Traffic Intersection Optimization: Queue Theory and Signal Timing

A signalised intersection is a queuing system

Vehicles arrive at a stop line, wait in queue, and are discharged at a finite rate during green — the textbook D/D/1 or M/D/1 queuing model that Danish engineer Agner Krarup Erlang pioneered for telephone exchanges in 1909. Three quantities govern the system: the arrival rate q (vehicles/hour), the saturation flow rate s — the maximum discharge rate through the stop line, around 1,800–1,900 vehicles per hour of green (vphg) for a standard lane — and the effective green ratio g/C. When arrivals exceed the effective service rate (q > s·g/C), the intersection is oversaturated and the queue grows without bound.

Poisson arrivals and the degree of saturation

At an isolated intersection with light-to-moderate volume, vehicles arrive independently and closely follow a Poisson process: the probability of exactly k arrivals in interval t is P(k) = (λt)ᵏ·e^(−λt)/k!, where λ is the mean arrival rate. At higher volumes or downstream of another signal, arrivals cluster into platoons and a binomial or Erlang-k distribution fits better. Either way, engineers track the degree of saturation x = q/(s·g/C): x < 1.0 means the queue clears every cycle, x = 1.0 is capacity, and x > 1.0 means the queue grows every cycle. Good design keeps critical lanes at x ≤ 0.85–0.90 as a buffer against random fluctuation.

P(k arrivals in time t) = (λt)^k · e^(−λt) / k!
Example: λ = 0.3 veh/s (1080 vph), t = 5 s
  P(0) ≈ 0.223   P(1) ≈ 0.335   P(2) ≈ 0.251

Webster's optimal cycle length

F. V. Webster's 1958 Road Research Laboratory report derived the cycle length that minimises total vehicle delay at a signalised intersection — a result that still stands, seventy years later, as the international default: C_opt = (1.5L + 5) / (1 − Y), where L is total lost time per cycle (start-up plus clearance loss, ≈ 3–5 s per phase) and Y is the sum of critical-lane volume-to-saturation-flow ratios. Counterintuitively, delay is fairly insensitive to cycle length near the optimum — a ±15% change raises delay by under 5% — but a cycle set far too long or short causes sharp delay spikes, because vehicles wait longer at red before each new cycle begins.

Level of Service and adaptive control

The Highway Capacity Manual grades intersections A through F by average control delay per vehicle: LOS A is under 10 seconds (free flow), LOS D (35–55 s) is the typical urban design threshold, and LOS F (over 80 s) means capacity failure. Fixed-time plans are tuned to historical averages, but real demand fluctuates — a school bell or a stalled car can shift flows within minutes. Adaptive systems like the UK's SCOOT and Australia's SCATS use real-time detector data to continuously retime splits, cycle length and offset, cutting average delay by 10–25% compared with optimised fixed timing. Coordinating offsets along a corridor creates a green wave, letting drivers clear consecutive signals without stopping.

Frequently asked questions

Try it live

Everything above runs in your browser — open Traffic Intersection, set green splits and cycle length against Poisson arrivals, and watch queues and delay grow or clear in real time. Nothing is installed, nothing is uploaded.