Traffic Engineering and Highway Capacity - Cheat Sheet
Your last-minute rapid-fire reference for traffic flow fundamentals, capacity calculations, and level-of-service concepts. All formulas, critical values, and exam pitfalls condensed for final review.
FUNDAMENTAL TRAFFIC FLOW RELATIONSHIPS
Formulas
q = k·u
Meaning
q = flow (veh/h); k = density (veh/km); u = space-mean speed (km/h)
Watch out
CRITICAL: Use SPACE-MEAN SPEED (harmonic mean), NOT time-mean speed. Most students use wrong speed type and get wrong answer.
When to use
Any time you need to relate flow, density, and speed; forms the basis of ALL traffic analysis
s = 1000/k
Meaning
s = average spacing (m); k = density (veh/km); factor 1000 converts km to m
Watch out
Denominator is k (density in veh/km), not flow. Easy to confuse with headway formula.
When to use
When you need distance between consecutive vehicle fronts
h = 3600/q
Meaning
h = average headway (s); q = flow (veh/h); factor 3600 converts hours to seconds
Watch out
Denominator is q (flow in veh/h), not density. If you use density here, you'll get spacing, not headway.
When to use
When you need time gap between successive vehicles passing a fixed point
PHF = V/(4·V₁₅)
Meaning
PHF = peak-hour factor; V = total volume in peak hour (veh); V₁₅ = highest 15-minute count (veh)
Watch out
Denominator is 4·V₁₅ (four 15-min periods in 1 hour). Students forget the '4' or use wrong count.
When to use
Assess how peaky the traffic demand is; always needed to convert observed volume to design flow
q_design = V/PHF
Meaning
q_design = design flow rate (veh/h); V = observed peak-hour volume; PHF = peak-hour factor
Watch out
Design flow > observed volume when PHF < 1.0 (which is always in real traffic). This inflates demand for design.
When to use
Convert observed hourly volume to equivalent uniform flow for capacity planning
u = q/k
Meaning
Space-mean speed derived from fundamental relation
Watch out
This is a rearrangement of q = k·u. Units must be consistent: q in veh/h, k in veh/km gives u in km/h.
When to use
When you know flow and density but need speed
Common values
Value
~2000 pc/h/lane
Symbol
C_ideal
Quantity
Ideal freeway lane capacity
Value
100–120 km/h
Symbol
u_ff
Quantity
Typical free-flow speed on divided highway
Value
0.80–0.95
Symbol
PHF
Quantity
Peak-hour factor (typical urban)
Value
2–3 seconds
Symbol
h_min
Quantity
Minimum safe headway (perception-reaction + braking)
Value
3600
Symbol
—
Quantity
Conversion: hours to seconds
Value
1000
Symbol
—
Quantity
Conversion: kilometers to meters
Important facts
- The fundamental relation q = k·u is the backbone of all traffic flow analysis.
- Space-mean speed (harmonic mean) is the correct speed for use in q = k·u; NOT time-mean speed.
- Spacing increases (vehicles more spread out) as density decreases.
- Headway increases (more time between vehicles) as flow decreases.
- PHF always ≤ 1.0; lower PHF means more peaky (concentrated) demand.
- Design flow rate q_design = V/PHF; always ≥ observed V because PHF < 1.0.
- Capacity is typically measured in passenger-car equivalents (pc/h) to account for heavy vehicles.
- On freeways, LOS is determined by density; on urban streets, typically by delay.
- Maximum flow (capacity) occurs at an intermediate optimum density; beyond that density, flow decreases (congested branch).
- Speed-density relationship: as density ↑, speed ↓ (inverse relationship).
Key definitions
Flow (Volume)
Example
A lane carries 1800 veh/h during peak hour.
Number of vehicles passing a fixed point per unit time; measured in veh/h (vehicles per hour).
Density (Concentration)
Example
A freeway lane has 25 veh/km during congestion.
Number of vehicles occupying a unit length of roadway at a given instant; measured in veh/km.
Space-Mean Speed
Example
For traffic engineering flow analysis, ALWAYS use space-mean speed.
Harmonic mean of spot speeds; average speed measured as total distance ÷ total travel time; used in q = k·u.
Spacing
Example
If k = 40 veh/km, spacing = 1000/40 = 25 m.
