PE Exam Exam > PE Exam Notes > Civil Engineering (PE Civil) > Cheatsheet: Traffic Flow Theory
Cheatsheet: Traffic Flow Theory
1. Fundamental Traffic Flow Parameters
1.1 Basic Definitions
| Parameter | Definition & Units |
|---|---|
| Flow (q) | Number of vehicles passing a point per unit time; vehicles/hour (veh/h) |
| Density (k) | Number of vehicles per unit length of roadway; vehicles/mile (veh/mi) |
| Speed (v or u) | Distance traveled per unit time; miles/hour (mph) |
| Spacing (s) | Distance between successive vehicles (front bumper to front bumper); feet or miles |
| Headway (h) | Time between successive vehicles passing a point; seconds |
| Time Mean Speed | Arithmetic mean of speeds observed at a point; TMS = Σv/n |
| Space Mean Speed | Harmonic mean of speeds; SMS = n/(Σ(1/v)) = total distance/total travel time |
1.2 Fundamental Relationships
| Equation | Description |
|---|---|
| q = k × v | Flow equals density times space mean speed |
| s = 1/k | Spacing equals reciprocal of density (in consistent units) |
| h = 1/q | Headway equals reciprocal of flow (in consistent units) |
| v = s/h | Speed equals spacing divided by headway |
| SMS = TMS - σ²/TMS | Relationship between space mean speed and time mean speed, where σ² is variance |
- Space mean speed is always less than or equal to time mean speed
- For traffic flow equations, use space mean speed, not time mean speed
- Unit consistency: if k is in veh/mi and v is in mph, then q is in veh/h
2. Speed-Density-Flow Relationships
2.1 Greenshields Model (Linear)
| Equation | Formula |
|---|---|
| Speed-Density | v = vf(1 - k/kj) |
| Flow-Density | q = vf × k(1 - k/kj) |
| Flow-Speed | q = kj × v(1 - v/vf) |
| Maximum Flow | qmax = vf × kj/4 |
| Optimum Density | kopt = kj/2 |
| Optimum Speed | vopt = vf/2 |
- vf = free-flow speed (speed at zero density)
- kj = jam density (density at zero speed)
- qmax = capacity (maximum flow rate)
- Parabolic flow-density curve with maximum at kj/2
2.2 Key Traffic States
| State | Characteristics |
|---|---|
| Free-Flow | Low density, speed ≈ vf, vehicles operate independently |
| Capacity | Maximum flow, k = kj/2, v = vf/2, unstable equilibrium |
| Jam Density | k = kj, v = 0, q = 0, bumper-to-bumper traffic |
| Uncongested | k <>opt, flow increases with density |
| Congested | k > kopt, flow decreases with increasing density |
3. Shockwave Analysis
3.1 Shockwave Fundamentals
| Parameter | Definition |
|---|---|
| Shockwave | Boundary between two different traffic states moving through the traffic stream |
| Wave Speed (vw) | Speed at which the shockwave propagates; vw = (q2 - q1)/(k2 - k1) |
- Positive vw: shockwave moves downstream (in direction of traffic)
- Negative vw: shockwave moves upstream (against traffic direction)
- Zero vw: stationary shockwave
3.2 Shockwave Equation
| Formula | Variables |
|---|---|
| vw = Δq/Δk = (q2 - q1)/(k2 - k1) | q1, k1 = flow and density upstream; q2, k2 = flow and density downstream |
3.3 Common Shockwave Scenarios
| Scenario | Description |
|---|---|
| Queue Formation | Vehicles arriving faster than departing; backward-forming shockwave (negative vw) |
| Queue Discharge | Vehicles departing faster than arriving; forward-recovery shockwave (positive vw) |
| Bottleneck Activation | Capacity drop creates upstream queue and downstream free-flow |
| Incident Clearance | Queue dissipates after incident removal; recovery wave moves upstream |
4. Queuing Theory Applications
4.1 Deterministic Queuing (D/D/1)
| Parameter | Formula |
|---|---|
| Queue Length at Time t | Q(t) = (arrival rate - departure rate) × t = (λ - μ) × t |
| Maximum Queue | Qmax = (λ - μ) × tduration (when λ > μ) |
| Queue Clearance Time | tclear = Qmax/(μ - λ) |
| Total Delay | D = Qmax × tclear/2 (for uniform arrival/departure) |
- λ = arrival rate (veh/h)
- μ = departure/service rate (veh/h)
- Queue forms when λ > μ
- Queue dissipates when μ > λ after demand period ends
4.2 Cumulative Arrival-Departure Curves
- Vertical distance between curves = queue length at that time
- Horizontal distance between curves = delay for individual vehicle
- Area between curves = total vehicle delay
- Slope of arrival curve = arrival rate
- Slope of departure curve = service/discharge rate
4.3 Key Queue Parameters
| Metric | Definition |
|---|---|
| Saturation Flow Rate | Maximum discharge rate from queue; vehicles per hour of green (vphg) |
| Oversaturation | Condition where arrival rate exceeds capacity (λ > μ) for extended period |
| Queue Storage | Available space for queued vehicles; critical for lane blockage analysis |
5. Gap Acceptance and Headway Distribution
5.1 Headway Definitions
| Term | Definition |
|---|---|
| Time Headway | Time interval between passage of successive vehicles at a point (seconds) |
| Space Headway | Distance between successive vehicles (spacing); measured front-to-front (feet) |
| Gap | Time or space between rear of leading vehicle and front of following vehicle |
| Critical Gap (tc) | Minimum time gap that driver will accept for maneuver (crossing, merging) |
| Follow-up Time (tf) | Time between successive vehicles completing same maneuver |
5.2 Headway Distribution Models
