Transportation Engineering

Traffic Engineering

Flow–speed–density relations, Webster's signal timing and capacity — the operational analysis of roads and intersections.

PART 1

Topic Breakdown & Traps

Traffic flow links three quantities by

q = k v

(flow = density × speed). Greenshields' linear model gives a maximum flow (capacity) at

v_{f} /2

and

k_{j} /2

, i.e.

q_{ma x} = \frac{v _{f} k _{j}}{4}

. Signalised intersections are timed with Webster's optimum cycle length, balancing delay against lost time.

Fundamental relation:

q = k v

Greenshields capacity:

q_{ma x} = \frac{v _{f} k _{j}}{4}

Webster optimum cycle:

C_{o} = \frac{1.5 L + 5}{1 - Y}

where

L

= total lost time,

Y = \sum y_{i}

= sum of critical flow ratios.

⚠
** $q = k v$ — flow is density times speed**, not speed alone. Maximum flow occurs at intermediate density, not maximum speed.
⚠
**Webster's $Y$ is the sum of flow ratios**, and the cycle blows up as $Y \to 1$ (oversaturation).
⚠
**Capacity is $v_{f} k_{j} /4$ ** in the Greenshields model — a quarter of the free-flow-speed × jam-density product.

PART 2

Q1BASIC1 Mark · NAT

A road has traffic density

25 veh/km

moving at

60 km/h

. The flow is _____ veh/h.

Q2MEDIUM2 Marks · NAT

Greenshields model: free-flow speed

60 km/h

, jam density

120 veh/km

. The capacity is _____ veh/h.

Q3HARD2 Marks · NAT

Using Webster's formula with total lost time

L = 12 s

and

Y = 0.6

, the optimum cycle length is _____ s.