Distance from front of one vehicle to front of the next vehicle; inverse relationship with density.
Headway
Example
If q = 1200 veh/h, average headway = 3600/1200 = 3 seconds.
Time interval between successive vehicles passing a fixed point; inverse relationship with flow.
Peak-Hour Factor (PHF)
Example
PHF = 0.90 means the peak hour is fairly uniformly distributed; PHF = 0.75 means demand is very peaky.
Ratio of peak-hour volume to 4 times the peak 15-minute volume; measure of demand concentration (0.25 to 1.0).
Capacity
Example
A freeway lane has ideal capacity ≈ 2000 pc/h (passenger-car equivalents per hour per lane).
Maximum sustainable flow rate a facility can accommodate under specified conditions (veh/h or pc/h).
Level of Service (LOS)
Example
LOS A: free flow, minimal interaction; LOS F: breakdown, unstable flow.
Qualitative measure of operational conditions (A = free flow to F = forced/breakdown) based on density (freeways) or delay (intersections).
Diagrams to know
- Fundamental traffic flow diagram: flow (q) vs. density (k) curve showing free-flow and congested branches, with capacity peak.
- Speed-density relationship: linear or curved relationship showing how speed decreases with increasing density.
- Headway vs. flow: inverse relationship; as flow increases, average headway decreases.
- Level-of-Service matrix for freeways: density ranges (veh/km/lane) for LOS A through F.
CAPACITY AND LEVEL OF SERVICE (LOS)
Formulas
LOS (freeways) = f(density k)
Meaning
Freeways: LOS determined by density (veh/km/lane); thresholds vary by highway class
Watch out
LOS for freeways is based on density, NOT speed or delay. Don't mix up with urban street LOS (which uses delay).
When to use
Analyzing freeway performance; look up density limits from HCM or local standards
LOS (urban streets/intersections) = f(delay or v/c ratio)
Meaning
Urban: LOS determined by average control delay (s) or volume-to-capacity ratio; varies by area type
Watch out
Urban LOS is delay-based or v/c-based, not density-based. Different metric from freeways.
When to use
Analyzing signalized intersections or urban arterials; use HCM tables
Capacity adjustment: C = C₀ × f_w × f_HV × f_g × ...
Meaning
C = adjusted capacity; C₀ = base capacity; f_i = adjustment factors for width, heavy vehicles, grade, etc.
Watch out
Each factor < 1.0 reduces capacity. Missing a factor or applying it incorrectly is common error.
When to use
When actual conditions differ from ideal (non-standard lane width, steep grades, trucks, etc.)
Common values
Value
7 veh/km/lane
Symbol
k_A
Quantity
LOS A (freeways): max density
Value
11 veh/km/lane
Symbol
k_B
Quantity
LOS B (freeways): max density
Value
16 veh/km/lane
Symbol
k_C
Quantity
LOS C (freeways): max density
Value
25 veh/km/lane
Symbol
k_D
Quantity
LOS D (freeways): max density
Value
30 veh/km/lane
Symbol
k_E
Quantity
LOS E (freeways): density at capacity
Value
2.0–2.5 passenger cars per truck
Symbol
E_T
Quantity
Heavy vehicle equivalency (trucks)
Value
1.5–2.0 passenger cars per RV
Symbol
E_R
Quantity
Heavy vehicle equivalency (RVs)
Important facts
- Freeways use DENSITY (veh/km/lane) as the LOS criterion; urban streets use DELAY or v/c ratio.
- LOS A (free flow) occurs at low density; LOS F (breakdown) at high density; LOS C is typically near-capacity conditions.
- Capacity is NOT the same as LOS E; capacity is at the inflection point; LOS E is slightly below capacity.
- Ideal capacity for a freeway lane ≈ 2000 pc/h under optimal conditions (good weather, no heavy vehicles, flat grade, etc.).
- Any adjustment factor in the capacity formula reduces capacity below ideal; combined effect is multiplicative.
- Level of Service thresholds vary by highway type (freeway, divided highway, undivided, urban arterial, etc.) per HCM.
- Peak-hour factor and adjustment factors for heavy vehicles (f_HV = 1/(1 + P_T(E_T - 1))) must be applied to get design flow.
