An Analytical Approximation of the Joint Distribution of Aggregate Queue-Lengths in an Urban Network

Abstract

Traditional queueing network models assume infinite queue capacities due to the complexity of capturing interactions between finite capacity queues. Accounting for this correlation can help explain how congestion propagates through a network. Joint queue-length distribution can be accurately estimated through simulation. Nonetheless, simulation is a computationally intensive technique, and its use for optimization purposes is challenging. By modeling the system analytically, we lose accuracy but gain efficiency and adaptability and can contribute novel information to a variety of congestion related problems, such as traffic signal optimization. We formulate an analytical technique that combines queueing theory with aggregation-disaggregation techniques in order to approximate the joint network distribution, considering an aggregate description of the network. We propose a stationary formulation. We consider a tandem network with three queues. The model is validated by comparing the aggregate joint distribution of the three queue system with the exact results determined by a simulation over several scenarios. It derives a good approximation of aggregate joint distributions.