Analytical evaluation of the use of left-turn phasing for single left-turn lane only

At most traditional signalized intersections, three types of phasing can be provided to left turns: protected only phasing, permitted only phasing, and a combination of protected and permitted phasing (protected/permitted phasing). While numerous guidelines for the selection of left-turn phasing have been developed, there is no widely recognized guideline or criterion for the use of left-turn phasing under specific traffic conditions especially for a single left-turn lane. Focusing on the inherent mechanism of the dynamic nature and uncertainty of left-turn queues under different left-turn phasing, this paper develops a left-turn queueing model with uncertain second vacation to analytically evaluate the performance of left turns for different left-turn phasing, which considers various factors including the type of left-turn phasing, signal timing, left-turn volume, opposing through volume, type of left-turn lane, number of opposing through lanes, and number of sneakers, etc. For different left-turn phasing, the left-turn queues formation and dissipation with different characteristics of vacations (i.e., server absences during the red time and uncertain blocked time caused by the opposing through flow during the permitted green time) are formulated, and the queue length distributions of left turns along with the time within one cycle and at an arbitrary time are derived. Furthermore, three sets of left-turn performance measures are obtained: primary queueing measures, fuel-consumption (or emissions) related measures, and safety related measure. On the basis of these performance measures, an analytical evaluation framework for left-turn queues is established, which can provide a more accurate and detailed basis for evaluating and improving the use of left-turn phasing. Model validation indicates that the proposed model can be an effective tool to evaluate the use of left-turn phasing under different traffic conditions. In addition, numerical experiments are also performed to theoretically identify the factors that could affect the performance of left turns and thus help to determine the left-turn phasing selection under different conditions.