- In the Philippines, refer to local traffic engineering guidelines or adopt HCM methodology adapted to Philippine conditions.
Key definitions
Level of Service A
Example
Early morning traffic on a wide freeway with light demand.
Free-flow operation; minimal vehicle interaction; high speed, low density.
Level of Service B
Example
Light peak-hour traffic on a freeway.
Reasonably free flow; slight back-pressure; speed and maneuverability slightly restricted.
Level of Service C
Example
Moderate congestion; traffic moving steadily but with restraint.
Stable flow; significant interaction; ability to change lanes restricted; speed approaching free-flow limit.
Level of Service D
Example
Heavy peak-hour traffic; speed still above 50 km/h on freeway.
Approaching unstable flow; high density; frequent lane changes restricted; minor incidents cause disruption.
Level of Service E
Example
Extreme congestion; flow highly volatile; stop-and-go conditions possible.
Unstable operation at or near capacity; any disturbance causes breakdown; very high density.
Level of Service F
Example
Gridlock; breakdown has occurred; flow is unstable.
Forced or breakdown flow; volume exceeds capacity; heavy congestion and stop-and-go traffic.
Diagrams to know
- LOS matrix table: density ranges (veh/km/lane) for LOS A–F on freeways.
- Flow-density diagram annotated with LOS boundaries.
- Capacity adjustment flowchart showing base capacity and sequential application of adjustment factors.
TRAFFIC SIGNAL TIMING & WEBSTER'S EQUATION
Formulas
C_opt = (1.5·L + 5) / (1 - Σ(y_i))
Meaning
C_opt = optimal cycle length (s); L = total lost time per cycle (s); y_i = flow ratio for movement i (q_i/s_i); Σ(y_i) = sum of critical flow ratios
Watch out
Numerator uses 1.5 and 5 (constants from Webster); denominator is (1 - sum of y-values). If Σ(y_i) ≥ 1.0, intersection is oversaturated; no feasible cycle exists.
When to use
Calculate optimal signal cycle length for isolated intersection using Webster's method
y_i = q_i / s_i
Meaning
y_i = flow ratio for movement i; q_i = volume for movement i (veh/h); s_i = saturation flow rate for that movement (veh/h)
Watch out
s_i varies by approach (straight, turn); typical base saturation flow ≈ 1800–2000 veh/h per lane under ideal conditions.
When to use
Determine how much of the saturation capacity is needed for each movement
L = n·l + (n-1)·I
Meaning
L = total lost time per cycle (s); n = number of signal phases; l = yellow + all-red time per phase (s); I = inter-green time (clearance)
Watch out
Lost time includes yellow time, all-red (red-light) intervals, and time to clear intersection between phases. Typical L ≈ 6–8 s for 2-phase, 8–12 s for 4-phase.
When to use
Calculate lost time for a given signal configuration
g_i = (C - L) · (y_i / Σ(y_i))
Meaning
g_i = green time for movement i (s); C = cycle length (s); L = lost time (s); y_i, Σ(y_i) = flow ratios
Watch out
Σ g_i + L = C (check sum of all green times plus lost time = cycle length). Students often forget to account for L.
When to use
Allocate available green time proportionally to each critical movement
Common values
Value
1800–2000 veh/h/lane
Symbol
s₀
Quantity
Base saturation flow rate (one lane, straight movement)
Value
3–4 seconds
Symbol
t_y
Quantity
Yellow time (typical)
Value
1–2 seconds
Symbol
t_ar
Quantity
All-red (typical, urban intersection)
Value
3–5 seconds
Symbol
l
Quantity
Lost time per phase (y + all-red + startup)
Value
6–8 seconds
Symbol
L
Quantity
Total lost time (2-phase signal)
Value
40–120 seconds (typical 50–80 s)
Symbol
C_opt
Quantity
Optimal cycle length range
Important facts
- Webster's method assumes isolated intersection; does not account for coordination with adjacent signals.
- Oversaturation (Σ(y_i) ≥ 1.0) means capacity is insufficient; no feasible signal timing can clear all demand.
- Optimal cycle length is a compromise: too short = high lost time %; too long = long delays.
- Saturation flow is typically reduced from base value for turns, grades, or narrow lanes; adjustment factors apply (s = s₀ · f_w · f_g · f_turn · ...).
