Webster's Optimum Cycle Length

In 1958 F. V. Webster of Britain's Road Research Laboratory published Technical Paper No. 39, Traffic Signal Settings. He simulated signalised intersections on the computing equipment of the day, measured the total delay at many combinations of cycle length and demand, and fitted a simple expression to where the delay came out lowest. That expression, $C_o = (1.5L + 5)/(1 - Y)$, is still the first thing anyone learns about timing a signal, and it is still a perfectly reasonable starting point.

Two quantities go in. $L$ is the total lost time per cycle: the startup lost time at the head of each green, where the first few drivers are reacting and accelerating rather than discharging at saturation, plus the clearance lost time at the end, being the part of the amber and all-red interval that no vehicle uses. Two to four seconds of each per phase is the usual order, so a four-phase intersection loses somewhere near 16 seconds every single cycle whatever the timings are. That is the strongest argument there is against adding phases: each one costs its lost time on every cycle, all day, forever. $Y$ is the sum of the critical flow ratios — for each phase, take the movement working hardest, compute its flow divided by its saturation flow, and add those one-per-phase numbers up. Not every movement at the intersection; one per phase, the critical one.

The numerator says a longer cycle amortises lost time over more seconds of useful green, so more lost time argues for a longer cycle. The denominator is where the drama is. As $Y$ approaches 1 the cycle blows up, because at $Y = 1$ the critical movements together are demanding every second the intersection has before any of it is paid to lost time. Watch how fast it moves: with $L = 12$ s, $Y = 0.70$ gives a 77-second cycle and $Y = 0.85$ gives 153 — a rise of 0.15 in demand doubles the cycle. At $Y \ge 1$ there is no answer at all, and the honest conclusion is not a longer cycle but that the intersection is oversaturated: it needs a lane, a phasing change, or less demand.

The 5 in that formula is five seconds, and the formula is not dimensionally homogeneous. The 1.5 is a pure number; the 5 carries units, because Webster fitted it to simulations run in seconds. This site keeps time in SI seconds internally so the arithmetic comes out right whatever unit you type into the box — but if you work the equation by hand with a lost time expressed in minutes, you will add five minutes where Webster meant five seconds, and no guard on any page can rescue you from that. Lost time in seconds, cycle in seconds, always. It is the single most common way this formula is got wrong, and it produces an answer that is off by a factor of sixty while looking like a number.

Webster also showed something that takes the pressure off: the delay curve is remarkably flat near its minimum. Anywhere from about three quarters to one and a half times the optimum gives total delay within roughly ten to twenty percent of the best available. So round the answer to something tidy, something your controller likes, something that divides into a coordinated arterial's common cycle — and do not defend the decimal place. In practice cycles are usually kept between about 40 and 120 seconds. Shorter and lost time eats the intersection alive. Longer and pedestrians and side-street drivers stop believing the signal is working and begin crossing against it, which is a safety problem the formula knows nothing about.

And a boundary that matters more than any of the arithmetic: this computes a cycle length, and a cycle length is not a warrant. Whether an intersection should be signalised at all is a code question, answered by the warrant analysis in your jurisdiction's manual — the MUTCD in the United States, the TAC manual in Canada, the equivalent standard elsewhere — on the basis of counts, collision history, pedestrian demand, delay to the minor approach and engineering judgement. Signals are not automatically an improvement: installing one where it is not warranted reliably increases rear-end collisions. Nothing this site prints substitutes for that analysis or for a design signed by someone qualified to sign it.