Formula Sheet: Traffic Flow Theory

Fundamental Traffic Flow Parameters

Basic Definitions

• Flow (q or v): The number of vehicles passing a point per unit time \[q = \frac{n}{t}\] where:
  • \(q\) = flow rate (vehicles/hour or veh/h)
  • \(n\) = number of vehicles
  • \(t\) = time interval (hours)
• Density (k or D): The number of vehicles occupying a given length of roadway at an instant in time \[k = \frac{n}{L}\] where:
  • \(k\) = density (vehicles/mile or veh/mi)
  • \(n\) = number of vehicles
  • \(L\) = length of roadway (miles)
• Speed (u or v): Distance traveled per unit time
  • Time mean speed (TMS): Arithmetic mean of speeds of vehicles passing a point \[\bar{u}_t = \frac{\sum_{i=1}^{n} u_i}{n}\]
  • Space mean speed (SMS): Harmonic mean of speeds over a section of roadway \[\bar{u}_s = \frac{n}{\sum_{i=1}^{n} \frac{1}{u_i}} = \frac{L}{\sum_{i=1}^{n} t_i / n}\]
  where:
  • \(u_i\) = speed of vehicle \(i\) (mph or mi/h)
  • \(n\) = number of vehicles
  • \(L\) = length of roadway section
  • \(t_i\) = travel time of vehicle \(i\)

Relationship Between Time Mean Speed and Space Mean Speed

\[\bar{u}_t = \bar{u}_s + \frac{\sigma_s^2}{\bar{u}_s}\] where:

• \(\bar{u}_t\) = time mean speed (mph)
• \(\bar{u}_s\) = space mean speed (mph)
• \(\sigma_s^2\) = variance of space mean speed (mph²)

Note: Time mean speed is always greater than or equal to space mean speed.

Fundamental Traffic Flow Equation

\[q = k \cdot u_s\] where:

• \(q\) = flow rate (veh/h)
• \(k\) = density (veh/mi)
• \(u_s\) = space mean speed (mph)

Note: This is the most fundamental relationship in traffic flow theory, relating the three primary traffic flow parameters.

Spacing and Headway

Spacing

Spacing (s): Distance between successive vehicles, measured from a common reference point (e.g., front bumper to front bumper) \[s = \frac{1}{k}\] where:

• \(s\) = average spacing (ft/veh or mi/veh)
• \(k\) = density (veh/mi)

Conversion: If spacing is in feet: \[s_{(ft)} = \frac{5280}{k}\]

Headway

Time headway (h): Time between successive vehicles passing a point \[h = \frac{1}{q}\] where:

• \(h\) = average time headway (hours/veh or seconds/veh)
• \(q\) = flow rate (veh/h)

Conversion: If headway is in seconds: \[h_{(sec)} = \frac{3600}{q}\]

Relationship Between Spacing and Headway

\[s = u_s \cdot h\] where:

• \(s\) = spacing (ft)
• \(u_s\) = space mean speed (ft/sec)
• \(h\) = time headway (sec)

Speed-Density-Flow Relationships

Greenshields Model (Linear Speed-Density)

Speed-Density Relationship: \[u = u_f \left(1 - \frac{k}{k_j}\right)\] where:

• \(u\) = speed at density \(k\) (mph)
• \(u_f\) = free-flow speed (mph) - speed at zero density
• \(k\) = density (veh/mi)
• \(k_j\) = jam density (veh/mi) - density at zero speed

Flow-Density Relationship: \[q = u_f k \left(1 - \frac{k}{k_j}\right)\] Flow-Speed Relationship: \[q = k_j u \left(1 - \frac{u}{u_f}\right)\] Maximum Flow (Capacity): \[q_{max} = \frac{u_f k_j}{4}\] occurring at:

• Optimal density: \(k_{opt} = \frac{k_j}{2}\)
• Optimal speed: \(u_{opt} = \frac{u_f}{2}\)

Greenberg Model (Logarithmic Speed-Density)

\[u = u_{opt} \ln\left(\frac{k_j}{k}\right)\] where:

• \(u\) = speed (mph)
• \(u_{opt}\) = speed at maximum flow (mph)
• \(k_j\) = jam density (veh/mi)
• \(k\) = density (veh/mi)

Note: This model is better suited for congested conditions.

