Some Problems in Traffic Delay

The problems that are considered here concern the delay to a single car waiting at a stop sign for a sufficiently large gap in the oncoming traffic to present itself so that the driver considers it safe to cross the highway. The systematic application of renewal theory techniques offers a method of solving these problems, which is much simpler than the combinatorial solutions published by Tanner and Mayne in the treatment of somewhat simpler versions. It is assumed that successive gaps in the main highway traffic are uncorrelated random variables with known probability density. We treat the case that the probability of crossing the highway is a known function, a ( t ) rather than a step function. It is shown that the distribution of delay times satisfies a convolution integral equation, and that the moments are easily found using Laplace transforms Integral equations are also found for the delay distribution when the gap distribution is non-stationary, e.g., when traffic light effects are important. Finally the case of semi-Markov correlated gaps are considered, and a formal expression for the delay distribution is given.