- Green time allocation follows weighted distribution: phases with higher demand get proportionally more green.
- All-red (red-light) time is clearance; prevents vehicles from colliding during phase transition.
- Startup loss: drivers don't move instantly; effective green time ≈ 0.5 s less than signal green time.
- For two-phase signal (N-S and E-W), Σ(y_i) = y_NS + y_EW; typical range 0.6–0.95 for good intersection.
Key definitions
Cycle Length
Example
A 60-second cycle has green times distributed among movements, plus lost time.
Total time for one complete cycle of signal indications (all phases); typically 40–120 seconds.
Lost Time
Example
Two phases with 3 s yellow each = 6 s lost time (ignoring startup lag).
Time per cycle during which no useful traffic movement occurs (yellow, all-red, startup lag); typically 3–5 s per phase.
Saturation Flow Rate
Example
Base saturation flow ≈ 1900 veh/h/lane; adjusted for lane width, grade, or turn movements.
Maximum flow (veh/h) that can pass through an intersection movement under ideal conditions (full green, no turning conflicts).
Flow Ratio (y)
Example
y = 500 veh/h / 1900 veh/h ≈ 0.26; higher y means more of the green time is needed.
Ratio of actual volume to saturation flow for a movement; measure of demand relative to capacity.
Critical Flow Ratio
Example
Phase 1 has 2 movements: y = 0.30 and y = 0.25; critical y = 0.30.
For each phase, the movement with the highest y-value determines how much total green time that phase needs.
Diagrams to know
- Signal timing diagram: timeline showing phase sequence, yellow, all-red, and effective green times.
- Phase diagram (2- or 4-way intersection) showing which movements are green in each phase.
- Flow ratio y vs. movement type chart (through, left-turn, right-turn).
TRAFFIC FLOW MODELS & RELATIONSHIPS
Formulas
Green-shield model: u = u_f(1 - k/k_j)
Meaning
u = space-mean speed (km/h); u_f = free-flow speed (km/h); k = density; k_j = jam density (veh/km when u = 0)
Watch out
At k = 0 (no cars), u = u_f. At k = k_j (bumper-to-bumper), u = 0. Model assumes linear relationship; real data often curved.
When to use
Linear speed-density model; simple representation of traffic behavior
Flow from Greenshields: q = k·u = k·u_f(1 - k/k_j)
Meaning
Substituting speed into fundamental relation gives parabolic flow-density curve
Watch out
Maximum flow q_max occurs at k = k_j/2 and u = u_f/2. Capacity is NOT at zero or jam density.
When to use
Find capacity (max flow) by differentiating dq/dk = 0 or by graphing
q_max = (u_f · k_j) / 4
Meaning
Maximum flow (capacity) from Greenshields model; occurs at optimal density k_opt = k_j/2
Watch out
This assumes linear speed-density; real relationships may differ. q_max is one-fourth the product of u_f and k_j.
When to use
Quick estimate of capacity given free-flow speed and jam density
Common values
Value
100–120 km/h
Symbol
u_f
Quantity
Typical free-flow speed (freeway)
Value
130–180 veh/km/lane
Symbol
k_j
Quantity
Typical jam density
Value
k_j/2 ≈ 65–90 veh/km/lane
Symbol
k_opt
Quantity
Optimal density (at capacity)
Value
u_f/2 ≈ 50–60 km/h
Symbol
u_opt
Quantity
Optimal speed (at capacity)
Important facts
- Greenshields assumes linear speed-density; real data often shows curved relationships (nonlinear).
- Capacity (max flow) occurs at mid-point of flow-density curve: k_opt = k_j/2 and u_opt = u_f/2.
- Beyond capacity, traffic enters congested branch: same flow at higher density and lower speed.
- Free-flow branch (low density): speed high, flow increases with density; stable.
- Congested branch (high density): speed low, flow decreases with density; unstable.
- Jam density varies by vehicle type and spacing; typically 130–200 veh/km for cars, lower for trucks.
- Real-world speed-density relationships (Pipes, Van Aerde models) better fit actual data but more complex.
Key definitions
Free-Flow Speed (u_f)
Example
On a freeway with posted 100 km/h limit, free-flow speed ≈ 95–105 km/h under ideal conditions.
Average speed of vehicles when traffic volume is low and drivers operate at desired speed (no congestion).
Jam Density (k_j)
Example
If average vehicle length + minimum gap ≈ 7 m, jam density k_j ≈ 1000/7 ≈ 143 veh/km.
Maximum density when traffic comes to complete stop (bumper-to-bumper); inverse of average spacing at standstill.
Greenshields Model
Example
u = 100(1 - k/150); if k = 75 veh/km, then u = 100(1 - 0.5) = 50 km/h.
Foundational linear model relating speed to density; assumes speed decreases linearly with increasing density.
Diagrams to know
- Fundamental diagram: flow-density curve with free-flow branch (upward), capacity peak, congested branch (downward), and annotations.
- Speed-density diagram: linear or curved relationship u vs. k showing u_f, k_j, and u = 0 at jam.
- Flow-speed diagram: concave curve showing q vs. u; often used in practice.
HEAVY VEHICLE EQUIVALENCY & ADJUSTMENT FACTORS
Formulas
q_e = (q_l + P_H·E_H·q_h) / 100
Meaning
q_e = equivalent passenger-car flow (pc/h); q_l = light vehicle flow; P_H = % heavy vehicles; E_H = equivalency factor; q_h = heavy vehicle flow
Watch out
E_H > 1; typical values E_truck ≈ 2.0–2.5, E_RV ≈ 1.5–2.0. Higher on grades or in curves.
When to use
Convert mixed traffic (cars + trucks + RVs) to passenger-car equivalents for capacity analysis
f_HV = 1 / (1 + P_H(E_H - 1))
Meaning
f_HV = heavy vehicle adjustment factor (< 1); reduces capacity proportional to heavy vehicle percentage
Watch out
If P_H = 0 (no trucks), f_HV = 1.0 (no adjustment). Common exam pitfall: forgetting to apply f_HV.
When to use
Adjust saturation flow or capacity downward when traffic includes significant heavy vehicles
f_g = 1 / (1 + 0.10·(G - 2)) for moderate grade
Meaning
f_g = grade adjustment factor (< 1 for upgrades); G = grade (% positive = upgrade)
Watch out
This formula is approximate; consult HCM for precise tables. Downgrades have less impact; use local guidelines.
When to use
Reduce capacity or saturation flow for uphill grades
Common values
Value
2.0–2.5 pc/truck
Symbol
E_T
Quantity
Truck equivalency (level grade)
Value
3.0–3.5 pc/truck
Symbol
E_T,uphill
Quantity
Truck equivalency (3% upgrade)
Value
1.5–2.0 pc/RV
Symbol
E_R
Quantity
RV equivalency (level)
Value
1.5–2.0 pc/bus
Symbol
E_B
Quantity
Bus equivalency (level)
Value
5–15%
Symbol
P_H
Quantity
Typical heavy vehicle percentage (urban)
Value
15–35%
Symbol
P_H
Quantity
Typical heavy vehicle percentage (intercity)
Important facts
- Heavy vehicles occupy more space, accelerate slower, and reduce overall traffic stream capacity.
- Passenger-car equivalency is a standard way to account for vehicle mix in capacity calculations.
- Equivalency factor increases on grades, curves, and in congestion; higher on upgrades than downgrades.
- f_HV adjustment factor is multiplicative: as % heavy vehicles ↑, f_HV ↓ (capacity decreases).
- In Philippines, heavy vehicles (trucks, buses, UV) are common; must account for them in any capacity analysis.
- Grade effect is largest for trucks on sustained upgrades (>3%); minimal on downgrades or short rises.
- Weather and road condition also affect equivalency; wet or rough surfaces increase E_H.
- Always convert mixed flow to pc/h before comparing to capacity standards (which are in pc/h).
Key definitions
Heavy Vehicle
Example
Buses, dump trucks, container trucks occupy 2–3 times the road space of a sedan.
Vehicle with more than 4 wheels (trucks, buses, RVs); requires more space and acceleration time than cars.
Passenger-Car Equivalent (PCE or pc)
Example
A lane with 1800 veh/h of mixed traffic (80% cars, 20% trucks) ≈ 1800 - 20%(1) + 20%(2.0) = 1800 - 360 + 720 = 2160 pc/h? No: recalculate.
