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Traffic Flow & Signals formula solvers

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Capacity of a Signalised Approach

c=s gCc = s\,\frac{g}{C}

Traffic Flow & SignalsAn approach discharges at its saturation flow while it is green and at nothing while it is red, so its capacity is the saturation flow scaled by the fraction of the cycle it owns. One line, and it is the bridge between a signal's timings and the vehicles per hour it actually delivers.

Degree of Saturation

x=qcx = \frac{q}{c}

Traffic Flow & SignalsDemand divided by capacity for one movement at a signal. It is the single number that says whether a timing plan works, and it is the term that makes every delay equation run away as it approaches one.

Deterministic Queue at the End of Red

Q=q rQ = q\,r

Traffic Flow & SignalsVehicles arrive at a steady rate and none leave while the light is red, so the queue at the moment the green comes on is simply the arrival rate multiplied by the red. It is the crudest queue model there is, and it is the one that decides how long a turn bay has to be.

Fundamental Traffic Flow Relation

q=k vq = k\,v

Traffic Flow & SignalsFlow equals density times space mean speed. Three quantities describe a stream of traffic and this identity ties them together, so measuring any two fixes the third — which is how a pair of loop detectors reports a speed it never measured.

Green Split by Critical Flow Ratio

gi=yiY (C−L)g_i = \frac{y_i}{Y}\,(C - L)

Traffic Flow & SignalsOnce a cycle length is chosen, the effective green left over after lost time is shared among the phases in proportion to their critical flow ratios. Splitting it this way gives every phase the same degree of saturation, which is the arrangement that minimises the worst delay at the intersection.

Greenshields Capacity

qmax=vf kj4q_{max} = \frac{v_f\,k_j}{4}

Traffic Flow & SignalsThe apex of the Greenshields parabola, found by setting its derivative to zero: capacity is a quarter of the free-flow speed times the jam density, and it happens at half the jam density and half the free-flow speed. Three of the tidiest halves in engineering, and all three are consequences of the straight line rather than facts about roads.

Greenshields Flow–Density Parabola

q=vf(k−k2kj)q = v_f\left(k - \frac{k^{2}}{k_j}\right)

Traffic Flow & SignalsSubstitute Greenshields' straight line into q = k v and the flow becomes a downward parabola in density: nothing on an empty road, nothing in a jam, and a maximum in between. It is the single most important picture in traffic theory, because it shows that the same flow is carried at two entirely different densities.

Greenshields Speed–Density Relation

v=vf(1−kkj)v = v_f\left(1 - \frac{k}{k_j}\right)

Traffic Flow & SignalsGreenshields' 1935 proposal that speed falls in a straight line from the free-flow speed at an empty road to zero at jam density. It is the simplest model that gets the two endpoints right, and everything else in the classical theory of traffic streams is built on it.

Webster Uniform Delay per Vehicle

d1=0.5 C (1−λ)21−λxd_1 = \frac{0.5\,C\,(1 - \lambda)^{2}}{1 - \lambda x}

Traffic Flow & SignalsThe first and largest term of Webster's delay equation: the average wait per vehicle when arrivals are perfectly uniform. It falls out of the area of the queue triangle over one cycle, and it is the part of delay that would remain even if traffic arrived like clockwork.

Webster's Optimum Cycle Length

Co=1.5L+51−YC_o = \frac{1.5L + 5}{1 - Y}

Traffic Flow & SignalsThe cycle length that minimises total delay at a fixed-time signal, from Webster's 1958 Road Research Laboratory paper. Lost time pushes the cycle up, because a longer cycle spends a smaller share of itself on startup and clearance; heavy demand pushes it up much harder, because the denominator collapses as Y approaches one.