Throughput analysis of a horizontal traffic queue under safe car following models

We consider a horizontal traffic queue (HTQ) on a periodic road segment, where vehicles arrive according to a spatio-temporal Poisson process, and depart after traveling a distance that is sampled independently and identically from a spatial distribution. When inside the queue, the motion of vehicles is governed by a car following model. We consider safe first and second order car following models. In the first order models, the speed of a vehicle is proportional to a power m > 0 of the distance to the vehicle in front. For the first order model, we show monotonicity of the service rate in between arrivals and departures. We extend the busy period calculations for M/G/1 queue to our setting, including for non-empty initial conditions in order to derive a probabilistic lower bound on the throughput for the m > 1 case. For the m < 1 case, we study throughput under a release control policy, where the additional expected waiting time caused by the release control policy is interpreted as the magnitude of the perturbation to the arrival process. We derive a lower bound on throughput for a given combination of maximum allowable perturbation and m < 1. For the second order model, we design a safe release control policy which guarantees a provable lower bound on the throughput. Illustrative simulation results are also presented.