Standardized unit for flow and capacity; one passenger car = 1 pc; one truck ≈ 2–2.5 pc depending on condition.
Equivalency Factor (E_H)
Example
E_truck = 2.0 on level road, E_truck = 3.0–4.0 on 5% upgrade; E_bus ≈ 1.5–2.0.
Number of passenger cars displaced by one heavy vehicle in traffic stream; depends on grade, curve, and vehicle type.
Diagrams to know
- Equivalency factor chart: E_H vs. grade (%) showing increase with uphill slope.
- Heavy vehicle adjustment factor f_HV vs. % heavy vehicles: shows capacity reduction.
- Vehicle type comparison: space occupied by car vs. truck vs. bus illustration.
PRACTICAL CAPACITY ANALYSIS (FREEWAYS & URBAN STREETS)
Formulas
C = C_i × N × f_w × f_HV × f_g × f_p
Meaning
C = capacity (pc/h); C_i = ideal flow (≈2000 pc/h/lane under standard conditions); N = number of lanes; f_i = adjustment factors
Watch out
Each factor < 1.0; combined effect multiplicative. Missing even one factor can cause major error. Order: width, HV, grade, driver population.
When to use
Calculate capacity for a freeway segment or urban arterial under non-ideal conditions
v/c ratio = V / C
Meaning
v/c = volume-to-capacity ratio (dimensionless); indicator of congestion severity and delay
Watch out
v/c > 1.0 means demand > capacity; breakdown occurs, traffic backs up, delay increases exponentially.
When to use
Quick check of whether a facility is under-loaded (v/c < 0.75), near capacity (0.75–0.95), or oversaturated (v/c > 1.0)
Delay (uniform): d = 0.5·C·(1 - g/C)² / (1 - (V/C)·(g/C))
Meaning
d = average delay (s) at signalized intersection; g = green time (s); V = volume; C = capacity
Watch out
Formula assumes uniform arrivals; actual delay higher if platoons arrive mid-red. LOS D, E, F typically have d > 35 s.
When to use
Estimate intersection delay for LOS determination in urban areas (simplified Webster formula)
Common values
Value
2000 pc/h/lane
Symbol
C_i
Quantity
Ideal capacity (per lane, freeway)
Value
f_w = 0.92–1.00
Symbol
f_w
Quantity
Typical lane width adjustment
Value
f_p = 0.85–1.00
Symbol
f_p
Quantity
Typical driver population adjustment
Value
0.75
Symbol
—
Quantity
v/c ratio threshold: acceptable
Value
1.0
Symbol
—
Quantity
v/c ratio threshold: at capacity
Important facts
- Ideal capacity 2000 pc/h/lane assumes: 12-foot (3.7 m) lanes, level grade, all passenger cars, ideal weather, no incidents.
- All real-world capacities are less than ideal; apply downward adjustment factors.
- For multi-lane facilities, total capacity = C_i × N × product of adjustment factors.
- v/c < 0.75: generally acceptable LOS; v/c = 0.75–0.95: approaching capacity, some congestion; v/c > 1.0: breakdown.
- Freeway LOS based on density; urban street LOS based on delay or v/c ratio.
- Delay increases exponentially as v/c approaches 1.0; small increases in volume cause large delay increases near capacity.
- Peak-hour factor and heavy vehicle factors must be applied to convert observed volume to design (pc/h).
- Signal timing optimization (Webster's method) is critical for intersection capacity and delay reduction.
Key definitions
Ideal Capacity (C_i)
Example
C_i ≈ 2000 pc/h/lane; real capacity always less due to adjustment factors.
Maximum flow per lane under ideal conditions: divided freeway, level grade, 100% passenger cars, good weather, no incidents.
Adjusted Capacity
Example
If f_w = 0.95, f_HV = 0.85, f_g = 0.90, then C = 2000 × 3 × 0.95 × 0.85 × 0.90 = 4600 pc/h (3-lane freeway).
Real-world capacity after accounting for lane width, heavy vehicles, grade, driver population, and other factors.
Volume-to-Capacity Ratio (v/c)
Example
v/c = 0.85 means facility is operating at 85% of its limit; some delay and queuing expected.
Ratio of demand flow to facility capacity; measure of congestion severity and reliability.
