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Traffic light queues as a generalization to queueing theory

Published online by Cambridge University Press:  14 July 2016

Hisashi Mine  and
Katsuhisa Ohno
Hisashi Mine
Affiliation:
Kyoto University
Katsuhisa Ohno
Affiliation:
Kyoto University
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Extract

Fixed-cycle traffic light queues have been investigated by probabilistic methods by many authors. Beckmann, McGuire and Winsten (1956) considered a discrete time queueing model with binomial arrivals and regular departure headways and derived a relation between the stationary mean delay per vehicle and the stationary mean queue-length at the beginning of a red period of the traffic light. Haight (1959) and Buckley and Wheeler (1964) considered models with Poisson arrivals and regular departure headways and investigated certain properties of the queue-length. Newell (1960) dealt with the model proposed by the first authors and obtained the probability generating function of the stationary queue-length distribution. Darroch (1964) discussed a more general discrete time model with stationary, independent arrivals and regular departure headways and derived a necessary and sufficient condition for the stationary queue-length distribution to exist and obtained its probability generating function. The above two authors, Little (1961), Miller (1963), Newell (1965), McNeil (1968), Siskind (1970) and others gave approximate expressions for the stationary mean delay per vehicle for fixed-cycle traffic light queues of various types. All of the authors mentioned above dealt with the queue-length.

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 3 , September 1971 , pp. 480 - 493
Copyright
Copyright © Applied Probability Trust 1971 

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References

Beckmann, M., Mcguire, C. B. and Winsten, C. B. (1956) Studies in the Economics of Transportation. Yale University Press.Google Scholar
Buckley, D. J. and Wheeler, R. C. (1964) Some results for fixed-time traffic signals. J. R. Statist. Soc. B 26, 133–140.Google Scholar
Chung, K. L. and Fuchs, W. H. J. (1951) On the distribution of values of sums of random variables. Mem. Amer. Math. Soc. 6, 1–11.Google Scholar
Cohen, J. (1967) On two integral equations of queueing theory. J. Appl. Prob. 4, 343–355.Google Scholar
Daley, D. J. (1965) General customer impatience in the queue GI/G/1. J. Appl. Prob. 2, 186–205.Google Scholar
Darroch, J. N. (1964) On the traffic light queue. Ann. Math. Statist. 35, 380–388.Google Scholar
Darroch, J. N., Newell, G. F. and Morris, R. W. (1964) Queues for a vehicle-actuated traffic light. Operat. Res. 12, 882–895.CrossRefGoogle Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Drew, D. R. (1968) Traffic Flow Theory and Control. McGraw-Hill, New York.Google Scholar
Feller, W. (1957) An Introduction to Probability Theory and its Applications, Vol. 1, Second Ed. Wiley, New York.Google Scholar
Finch, P. D. (1959) A probability limit theorem with application to a generalization of queueing theory. Acta. Math. Acad. Sci. Hung. 10, 317–325.Google Scholar
Gaver, D. P. (1959) Imbedded Markov chain analysis of a waiting-line process in continuous time. Ann. Math. Statist. 30, 698–720.Google Scholar
Haight, F. (1959) Overflow at a traffic light. Biometrika, 46, 420–424.CrossRefGoogle Scholar
Jánossy, L., Rényi, A. and Aczel, J. (1950) On composed Poisson distributions, I. Acta. Math. Acad. Sci. Hung. 1, 209–224.Google Scholar
Khintchine, A. Y. (1960) Mathematical Methods in the Theory of Queueing. Griffin, London.Google Scholar
Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 277–289.Google Scholar
Little, J. D. C. (1961) Approximate expected delays for several manoeuvres by a driver in Poisson traffic. Operat. Res. 9, 39–52.Google Scholar
LoèVe, M. (1963) Probability Theory. Third Ed. Van Nostrand, Princeton, New Jersey.Google Scholar
Mcneil, D. R. (1968) A solution to the fixed-cycle traffic light problem for compound Poisson arrivals. J. Appl. Prob. 5, 624–635.Google Scholar
Miller, A. J. (1963) Settings for fixed-cycle traffic signals. Operat. Res. Quart. 14, 373–386.Google Scholar
Newell, G. F. (1960) Queues for a fixed-cycle traffic light. Ann. Math. Statist. 31, 589–597.Google Scholar
Newell, G. F. (1965) Approximation methods for queues with application to the fixed-cycle traffic light. SIAM Rev. 7, 223–240.CrossRefGoogle Scholar
Parzen, E. (1962) Stochastic Processes. Holden-Day, San Francisco.Google Scholar
Siskind, V. (1970) The fixed cycle traffic light problem: a note on a paper by McNeil. J. Appl. Prob. 7, 245–248.Google Scholar
Spitzer, F. (1956) A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82, 323–386.Google Scholar
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar