Traffic Capacity & LOS · Study · PE Civil: Transportation · FE → PE Prep
Traffic Capacity & LOS
9% of exam
Uninterrupted and interrupted flow: peak-hour factor, flow rate, density and speed relationships, freeway/segment level of service, and signalized-intersection and roundabout capacity.
5 concepts
A. Uninterrupted flow
Flow, Speed, Density & the Peak-Hour Factor
The fundamental q = k·v relation, the Greenshields speed-density line, jam density and capacity, and the peak-hour factor that scales an hourly volume to a 15-minute flow rate.
Three numbers describe any stream of traffic at a point: how many vehicles pass per hour (flow q), how fast they move (speed v), and how tightly they are packed (density k
). They are not independent — they are locked together by one identity,
q=kv
, and everything downstream in capacity and level-of-service analysis is built on it. The PE Civil Transportation exam tests this relentlessly in the uninterrupted-flow area, and the points that get lost are almost always unit slips (vehicles per hour versus per 15 minutes) or a confusion between the two kinds of average speed. In the NCEES PE Civil Reference Handbook the fundamental relation and the speed-density/flow-density material sit under §5.1.1 'Uninterrupted Flow' (incl. §5.1.1.2 Space Mean Speed and §5.1.1.4 Greenshields Maximum Flow Rate) and §5.1.2 'Street Segment Interrupted Flow' (§5.1.2.1 Speed-Density Model, §5.1.2.2 Flow-Density Model), with the peak-hour factor in §5.1.3 'Traffic Analysis'; this concept makes you fluent in all of it.
The fundamental relation q = k·u
Flow is the product of density and space-mean speed. Read the equation dimensionally and it is obvious: [miveh]×[hmi]=[hveh]. Density is the snapshot quantity — vehicles per mile at one instant — while flow is the temporal quantity — vehicles per hour past one point. The single most important habit on this topic is to keep these two views separate: a detector measures flow and speed directly, but it must infer density from k=q/us. The speed in this identity must be the space-mean speed us, never the time-mean speed.
q=kus
Space-mean versus time-mean speed
Average speed is ambiguous until you say which average. The time-mean speed ut is the arithmetic mean of the spot speeds of individual vehicles passing a point; the space-mean speed us is the harmonic mean — the length of a section divided by the average travel time across it. Because slower vehicles spend more time in the section, they are weighted more heavily in the space average, so us≤ut always, with the gap growing as speeds scatter. Only us is consistent with q=kus; using ut there is a classic exam trap.
us=i=1∑ntinL,ut=n1i=1∑nui,us≤ut
The Greenshields linear model
Greenshields proposed the simplest realistic closure: speed falls linearly as density rises, from the free-flow speed uf at near-zero density to zero at the jam density kj where vehicles are bumper to bumper. With this one straight line the whole stream is determined. Substituting the line into q=kus gives a parabola in density — flow climbs from zero, peaks, and falls back to zero at jam — which is the shape every fundamental diagram takes.
us=uf(1−kjk)
Capacity from the parabola
Put the Greenshields line into the flow identity, q=ufk(1−k/kj), and maximize: the peak sits exactly halfway along, at the optimum density kcap=kj/2 and the optimum speed ucap=uf/2. The peak value is the capacity of the Greenshields stream, and the handbook gives it directly as qmax=ufkj/4. The lesson examinees must internalize is that maximum flow does NOT happen at maximum speed — it happens at half the free-flow speed, where speed and density are balanced.
qmax=4ufkj,kcap=2kj,ucap=2uf
Two regimes, one diagram
Every flow below capacity can be served at two different densities — one on the free-flow (uncongested) branch to the left of the peak and one on the congested branch to the right. The left branch is the stable operating side: low density, high speed. The right branch is breakdown: high density, crawling speed, the same vehicles per hour but moving in a queue. Knowing which branch you are on is what distinguishes a healthy facility from a jammed one carrying identical flow.
The peak-hour factor
Demand is never uniform within an hour. The peak-hour factor (PHF) measures how peaked it is: it is the hourly volume divided by four times the peak 15-minute count, PHF=V/(4V15). A perfectly uniform hour gives PHF=1.00; real urban peaks run 0.85 to 0.95. Design and capacity work uses the flow RATE within the peak 15 minutes, not the hourly average, so you convert by dividing the hourly volume by the PHF: v=V/PHF. Forgetting this step under-sizes a facility for the surge it must actually handle.
PHF=4V15V,v=PHFV
Exam strategy
Anchor on q=kus and always write the units beside each symbol before substituting; a detector gives you flow and speed, so density is k=q/us. For any Greenshields question, immediately note the three anchors qmax=ufkj/4, kcap=kj/2, ucap=uf/2, and remember capacity is at half the free-flow speed. When a problem gives a 15-minute count, decide at once whether it wants the hourly volume (×4) or the equivalent flow rate (V/PHF). If a speed is described as a spot-speed average, it is time-mean and must NOT be put into q=kus.
Key equations
Fundamental relationq=kus
Flow q (veh/h) equals density k (veh/mi) times space-mean speed us (mph). The backbone of all uninterrupted-flow analysis.
Density from a detectork=usq
Density inferred from measured flow and space-mean speed; vehicles per mile per lane (or per roadway).
Equivalent hourly flowq=T3600n
Converts n vehicles counted in T seconds to an equivalent hourly rate (vph). Same identity scales a 15-min count by 4.
