Service Rate, Busy Period & Throughput Analysis of a Horizontal Traffic Queue

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 speed of a vehicle is proportional to a power $m >0$ of the distance to the vehicle in front. The service rate of HTQ is equal to the sum of the speeds of the vehicles, and has a complex dependency on the state (vehicle locations) of the system. We show that the service-rate increases (resp., decreases) in between arrivals and departures for $m<1$ (resp., $m>1$) case. For a given initial condition, we define the throughput of such a queue as the largest arrival rate under which the queue length remains bounded. We extend the busy period calculations for M/G/1 queue to our setting, including for non-empty initial condition. These calculations are used to prove that the throughput for $m=1$ case is equal to the inverse of the time required to travel average total distance by a solitary vehicle in the system, and also to derive a probabilistic upper bound on the queue length over a finite time horizon for the $m>1$ case. Finally, we study throughput under a release control policy, where the additional expected waiting time caused by the 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, for $m<1$ and $m>1$ cases. In particular, if the allowable perturbation is sufficiently large, then this lower bound grows unbounded as $m \to 0^+$. Illustrative simulation results are also presented.