Deterministic Queue at the End of Red

The crudest queue model there is: vehicles arrive at a steady rate $q$, nobody leaves while the light is red, so at the instant the green appears the queue is $Q = q\,r$. It is a rectangle of arrivals with a flat departure line under it, and it is one line of arithmetic. It is also the calculation that decides how long a left-turn bay has to be, so its crudeness is worth understanding rather than apologising for.

Get the units right and it is exact. A flow of 600 veh/h is one vehicle every six seconds, so a 48-second red accumulates eight of them. Multiply by a storage length — about 7.5 m per vehicle including the gap, more where trucks are common — and eight vehicles is a 60 m queue. That is the figure a turn bay is checked against, and a queue estimate always rounds up, because you cannot store four fifths of a car and the consequence of being one space short is not proportional.

It is not proportional because of what happens when a bay overflows. A left-turn queue longer than its bay does not simply inconvenience the turners: it spills back into the adjacent through lane and blocks it, taking the through movement's capacity with it. The reverse failure is just as bad — a through queue past the mouth of the bay prevents turners from reaching the storage that is sitting empty in front of them. Either way, a bay a few metres short can cost far more capacity than it would have cost to build it properly, which is why this very simple calculation gets used far more often than its sophistication deserves.

What the model assumes, and where each assumption fails.

Uniform arrivals. Real arrivals are random, so this is an average cycle and roughly half of your cycles will be worse than it. For design, use a percentile — the 95th is the common choice — which means a queueing model that carries the variance rather than this one. Randomness matters most when flows are light, where a Poisson-like arrival process can easily deliver twice the average in a single cycle.

The queue cleared last cycle. If it did not, the leftover is added to this cycle's arrivals, and if the approach is oversaturated the residual grows every cycle. In that condition no single number describes the queue at all: it is a function of how long the oversaturation lasts, and it must be modelled with time in it.

Arrivals are independent of the upstream signal. On a coordinated arterial they are not. Vehicles arrive in platoons, and the answer depends almost entirely on whether the platoon lands on your red or your green. A well-offset signal can have a queue near zero while this formula predicts a substantial one. That is the whole point of offsets, and it is the largest single error in a deterministic queue estimate on an urban arterial.

Two definitions to be careful with. The $r$ here is effective red — the cycle length minus the effective green — which includes the all-red and the unused part of the amber, not just the red indication. And the queue this returns is the number waiting when the green appears, which is not the same as the maximum back of queue: vehicles keep joining after the green starts and before the front of the queue has begun to move, so the furthest upstream the queue actually reaches is somewhat greater. For storage design, that back-of-queue figure is the one that matters.

Run the formula the other way and it answers a storage question — given the arrival rate and the queue a bay can hold, how much red can this movement tolerate? That is a genuine constraint on cycle length, and notably it argues for a shorter cycle than Webster's optimum. A long cycle means a long red for everyone not currently green, and a long red means a long queue. An intersection with short turn bays can be forced down to a cycle that costs a little delay in exchange for not blocking its through lanes, and that trade is usually worth making.