TRN-26

Webster delay

d = 0.9 [C(1−λ)²/(2(1−λx)) + x²/(2q(1−x))]. Uniform plus random delay.

Reading speed

SignalsWebster 1958

Governing equation

d=0.9\left[\dfrac{C(1-\lambda)^2}{2(1-\lambda x)}+\dfrac{x^2}{2q(1-x)}\right]

where

C: Cycle (s)
g: Effective green (s)
q: Arrival rate (veh/s)
s: Saturation flow (veh/s)
d: Delay per veh (s)

Lecture brief

Historical brief

Highway and traffic theory crystallized with Greenshields’ 1935 speed–density line, then AASHTO stopping sight and equivalent axles. The lab is that operational arithmetic of flow, braking and pavement load. This sheet (TRN-26 — Webster delay) is the form associated with Webster 1958. Working symbols: $C$ , $g$ , $q$ , $s$ $\rightarrow$ $d$ . Webster combined a deterministic red wait with a Pollaczek–Khinchine overflow. C* ≈ (1.5L+5)/(1−Y).

Purpose

Purpose: compute

d

from

C

g

q

s

in Transport via

d=0.9\left[\dfrac{C(1-\lambda)^2}{2(1-\lambda x)}+\dfrac{x^2}{2q(1-x)}\right]

d = 0.9 [C(1−λ)²/(2(1−λx)) + x²/(2q(1−x))]. Uniform plus random delay. Use it when a real transport question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

C = 90.000\,\mathrm{s}

g = 35.000\,\mathrm{s}

q = 0.180\,\mathrm{veh/s}

s = 0.500\,\mathrm{veh/s}

, the governing relation

d=0.9\left[\dfrac{C(1-\lambda)^2}{2(1-\lambda x)}+\dfrac{x^2}{2q(1-x)}\right]

yields

d = 52.47\,\mathrm{s}

. A signal, a cycle, a delay per vehicle. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Cycle C90 sEffective green g35 sArrival rate q0.18 veh/sSaturation flow s0.5 veh/s

Outputs

Delay per veh d52.47 s

Reading speed

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Narration of this film

A signal, a cycle, a delay per vehicle.

Webster combined a deterministic red wait with a Pollaczek–Khinchine overflow. C* ≈ (1.5L+5)/(1−Y).

Reading speed

Watch on YouTube