The queue dynamics of protected/permissive left turns at pre-timed signalized intersections

Queue length is a very fundamental indicator for evaluating the efficiency of traffic control strategy for signalized intersections, which is generally regarded as the basis for design and control of traffic operations. As the protected/permissive left-turn queue at traffic signals has significant randomness and unpredictability, the average value itself is often not sufficient to reflect the variation and uncertainty of queueing process, thus it is a less useful guide as the performance evaluation for the use of protected/permissive left-turn operation. In this paper, a Markov queueing model with server fixed vacation and multiple uncertain interruptions is proposed, and a numerical solution method is formulated to measure the queue dynamics of protected/permissive left turns at signalized intersections. Overcome the dependence on the average value of queue length, a more efficient version of the queue length along the time axis is explored by considering various practical constraints. The obtained performance indicators can be used to measure the queue length at any chosen time point during a signal cycle only by inputting some basic information. The proposed model is evaluated using the well-known Webster formula with different signal timings, left-turn demands and opposing through demands. In addition, numerical results show that the indicator of queue length along the time axis within a cycle can provide more sufficient information to reflect the variation and uncertainty of queue length, thus it can help to establish a basic queue profile on evaluating the use of protected/permissive left-turn operation.