop, J. G. (1952) Some theoretical aspects of road traffic research. Proc. Inst. Civ. Engrs. II 1 (2), 325–362.Google Scholar
[15] Miller, A. J. (1961) A queueing model for road traffic flow. J. Royal Statist. Soc. B 23, 63–75.Google Scholar
[16] Oliver, R. M. (1962) A note on a traffic counting distribution. Operat. Res. Quart. 13, 171–178.Google Scholar
[17] Miller, A. J. (1963) Analysis of bunching in rural two-lane traffic Operations. Res. 11, 236–247.Google Scholar
[18] Gaver, D. P. Jr. (1963) Accommodation of second-class traffic. Operations Res. 11, 72–87.CrossRefGoogle Scholar
[19] Buckley, D. J. Road traffic headway distributions. Proc. first conference Australian Road Research Board 1962, 153–187.Google Scholar
[20] Oliver, R. M. and Thibault, B. (1962) A high-flow traffic-counting distribution. Highway Res. Bd. Bull. 356, 15–27.Google Scholar
[21] May, A. D. Jr. (1965) Gap availability studies. Highway Res. Record 72, 101–136.Google Scholar