Green Split by Critical Flow Ratio

Choosing the cycle length is the first half of timing a fixed-time signal. Dividing it up is the second. Of the $C$ seconds in the cycle, $L$ are lost to startup and clearance no matter what you do, so $C - L$ seconds of effective green are available to share. Share them in proportion to each phase's critical flow ratio and you get $g_i = (y_i/Y)(C - L)$.

The proportional split is not an arbitrary convention; it has a specific optimality property worth knowing. Each phase's degree of saturation is $x_i = y_i C / g_i$, and substituting the split above gives $x_i = YC/(C-L)$ — the same value for every phase. Splitting in proportion to demand equalises the degree of saturation across the intersection, so no phase fails before the others do. That is the arrangement that minimises the worst delay, and it is why any hand adjustment favouring one phase is favouring it at the direct, quantifiable expense of the rest.

The critical flow ratio for a phase, $y_i$, is the flow divided by the saturation flow for the movement working hardest in that phase — not the sum over the phase's movements, and not the busiest movement at the intersection. Getting that wrong inflates $Y$, which inflates the cycle, which increases delay for everybody. Identifying the critical movements correctly is genuinely the hard part of the exercise, particularly with overlapping phases or a protected-permitted left, where the critical path through the phase diagram is not always the obvious one.

Three things this arithmetic does not know, and all three will override it.

First, the answer is effective green, and a controller is programmed in displayed green. Convert before you enter anything: add the startup lost time and subtract the portion of the amber that drivers actually use.

Second, it has no floor. A phase with a small $y$ can come out at three seconds, and no standard on earth permits a three-second green. Vehicle minimum greens exist, and far more often the governing constraint is the pedestrian interval — the walk indication plus the flashing-don't-walk clearance, sized from the crossing distance and an assumed walking speed that has been revised downward over the years as agencies have taken older pedestrians seriously. On a wide crossing that interval can exceed anything the vehicle arithmetic asks for, and it is not negotiable. The practical procedure is to impose the minimums first and split what remains proportionally among the phases that still have slack.

Third, it says nothing about the order of the phases or about coordination with the next signal. On an arterial, phase sequence and offset routinely matter more to total delay than the split does, because a platoon that arrives on the green experiences almost no delay and the same platoon arriving on the red experiences a great deal. Bandwidth and progression analysis address that, and they sit above this equation rather than inside it.

A final check worth doing every time: the greens plus the lost time must add back to the cycle. $\sum g_i = C - L$ by construction, so if your numbers do not reconcile, either a phase's minimum has been imposed without taking the seconds from somewhere, or a critical movement has been counted in two phases.