On the Queueing Time Analysis for State-Dependent Fixed-Cycle Traffic Light Queues (Contemporary Mathematics, 2025)
We analyze a Fixed-Cycle Traffic Light (FCTL) intersection model. Vehicles arrive according to a Poisson process and must wait for a green signal. Each signal period (red or green) consists of a number of phases. Exactly one waiting vehicle is released (passes through the intersection) per green signal period phase, while vehicles remain waiting during red signal periods phases. The lengths of red and green signal periods are not constants, rather they depend on the number of vehicles in the queue. That is, we propose a state-dependent scheduling mechanism for green and red signal periods in an FCTL intersection. The number of green phases increases if the number of vehicles waiting in the intersection is greater than or equal to a threshold N(> 0). The number of green phases increases from g(> 0) to g1(≥ g) and the number of red phases decreases from r(> 0) to r1(≤ r) in such a way that the total length of a cycle period, c = g + r = g1 + r1, is fixed. This mechanism allows one to control the waiting time of vehicles through the FCTL intersection. We analyze the distributions of queue length and vehicle waiting time during each phase of the green signal period. We provide several numerical examples to gain insight into the performance of our proposed FCTL scheduling mechanism. The proposed state-dependent FCTL queueing model dynamically adjusts green and red light durations based on the volume of traffic in queues. This FCTL model with state-dependent scheduling is ideal for smart city traffic optimization, improves traffic flow, reduces delays, and minimizes fuel consumption in busy urban areas.
Keywords: fixed-cycle traffic light queue; probability generating function; root finding; waiting-time analysis