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In their book Studies in the Economics of Transportation, Beckmann, McGuire and Winsten (BMW) (2, pp. 11-13, 40-42) proposed a simple queuing model for traffic flow through a fixed-cycle traffic light. Although they derived a relation between the average delay per car and the average length of the queue at the beginning of a red phase of the light, they only indicated some possible numerical schemes for evaluating the latter. Here we shall derive analytic expressions for the average queue length and consequently also the average delay under equilibrium conditions for the BMW model.
G. F. Newell (Thu,) studied this question.