Diagrams to know
- Capacity calculation flowchart: start with C_i, apply N, then sequentially apply each adjustment factor.
- Delay vs. v/c ratio curve: shows exponential increase in delay near v/c = 1.0.
- LOS thresholds: density ranges (freeways) or delay ranges (urban) for each LOS A–F.
Must remember
- FUNDAMENTAL RELATION: q = k·u (flow = density × SPACE-MEAN speed). This is the backbone of all traffic analysis. Using time-mean speed instead is the #1 exam killer.
- PEAK-HOUR FACTOR: PHF = V / (4·V₁₅). Design flow = V / PHF. Always > V when PHF < 1.0. Numerator is total hourly volume; denominator is 4 times the peak 15-minute count.
- SPACING vs. HEADWAY: Spacing s = 1000/k (meters, uses density). Headway h = 3600/q (seconds, uses flow). Reversing these formulas is a classic board-exam error.
- WEBSTER'S CYCLE LENGTH: C_opt = (1.5·L + 5) / (1 - Σy_i). Oversaturation when Σy_i ≥ 1.0 (no solution exists). Check that Σg_i + L = C.
- GREENSHIELDS MODEL: u = u_f(1 - k/k_j); capacity q_max = (u_f·k_j)/4 occurs at k = k_j/2. Maximum is always at midpoint, not at zero or jam density.
- HEAVY VEHICLE ADJUSTMENT: f_HV = 1/(1 + P_H(E_H - 1)). PCE equivalency E_T ≈ 2.0–2.5 for trucks on level grade; higher on upgrades. Convert all mixed traffic to pc/h.
- LEVEL OF SERVICE: Freeways use DENSITY (veh/km/lane) for LOS; urban streets use DELAY (seconds) or v/c ratio. Do NOT mix these criteria.
- CAPACITY FORMULA: C = C_i × N × f_w × f_HV × f_g × f_p. Each factor < 1.0 reduces capacity. Ideal capacity C_i ≈ 2000 pc/h/lane. Multiplicative effect of all factors.
- v/c RATIO: v/c = V / C. Threshold v/c ≈ 0.75 generally acceptable; 0.75–0.95 approaching capacity; > 1.0 breakdown/oversaturation. Delay increases exponentially near v/c = 1.0.
- UNITS CRITICAL: Flow q in veh/h. Density k in veh/km. Spacing s in meters (use 1000). Headway h in seconds (use 3600). Cycle length & delay in seconds. Mixing units = instant failure.
Last minute tips
- SPACE-MEAN SPEED CHECK: If exam gives spot speeds and asks for flow, ALWAYS use harmonic mean (space-mean speed), not arithmetic mean. This is tested every licensing exam.
- PHF DENOMINATOR: The denominator is 4·V₁₅, not just V₁₅. Four 15-minute periods make one hour. Students commonly forget the 4.
- OVERSAT CHECK: Before solving Webster's equation, quickly add Σy_i. If ≥ 1.0, intersection is oversaturated; no feasible cycle exists. State this; don't force a solution.
- CAPACITY ALWAYS PRODUCT: All adjustment factors multiply together. If you have 3 factors (f_w, f_HV, f_g) = 0.92 × 0.85 × 0.90, multiply all three; do NOT add them.
- SIGN CONVENTION: In Greenshields and delay formulas, be careful with signs and parentheses. A single error in the denominator (1 - k/k_j) vs. (1 + k/k_j) flips the entire result.
Comparison tables
SPACING vs. HEADWAY — Common Exam Confusion
| Aspect | Spacing (s) | Headway (h) |
|---|---|---|
| Definition | Distance between front of successive vehicles | Time between successive vehicles passing a point |
| Units | Meters (m) | Seconds (s) |
| Formula | s = 1000/k | h = 3600/q |
| Depends on | Density k (veh/km); at a specific time | Flow q (veh/h); at a fixed location over time |
| Example | If k = 40 veh/km, then s = 25 m between cars | If q = 1200 veh/h, then h = 3 seconds between cars passing observer |
| Common mistake | Using flow (q) instead of density (k); reverses formula | Using density (k) instead of flow (q); gives wrong time units |
FREEWAY LOS vs. URBAN STREET LOS — Different Criteria
| Criterion | Freeway LOS (HCM) | Urban Street LOS (HCM) |
|---|---|---|
| Measure used | Density (veh/km/lane) | Average delay per vehicle (seconds) or v/c ratio |
| LOS A threshold | k ≤ 7 veh/km/lane | d < 10 s (or v/c < 0.50) |
| LOS C threshold | k ≤ 16 veh/km/lane | d < 20 s (or v/c < 0.70) |
| LOS E threshold | k ≤ 30 veh/km/lane (at capacity, ~2000 pc/h/lane) | d < 35 s (or v/c < 0.85) |
| Typical exam error | Using delay instead of density; wrong LOS result | Using density instead of delay; inappropriate for urban streets |
SPEED-MEAN TYPES — Which to Use When?
