Distribution of Delay in Signalized Intersections: Day-to-Day Variability in Peak-Hour Volumes

The Highway Capacity Manual (HCM) and other analytical methods provide a point estimate of average vehicle delay for a specified lane group over a defined time period (e.g., peak hour). The estimated delay is a function of a number of factors, including geometrAy, signal timings, and approach volumes. Some of these factors, particularly, peak hour volume, can be considered to be random variable, and therefore the distribution of intersection delay is a function of the distribution of these random factors. This paper presents findings of a study that addresses the variability in estimated HCM delay as a function of the variation of peak hour volumes. For protected movements, the distribution of delay is analytically determined as a function of the distribution of the peak hour volume and the HCM delay equation characteristics. A numerical method is proposed to find the distribution of delay for permitted movements for which capacity changes as a result of variability in the opposing volume. The proposed methods are demonstrated through application to a hypothetical intersection. The results indicate that when the degree of saturation for the average peak hour volume is not far from 1, the HCM point estimate of average delay is not sufficient for evaluating the quality of service or designing the intersection signal timing plan. For example, the results from the hypothetical intersection examined in this paper suggest that for a permitted left turn with an average degree of saturation of 0.77 and an average delay of $37.8\mathrm{s}/\mathrm{vehicle}$ ($\text{level}\text{of}\text{service}=D$), there is approximately 20% probability that the lane group operates at a level of service of $E$ or worse (i.e., $\text{delay}>55\mathrm{s}/\text{vehicle}$).