TRN-33

Two-regime fundamental diagram

q = vf k (1 − k/kc) for k ≤ kc, else q = vw (kj − k). A piecewise q–k.

Reading speed

TrafficEdie two-regime

Governing equation

q=\begin{cases}v_f k(1-k/k_c)&k\le k_c\\v_w(k_j-k)&k>k_c\end{cases}

where

k: Density (veh/km)
v_f: Free speed (km/h)
k_c: Critical density (veh/km)
k_j: Jam density (veh/km)
v_w: Wave speed (km/h)
q: Flow (veh/h)

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-33 — Two-regime fundamental diagram) is the form associated with Edie two-regime. Working symbols: $k$ , $v_f$ , $k_c$ , $k_j$ , $v_w$ $\rightarrow$ $q$ . Edie (and later Newell) split free-flow and congested branches. Capacity sits at the joint kc.

Purpose

Purpose: compute

q

from

k

v_f

k_c

k_j

v_w

in Transport via

q=\begin{cases}v_f k(1-k/k_c)&k\le k_c\\v_w(k_j-k)&k>k_c\end{cases}

q = vf k (1 − k/kc) for k ≤ kc, else q = vw (kj − k). A piecewise q–k. Use it when a real transport question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

k = 30.000\,\mathrm{veh/km}

v_f = 90.000\,\mathrm{km/h}

k_c = 25.000\,\mathrm{veh/km}

k_j = 140.000\,\mathrm{veh/km}

v_w = 18.000\,\mathrm{km/h}

, the governing relation

q=\begin{cases}v_f k(1-k/k_c)&k\le k_c\\v_w(k_j-k)&k>k_c\end{cases}

yields

q = 1980\,\mathrm{veh/h}

. A triangular / two-piece q–k, a capacity peak. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Density k30 veh/kmFree speed v_f90 km/hCritical density k_c25 veh/kmJam density k_j140 veh/kmWave speed v_w18 km/h

Outputs

Flow q1980 veh/h

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A triangular / two-piece q–k, a capacity peak.

Edie (and later Newell) split free-flow and congested branches. Capacity sits at the joint kc.

Reading speed

Watch on YouTube