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The vectorial representation of the frequency of encounters of freely flowing vehicles

Published online by Cambridge University Press:  14 July 2016

F. Garwood
F. Garwood*
Affiliation:
Transport and Road Research Laboratory, Crowthorne, Berkshire
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Abstract

It is known that if q vehicles at speed v pass a fixed point per unit time, randdomly and independently, and mix with another set q', v', then the expected number of overtakings per unit length of road per unit time is qq' (v' – v)/vv'. If k, k' are the concentrations, i.e., number per unit length of road, this rate can also be expressed as kk' (v'– v) and as q'k – qk'. It is also the expected number of intersections per unit area on the space-time diagram, where the vehicle paths are represented by sets of random parallel lines. If each set is represented by a vector with magnitude equal to the density (i.e., inverse of the spacing), the rate is equal to the magnitude of the vector product. This is extended to the general case of a stream of vehicles with distributed speeds. This has an equivalent vector, whose components are the total concentration (in the direction of the time axis) and total flow (distance direction). This leads to the concept of the cumulative vector curve in problems of geometrical probability. The extension to the case of variable flows and speeds is indicated. The vector representing the distribution of speeds is then variable, but satisfies a condition of continuity which makes its divergence vanish.

Keywords

GEOMETRICAL PROBABILITYRANDOM LINES IN A PLANEDENSITY OF INTERSECTIONS OF RANDOM LINESREPRESENTATIONS OF DISTRIBUTIONS BY VECTORSAPPLICATIONS TO VEHICLE OVERTAKINGS ON ROADSVEHICLE FLOW AND CONCENTRATIONTRAFFIC ENGINEERING
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 4 , December 1974 , pp. 797 - 808
Copyright
Copyright © Applied Probability Trust 1974 

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References

Ashton, W. D. (1966) The Theory of Road Traffic Flow. Methuen, London.Google Scholar
Bartlett, M. S. (1967) The spectral analysis of line processes. 5th Berkeley Symposium Math. Statist. Prob. Vol. III, 135–153.Google Scholar
Carleson, L. (1957) En mathematisk modell för landsvägstrafik, Nord. Mat. Tidskrift 5, 176–180.Google Scholar
Haight, F. A. (1963) Mathematical Theories of Traffic Flow. Academic Press, London and New York.Google Scholar
Kendall, M. G. and Moran, P. A. P. (1962) Geometrical Probability. Griffin, London.Google Scholar
Lighthill, M. J. and Whitham, G. B. (1955) On kinematic waves II. A theory of traffic on long crowded roads. Proc. Roy. Soc. A 229, 317–345.Google Scholar
Mardia, K. V. (1972) Statistics of Directional Data. Academic Press, London and New York.Google Scholar
Miles, R. E. (1969) Poisson flats in Euclidean spaces. Part I: A finite number of random uniform flats. Adv. Appl. Prob. 1, 211–237.Google Scholar
Miles, R. E. (1971) Poisson flats in Euclidean spaces. Part II. Homogeneous Poisson flats. Adv. Appl. Prob. 3, 1–43.Google Scholar
Prigogine, I. and Herman, R. (1971) Kinetic Theory of Vehicular Traffic. Elsevier, London.Google Scholar
Wright, C. C. (1973) A theoretical analysis of the moving observer method. Transportation Res. 7, 293–311.Google Scholar
Wright, C. C., Hyde, T., Holland, P. J. and Jackson, B. J. (1973) A method of estimating traffic speeds from flows observed at the end of a road link. Traffic Eng. and Control 14, 472–475.Google Scholar
Wardrop, J. G. (1952) Some theoretical aspects of road traffic research. Proc. Inst. of C.E. Part II 1, 325–378.Google Scholar