Underwood Model (Exponential Speed-Density)

\[u = u_f e^{-k/k_{opt}}\] where:

• \(u\) = speed (mph)
• \(u_f\) = free-flow speed (mph)
• \(k\) = density (veh/mi)
• \(k_{opt}\) = optimal density at capacity (veh/mi)

Note: This model is better suited for uncongested conditions.

Shock Wave Analysis

Shock Wave Speed

Shock wave: A boundary between two different traffic states that propagates through traffic \[u_w = \frac{q_2 - q_1}{k_2 - k_1}\] where:

• \(u_w\) = shock wave speed (mph)
• \(q_1, q_2\) = flow rates in states 1 and 2 (veh/h)
• \(k_1, k_2\) = densities in states 1 and 2 (veh/mi)

Alternative form using fundamental equation: \[u_w = \frac{k_2 u_2 - k_1 u_1}{k_2 - k_1}\] Sign convention:

• Positive \(u_w\): shock wave moves downstream (in direction of traffic)
• Negative \(u_w\): shock wave moves upstream (against traffic direction)

Queue Formation and Dissipation

Queue length at time t: \[L_q(t) = -u_w \cdot t\] where:

• \(L_q(t)\) = queue length at time \(t\) (miles)
• \(u_w\) = shock wave speed (negative for upstream movement) (mph)
• \(t\) = time (hours)

Number of vehicles in queue: \[N_q = (k_2 - k_1) \cdot L_q\] where:

• \(N_q\) = number of vehicles in queue
• \(k_2\) = density in queued region (veh/mi)
• \(k_1\) = density upstream of queue (veh/mi)
• \(L_q\) = queue length (miles)

Gap and Gap Acceptance

Gap Definitions

Gap (g): Time interval between successive vehicles in the same lane

• Same as time headway
• Units: seconds

Lag (l): Time interval from when an observer begins measuring to when the next vehicle arrives Critical gap (t_c): Minimum time interval in the major stream that allows one minor stream vehicle to make a maneuver

Gap Distribution

Negative exponential distribution (random arrivals): \[P(h > t) = e^{-\lambda t}\] where:

• \(P(h > t)\) = probability that headway exceeds time \(t\)
• \(\lambda\) = average arrival rate (veh/sec)
• \(t\) = time interval (sec)

Probability of acceptable gap: \[P_{accept} = e^{-q \cdot t_c / 3600}\] where:

• \(P_{accept}\) = probability of an acceptable gap
• \(q\) = flow rate in major stream (veh/h)
• \(t_c\) = critical gap (sec)

Traffic Stream Models

Poisson Distribution (Random Arrivals)

Probability of exactly n vehicles arriving in time t: \[P(n) = \frac{(\lambda t)^n e^{-\lambda t}}{n!}\] where:

• \(P(n)\) = probability of \(n\) vehicles arriving
• \(\lambda\) = average arrival rate (veh/sec)
• \(t\) = time interval (sec)
• \(n\) = number of vehicles

Mean and variance:

• Mean = \(\lambda t\)
• Variance = \(\lambda t\)

Vehicle Following Models

Simplified car-following model: \[a_n(t + \Delta t) = \alpha \left[\frac{v_n(t) - v_{n-1}(t)}{x_{n-1}(t) - x_n(t)}\right]\] where:

• \(a_n(t + \Delta t)\) = acceleration of following vehicle \(n\) at time \(t + \Delta t\)
• \(\alpha\) = sensitivity coefficient
• \(v_n(t)\) = speed of vehicle \(n\) at time \(t\)
• \(v_{n-1}(t)\) = speed of leading vehicle at time \(t\)
• \(x_{n-1}(t) - x_n(t)\) = spacing between vehicles
• \(\Delta t\) = reaction time