Space-mean speedus=∑tinL
Greenshields speed-densityus=uf(1−kjk)
Greenshields flow-densityq=ufk(1−kjk)
Greenshields capacityqmax=4ufkj
Optimum density and speedkcap=2kj,ucap=2uf
Peak-hour factorPHF=4V15V
Hourly volume V
Peak flow ratev=PHFV
Equivalent hourly flow rate during the peak 15 minutes; the value used in capacity and LOS work.
Worked examples
Capacity of a Greenshields stream
Problem. A freeway lane has a free-flow speed of 60mph and a jam density of 240veh/mi. Using the Greenshields model, find the capacity, the density and speed at capacity, and the flow when the density is 80veh/mi.
Solution. Capacity: qmax=ufkj/4=(60)(240)/4=3600vph.
At capacity: kcap=kj/2=120veh/mi and ucap=uf/2=30mph (check: 120×30=3600vph, consistent).
At k=80veh/mi: us=60(1−24080)=60(0.667)=40.0mph, so q=kus=80×40.0=3200vph.
Sanity check: 80veh/mi<kcap, so we are on the uncongested branch and the flow (3200) is below capacity (3600) — consistent.
qmax=4(60)(240)=3600vph
Density from detector data
Problem. A loop detector reports a flow rate of 1500vph in a single lane at a space-mean speed of 50mph. Find the density, and the average spacing between vehicles.
Solution. Density: k=q/us=1500/50=30.0veh/mi
Peak-hour factor and design flow rate
Problem. During the analysis hour a count station records 2900veh, with the peak 15-minute count being 820veh. Find the peak-hour factor and the equivalent peak flow rate.
•Putting time-mean speed into q=kus. Only space-mean (harmonic) speed is consistent with the fundamental relation; a spot-speed average will understate density (ut ≥ us, so k = q/ut comes out low).
•Assuming capacity occurs at the free-flow speed. In Greenshields capacity is at HALF the free-flow speed (ucap=uf/2), where the flow-density parabola peaks.
•Confusing volume with flow rate. A 15-min count ×4 gives an hourly rate; the hourly volume must be divided by PHF to recover the peak flow rate — they are not the same number.
•Mixing per-lane and per-roadway density. Greenshields kj and the resulting capacity are per lane unless stated; multiply by the number of lanes only at the end.
•Reading off the wrong branch of the fundamental diagram. A given sub-capacity flow corresponds to two densities; high density at low speed means you are in the congested (queued) regime, not free flow.
•Forgetting that PHF≤1 always. A computed PHF above 1.0 means the 15-min count or the hourly volume was transcribed wrong.
References
NCEES PE Civil Reference Handbook — §5.1.1 Uninterrupted Flow (§5.1.1.2 Space Mean Speed, §5.1.1.4 Greenshields Maximum Flow Rate, $q_{max}=u_f k_j/4$)
NCEES PE Civil Reference Handbook — §5.1.2 Street Segment Interrupted Flow (§5.1.2.1 Speed-Density Model, §5.1.2.2 Flow-Density Model)
Garber & Hoel, Traffic and Highway Engineering, 5th ed. — Source of the Greenshields maximum-flow relationship cited by the handbook.
Uninterrupted-Flow Level of Service
Convert a measured volume to an equivalent passenger-car flow rate, divide by speed to get density, and read the A-F level of service for a basic freeway segment.
Unlocks with an access pass — one-time payment, no auto-renew. View passes
Pass holders
B. Street segment interrupted flow
Urban Street Segment Running Time & LOS
Segment running time and travel speed, how signal spacing and control delay erode it, and the travel-speed level of service for an urban street segment.
Unlocks with an access pass — one-time payment, no auto-renew. View passes
Pass holders
C. Intersection capacity
Signalized Intersection Capacity & Delay
Lane-group capacity from saturation flow and green ratio, the volume-to-capacity ratio, and HCM/Webster control delay with its A-F level-of-service grade.
Unlocks with an access pass — one-time payment, no auto-renew. View passes
Pass holders
Roundabout Capacity & Control Delay
The gap-acceptance exponential capacity model driven by circulating flow, the entry volume-to-capacity ratio, and roundabout control delay with its level-of-service grade.
Unlocks with an access pass — one-time payment, no auto-renew. View passes
Pass holders
Harmonic-type average: section length L over mean travel time. The only speed valid in q=kus.
Linear model; uf = free-flow speed (mph), kj = jam density (veh/mi). Speed is zero at jam.
Parabola obtained by substituting the speed-density line into q=kus; peak gives capacity.
Maximum flow of the Greenshields stream (vph), occurring at kcap=kj/2 and ucap=uf/2.
Density and speed at capacity. Capacity is reached at half the free-flow speed, not at maximum speed.
over four times the peak 15-min count
V15
. Range 0.25–1.0; uniform hour = 1.0.
.
Average spacing is the inverse of density:
s=1/k=1/30mi=5280/30=176ft per vehicle
.
Sanity check:
30veh/mi
at
50mph
is light, free-flowing traffic, and
176ft
headway is a comfortable gap — consistent with uncongested operation.
k=501500=30.0veh/mi
.
Peak flow rate:
v=V/PHF=2900/0.884=3280vph
(equivalently
4×820
, the 15-min count scaled to an hourly rate).
Sanity check:
PHF<1
as required, and the flow rate (
3280
) exceeds the hourly volume (
2900
) because demand surges within the hour — consistent.