Statistics of delay for a driver population with step and distributed gap acceptance functions

A model generally used for the reduction of gap acceptance data is one in which each driver has a unit step gap acceptance function, but the time of the step is distributed over the driver population. Data is now available to show that individual gap acceptance functions do not take the form of a step. In this investigation we assume that the individual gap acceptance function takes the form $\text{α(t)}\text{= 0}\text{t ⩽ T}\text{= 1 − exp [−β(t−T)}\text{t ⩾ T}$ but the data customarily available is reduced as if they came from a distribution of step functions. Under these circumstances we show that for a single car waiting time problem, the mean delay is calculated correctly using either assumption of the individual gap acceptance functions. The variance of the calculated waiting time would be overestimated as would be the probability of zero delay. The capacity would be overestimated. Data available from measurements by Bottom and Ashworth suggest that the discrepancies caused by assuming step gap acceptance functions may not be too large.