Delay and Travel Time

Types of Delay

Total delay: \[d_{total} = t_{actual} - t_{ideal}\] where:

• \(d_{total}\) = total delay (sec)
• \(t_{actual}\) = actual travel time (sec)
• \(t_{ideal}\) = ideal travel time at free-flow speed (sec)

Stopped time delay: Time vehicle is completely stopped Time-in-queue delay: Time from when vehicle joins queue to when it departs

Queue Delay (D/D/1 Queue)

For undersaturated conditions (arrival rate < service=""> \[d_{avg} = \frac{\lambda}{2\mu(\mu - \lambda)}\] where:

• \(d_{avg}\) = average delay per vehicle (time units)
• \(\lambda\) = arrival rate (veh/time)
• \(\mu\) = service rate (veh/time)

Deterministic Queue Analysis

Maximum queue length (uniform arrivals and departures): \[Q_{max} = \frac{(\lambda - \mu) \cdot T \cdot \mu}{\lambda}\] where \(\lambda > \mu\) during interval \(T\) Total vehicle delay (deterministic): \[D_{total} = \frac{Q_{max}^2}{2(\lambda - \mu)}\]

Level of Service and Capacity

Service Flow Rate

Maximum service flow rate for a given LOS: \[SF_i = c \cdot (v/c)_i\] where:

• \(SF_i\) = service flow rate for LOS \(i\) (veh/h)
• \(c\) = capacity (veh/h)
• \((v/c)_i\) = volume-to-capacity ratio threshold for LOS \(i\)

Density-Based LOS Criteria

Density is the primary measure for freeways LOS thresholds are typically defined by density ranges (veh/mi/ln)

Platoon Flow and Dispersion

Platoon Characteristics

Platoon ratio: \[R_p = \frac{\text{Number of vehicles in platoons}}{\text{Total number of vehicles}}\] Percent followers: Percentage of vehicles traveling in platoons (headway < threshold,="" typically="" 3="" seconds)="">

Platoon Dispersion

Pacey's platoon dispersion model: Describes how platoons spread out as they travel downstream The distribution of arrival times changes from concentrated to dispersed Travel time distribution variance increases with distance: \[\sigma_t^2(x) = \sigma_{t0}^2 + \beta x\] where:

• \(\sigma_t^2(x)\) = variance of travel time at distance \(x\)
• \(\sigma_{t0}^2\) = initial variance
• \(\beta\) = dispersion coefficient
• \(x\) = distance from origin

Queuing Theory Applications

M/M/1 Queue (Poisson arrivals, exponential service, 1 server)

Average number of vehicles in system: \[L = \frac{\rho}{1 - \rho} = \frac{\lambda}{\mu - \lambda}\] Average number in queue: \[L_q = \frac{\rho^2}{1 - \rho} = \frac{\lambda^2}{\mu(\mu - \lambda)}\] Average time in system: \[W = \frac{1}{\mu - \lambda}\] Average time in queue: \[W_q = \frac{\lambda}{\mu(\mu - \lambda)}\] Utilization factor: \[]\rho = \frac{\lambda}{\mu}\] where:

• \(\lambda\) = arrival rate (veh/time)
• \(\mu\) = service rate (veh/time)
• \(\rho\) = utilization factor (must be < 1="" for="">
• \(L\) = average number of vehicles in system
• \(L_q\) = average number in queue
• \(W\) = average time in system
• \(W_q\) = average waiting time in queue

M/M/N Queue (N servers)

Probability of zero vehicles in system: \[P_0 = \left[\sum_{n=0}^{N-1} \frac{(N\rho)^n}{n!} + \frac{(N\rho)^N}{N!(1-\rho)}\right]^{-1}\] Average number in queue: \[L_q = \frac{P_0 (N\rho)^N \rho}{N!(1-\rho)^2}\] where:

