🚦 Traffic Intersection — Signal Timing & Queue Theory

Saturation flow rate is the maximum rate at which queued vehicles can depart through a green signal, measured in vehicles per hour of green per lane. This simulation uses 1800 veh/hr (30 veh/min), typical for a single through lane. The ratio of demand to saturation flow (the flow ratio) determines how much green time each phase needs.

When the sum of critical flow ratios Y exceeds 1, demand exceeds capacity regardless of how signal timing is optimised. Webster's formula is no longer valid and queues grow without bound. The simulation shows this by locking the cycle at 120 s and using equal green splits, but LOS F and growing queues indicate that geometric improvements (extra lanes, turn restrictions) are needed rather than timing changes alone.

Green split determines what fraction of each cycle is allocated to each phase. Moving it toward N-S reduces waiting time on those approaches but increases it on E-W. With balanced demand the optimal split is 50/50; with unbalanced demand (e.g., heavy N-S) the optimal split weights toward the heavier direction proportionally to its flow ratio, exactly as Webster's formula prescribes.

Lost time accounts for the seconds at the start of each green phase (start-up lost time, about 2 s) when queued vehicles are reacting and accelerating, plus the clearance interval at the end. These seconds provide no effective capacity. Total lost time L is typically 3–5 s per phase, summed across all phases in one cycle, and directly increases the optimal cycle length.

The simulation records the arrival time and departure time of every vehicle. Average delay is the mean of all (departure − arrival) values over a rolling window of the last 500 departed vehicles. The theoretical Webster uniform delay formula d = C(1 − g/C)² / [2(1 − ρ)] gives a closed-form estimate, where g is effective green and ρ is the degree of saturation (v/c ratio).

Throughput is the number of vehicles successfully passing through the intersection per hour, calculated over rolling 60-second simulation windows and scaled to an hourly rate. It differs from demand (arrival rate) because queued vehicles depart only during green phases at the saturation flow rate. Under congestion, throughput saturates at the intersection capacity and additional arrivals simply extend queues.

This simulation uses a two-phase cycle (N-S green then E-W green) for clarity. Real-world intersections often use four or more phases to handle dedicated left-turn movements, pedestrian crossings, or complex geometry. The underlying Webster framework extends to multiple phases by summing flow ratios across all critical movements, with proportionally longer cycles for more phases.