How long does one wait at a traffic congestion?

Suppose a wide road is somehow constricted at one place, allowing only one car at a time. Unless the traffic is very sparse, the vehicles will gather before the constriction and a given car will have to wait till it gets a chance to pass through. At a time which car will exit? Perhaps it is the one, with a driver who can jostle and aggressively put its nose into the constriction. We assume that the quantitative measure of aggressiveness depends on the nature of the individual and increases monotonically with waiting time. Our precise hypothesis is that the aggressiveness of a given driver who has been stranded for time τ, is the product of two quantities N, and τσ, where N is fixed for a given individual but varies randomly from person to person (in the range 0 to 1), while σ is the same for all individuals. We show that this hypothesis leads to the conclusion that the probability of waiting for a time τ is proportional to τσ+1. Although the applicability of our hypothesis to real situations is difficult to verify at microscopic level, we note that empirical studies confirm such algebraic decay with an exponent σ=1.5 and 2.5 in two cities of India and 0.5 for a traffic intersection in Germany. We shall present some justifications for our hypothesis along with some variants and limitations of our model.