• \(N\) = number of servers (lanes)
• \(\rho = \frac{\lambda}{N\mu}\) = utilization per server

Flow Breakdown and Capacity

Capacity Drop Phenomenon

Capacity drop: When flow breaks down from free-flow to congested conditions, the maximum flow in congested state is typically lower than pre-breakdown capacity \[q_{congested} \approx 0.85 \text{ to } 0.95 \times q_{capacity}\]

Two-Capacity Phenomenon

• Pre-queue discharge capacity: Maximum flow before breakdown
• Queue discharge capacity: Maximum flow from standing queue (typically 5-15% lower)

Cumulative Flow Curves

Cumulative Vehicle Count (N-curve)

\[N(t) = \int_0^t q(\tau) \, d\tau\] where:

• \(N(t)\) = cumulative number of vehicles passing a point by time \(t\)
• \(q(\tau)\) = flow rate at time \(\tau\) (veh/h)

Properties:

• Slope of N-curve = instantaneous flow rate
• Horizontal difference between arrival and departure curves = delay
• Vertical difference between curves = queue length (vehicles)

Vehicle Delay from Cumulative Curves

Total vehicle delay: \[D_{total} = \int_{t_1}^{t_2} [N_A(t) - N_D(t)] \, dt\] where:

• \(N_A(t)\) = cumulative arrival curve
• \(N_D(t)\) = cumulative departure curve
• \(t_1, t_2\) = start and end times

Average delay per vehicle: \[d_{avg} = \frac{D_{total}}{N_{total}}\]

Traffic Stream Variability

Coefficient of Variation

\[CV = \frac{\sigma}{\mu}\] where:

• \(CV\) = coefficient of variation (dimensionless)
• \(\sigma\) = standard deviation
• \(\mu\) = mean

Application: Measures relative variability in headways, speeds, or gaps

Variance-to-Mean Ratio

\[VMR = \frac{\sigma^2}{\mu}\] Interpretation:

• \(VMR = 1\): Random (Poisson) arrivals
• \(VMR < 1\):="" more="" uniform="" than="">
• \(VMR > 1\): More clustered than random (platooned)

Lane Distribution and Utilization

Lane Flow Distribution

Lane utilization factor: \[f_i = \frac{q_i}{q_{avg}}\] where:

• \(f_i\) = utilization factor for lane \(i\)
• \(q_i\) = flow in lane \(i\) (veh/h)
• \(q_{avg}\) = average flow per lane (veh/h)

Effective number of lanes: \[N_{eff} = \frac{q_{total}}{q_{max,lane}}\] where:

• \(N_{eff}\) = effective number of lanes
• \(q_{total}\) = total flow across all lanes (veh/h)
• \(q_{max,lane}\) = maximum flow in any single lane (veh/h)

Kinematic Wave Theory

Wave Propagation Speed

\[u_w = \frac{dq}{dk}\] where:

• \(u_w\) = kinematic wave speed (mph)
• \(q\) = flow (veh/h)
• \(k\) = density (veh/mi)

For Greenshields model: \[u_w = u_f \left(1 - \frac{2k}{k_j}\right)\] At capacity: \[u_w = 0\] (wave is stationary)

Merging and Weaving

Merge Capacity

Total capacity at merge: \[c_{merge} = c_1 + c_2 - I\] where:

• \(c_{merge}\) = merge capacity (veh/h)
• \(c_1, c_2\) = capacities of merging streams (veh/h)
• \(I\) = interaction factor/capacity loss (veh/h)

Weaving Section Analysis

Weaving intensity: \[W = \frac{v_{weaving}}{v_{total}}\] where:

• \(W\) = weaving intensity (dimensionless)
• \(v_{weaving}\) = volume of weaving vehicles (veh/h)
• \(v_{total}\) = total volume in weaving section (veh/h)

Volume ratio: \[VR = \frac{v_{major}}{v_{total}}\] where \(v_{major}\) is the larger of the two directional flows