Group-based approach to derivatives of delay models based on incremental queue accumulations for isolated signalized junctions

We devise an approximation method for the derivatives of group-based variables in individual signal groups during each cycle of a traffic signal, based on the estimated control delay formulas and queue formation patterns. The key challenges include establishing the most appropriate formulations to estimate the derivatives of the control delay toward group-based variables for individual lanes on each cycle, according to the queuing formation pattern, and identifying the vehicular arrivals and departures occurring in the extended or the setback time duration. The derivatives of the control delay computed by the proposed methods for the variables are compared with numerical differentiations collected by simulation. The results of the two methods show good concurrence, with a few outliers for both the start and the duration of the effective green time. The proposed methods lay the foundation stone of more efficient non-linear mathematical programs to optimize group-based signal plans in real time.