Polling Models: A Short Survey and Some New Results

Abstract

In this paper, we review the research history of polling models, consolidate relevant literature, and explore a polling model aimed at minimizing waiting times at traffic signals. This is achieved by regulating the duration a signal remains in the ON position using a random clock. Suppose the signal is active for vehicles to proceed in a specific direction, and all vehicles in that queue have been served, but there is still time remaining for the transition from Green to Red. In this scenario, once the last vehicle departs, and there are no more vehicles in sight requiring service, a random-duration clock initiates. This clock has a stochastically much shorter lifespan than the remaining time needed for the transition from green to red. If no vehicle comes for service in this queue during the ON time of this clock, the signal is turned red the moment the clock realizes it. Subsequently, the signal turns green for the next waiting line in a cyclic order. These modifications have significantly reduced traffic congestion at junctions. We analyze this system as a continuous-time Markov chain (CTMC), treating different arrival streams at the junction as independent Poisson processes. Service times for customers are assumed to be independent, exponentially distributed, nonidentical random variables for distinct queues, while within the same queue, the service times are identical. We assume an infinite capacity polling system with a First-Come-First-Served (FCFS) queue discipline and employ Matrix-Geometric methods for analysis. Finally, we compute several performance measures to evaluate the system’s efficiency. The numerical work we investigated has two nodes; the queue at one of these nodes is of infinite capacity, and that of the other is finite but arbitrarily large.