| Distribution | Application & Formula |
|---|---|
| Negative Exponential | Random arrivals (low flow); P(h ≥ t) = e-λt where λ = flow rate |
| Shifted Exponential | Accounts for minimum headway (τ); P(h ≥ t) = e-λ(t-τ) for t ≥ τ |
| Erlang | Platoon flow conditions; more uniform than exponential |
5.3 Gap Acceptance Capacity
| Formula | Description |
|---|---|
| C = (3600/tf) × P(h ≥ tc) | Capacity for minor movement with gap acceptance |
| P(h ≥ tc) = e-λtc | Probability of acceptable gap (exponential distribution) |
- C = capacity for minor stream (veh/h)
- tf = follow-up time (seconds)
- tc = critical gap (seconds)
- λ = major stream flow rate (veh/s)
6. Macroscopic vs. Microscopic Models
6.1 Model Classifications
| Type | Characteristics |
|---|---|
| Macroscopic | Aggregate traffic stream; uses flow, density, speed; fluid dynamics analogy; Greenshields model |
| Microscopic | Individual vehicle behavior; car-following, lane-changing; simulation models (VISSIM, CORSIM) |
| Mesoscopic | Hybrid approach; individual vehicles with aggregate relationships; probability-based |
6.2 Car-Following Models (Microscopic)
| Model | Formula/Description |
|---|---|
| Pipes Model | Minimum spacing = L + dv where L = vehicle length, d = reaction distance, v = speed |
| GM Model | an+1(t + T) = α[vn(t) - vn+1(t)]/xn(t) - xn+1(t) |
| Gipps Model | Safe speed approach; acceleration limited by vehicle capabilities and collision avoidance |
- an+1 = acceleration of following vehicle
- T = reaction time
- α = sensitivity coefficient
- vn = speed of leading vehicle
- xn = position of vehicles
7. Level of Service and Capacity
7.1 Level of Service Definitions
| LOS | Density Range (pc/mi/ln) |
|---|---|
| A | 0-11 |
| B | 11-18 |
| C | 18-26 |
| D | 26-35 |
| E | 35-45 |
| F | >45 |
- LOS A: Free-flow operations, no restrictions
- LOS E: At or near capacity, unstable flow
- LOS F: Forced flow, breakdown conditions, demand exceeds capacity
- pc/mi/ln = passenger cars per mile per lane
7.2 Capacity Concepts
| Term | Definition |
|---|---|
| Capacity | Maximum sustainable flow rate; typically 2200-2400 pc/h/ln for freeways |
| Service Flow Rate | Maximum flow rate for specified LOS |
| Volume-to-Capacity (v/c) | Ratio of demand flow to capacity; indicates degree of saturation |
| Peak Hour Factor (PHF) | PHF = hourly volume/(4 × peak 15-min volume); measures flow variation |
8. Traffic Flow Measurement
8.1 Data Collection Methods
| Method | Measures |
|---|---|
| Loop Detectors | Count, occupancy, speed (dual loops); point measurement |
| Radar/Lidar | Speed, volume; point measurement |
| Video Detection | Volume, speed, classification, density; area coverage |
| Probe Vehicles | Travel time, speed; floating car method |
| Pneumatic Tubes | Volume, axle count, classification; temporary installation |
8.2 Key Calculations from Field Data
| Parameter | Calculation Method |
|---|---|
| Occupancy | Occupancy (%) = (time detector occupied/total time) × 100 |
| Density from Occupancy | k = (occupancy × 5280)/(average vehicle length + detector length) |
| Flow Rate | q = count/time interval (convert to hourly rate) |
| Space Mean Speed | SMS = total distance traveled by all vehicles/total travel time |
9. Platoon Analysis
9.1 Platoon Characteristics
| Parameter | Definition |
|---|---|
| Platoon | Group of vehicles traveling together, influenced by signal or slow vehicle |
| Platoon Dispersion | Spreading of platoon as it moves downstream; vehicles travel at different speeds |
| Platoon Ratio | Proportion of vehicles arriving in platoons vs. random arrivals |
9.2 Signal Progression and Platooning
| Concept | Description |
|---|---|
| Bandwidth | Width of green band in time-space diagram allowing continuous flow |
| Offset | Time difference between start of green at successive signals |
| Progression Speed | Optimal speed for vehicles to encounter successive green signals |
- Time-space diagrams show vehicle trajectories and signal timing
- Ideal offset = distance/progression speed
- Platoon dispersion increases with distance from signal
10. Advanced Flow Concepts
10.1 Hysteresis in Traffic Flow
- Different speed-density relationships for accelerating vs. decelerating traffic
- Flow-density curve shows different paths during congestion onset and recovery
- Capacity drop phenomenon: discharge flow from queue less than pre-queue capacity
10.2 Three-Phase Traffic Theory
| Phase | Characteristics |
|---|---|
| Free Flow | Low density, high speed, no interaction between vehicles |
| Synchronized Flow | High density, reduced speed, vehicles move together but not stop-and-go |
| Wide Moving Jam | Stop-and-go waves propagating upstream through traffic |
10.3 Fundamental Diagram Characteristics
- Flow-density curve: parabolic shape with maximum at capacity
- Speed-density curve: linear (Greenshields) or other non-linear forms
- Flow-speed curve: parabolic, two speeds possible for same flow (except at capacity)
- Slope of flow-density curve at any point equals space mean speed
- Maximum slope from origin to flow-density curve represents critical density
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