| Speed Type | Definition & Calculation | Use in Traffic Flow? | Formula |
|---|---|---|---|
| Space-Mean Speed (u̅_s) | Harmonic mean of spot speeds; average over distance (total distance ÷ total time) | YES – ALWAYS for q = k·u | u̅_s = n / Σ(1/u_i) or total distance / total time |
| Time-Mean Speed (u̅_t) | Arithmetic mean of spot speeds; average over time (sum of speeds ÷ count) | NO – incorrect for q = k·u; results in error | u̅_t = Σ(u_i) / n |
| Relationship | u̅_t > u̅_s (time-mean always exceeds space-mean) | Always use space-mean in flow relationships | u̅_t / u̅_s ≈ 1.05–1.20 (varies with speed variance) |
PEAK-HOUR FACTOR (PHF) INTERPRETATION
| PHF Value | Traffic Pattern | Design Flow Rate vs. Observed V | Implication |
|---|---|---|---|
| PHF = 1.0 | Uniform (all 15-min counts equal) | Design flow = V (no inflation) | Rare; typically not real traffic |
| PHF = 0.90 | Fairly uniform (slight variation) | Design flow = V/0.90 = 1.11·V | Typical urban/suburban; moderate peaking |
| PHF = 0.75 | Very peaky (one dominant 15-min interval) | Design flow = V/0.75 = 1.33·V | Heavy peaking; significant design inflation |
| PHF = 0.25 | Extreme peaking (one 15-min dominates, others near zero) | Design flow = V/0.25 = 4.0·V | Physically impossible in reality; error if calculated |
CAPACITY ADJUSTMENT FACTORS — Quick Reference
| Factor | Symbol | Typical Range | When < 1.0 (reduces capacity) | Example |
|---|---|---|---|---|
| Lane width | f_w | 0.87–1.00 | Narrow lanes (< 3.6 m) or shoulder encroachment | f_w = 0.92 for 3.3 m lanes on urban street |
| Heavy vehicles | f_HV | 0.75–1.00 | High % trucks or RVs (especially on grades) | f_HV = 0.85 with 20% trucks, E_T = 2.5 |
| Grade | f_g | 0.70–1.00 | Sustained uphill grade > 3% | f_g = 0.85 for 4% upgrade |
| Driver population | f_p | 0.85–1.00 | Unfamiliar drivers (tourist season, construction zone) | f_p = 0.85 in unfamiliar area |
| Access point density | f_a | 0.80–1.00 | High driveway/intersection density (urban) | f_a = 0.90 for 8 access points per km |
Diagrams
Traffic Flow Problem-Solving Decision Tree
Decision flowchart for determining which traffic formula to apply based on the given problem statement. Helps students quickly identify whether they need the fundamental relation, spacing/headway, PHF, signal timing, or capacity calculations.
Traffic Engineering Formulas & Relationships Map
Capacity Calculation Step-by-Step Process
Systematic flowchart for calculating real-world capacity, showing each step from ideal capacity through adjustment factors to final verification. Helps students avoid skipping critical steps or applying factors in wrong order.
Traffic Flow States & Transitions (Greenshields Model)
State diagram showing traffic flow evolution from free flow through capacity to breakdown. Illustrates how density, speed, and flow change as demand increases and transitions from stable (free-flow branch) to unstable (congested branch) conditions.
Common Exam Question Distribution (Traffic Engineering)
Approximate distribution of question types in PRC Civil Engineer Licensure Exams for traffic engineering, based on past exam patterns. Shows that capacity, fundamental relation, and LOS are the top tested topics; focus study accordingly.