3.3 Signalized Intersection Capacity & Delay Modeling (HCM)

Signalized Intersection Capacity & Delay Modeling (HCM)

Signalized intersections represent the primary form of interrupted flow in urban traffic systems. Capacity and Level of Service are evaluated on a lane group basis—grouping lanes with shared vehicular movements (e.g., exclusive left-turn lane, exclusive right-turn lane, or combined through-and-right lanes).

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Base Saturation Flow Rate ($s_0$) and Adjustment Factors

The saturation flow rate ($s$) represents the maximum equivalent hourly rate at which vehicles can pass through a signalized lane group during continuous green signal indication, assuming a constant queue of vehicles.

• Standard Base Saturation Flow Rate: $s_0 = 1,900\text{ pc/h/ln}$ (passenger cars per hour of green per lane under ideal conditions).

Full 11-Factor HCM Saturation Flow Formula

$$s = s_0 \times N \times f_w \times f_{HV} \times f_g \times f_p \times f_{bb} \times f_a \times f_{LU} \times f_{LT} \times f_{RT} \times f_{Lpb} \times f_{Rpb}$$

Where:

• $N$ = number of lanes in the lane group.
• $f_w$ = Lane width adjustment factor: $$f_w = 1 + \frac{W - 12}{30} \quad [\text{for lane width } W \ge 8.0\text{ ft}]$$ (For standard 12-ft lanes, $f_w = 1.00$; for 10-ft lanes, $f_w = 0.933$; for 14-ft lanes, $f_w = 1.067$).
• $f_{HV}$ = Heavy-vehicle adjustment factor: $$f_{HV} = \frac{100}{100 + \%HV(E_T - 1)} = \frac{1}{1 + P_{HV}(E_T - 1)}$$ (Using $E_T = 2.0\text{ passenger car equivalents}$ per heavy vehicle).
• $f_g$ = Approach grade adjustment factor: $$f_g = 1 - \frac{\%G}{200}$$ (Where $\%G$ is approach grade; $+4\%\text{ uphill} \implies f_g = 0.98$; $-4\%\text{ downhill} \implies f_g = 1.02$).
• $f_p$ = On-street parking adjustment factor: $$f_p = \frac{N - 0.1 - \frac{18 N_m}{3,600}}{N}$$ ($N_m$ = number of parking maneuvers per hour; with no parking, $f_p = 1.00$).
• $f_{bb}$ = Bus blockage adjustment factor: $$f_{bb} = \frac{N - \frac{14.4 N_B}{3,600}}{N}$$ ($N_B$ = number of local transit buses stopping within 250 ft of the stop line per hour).
• $f_a$ = Area type adjustment factor: $0.90$ for Central Business Districts (CBD); $1.00$ for all other non-CBD areas.
• $f_{LU}$ = Lane utilization adjustment factor: Accounts for unequal lane loading in multi-lane groups ($1.00$ for 1 lane; $0.952$ for 2 lanes; $0.908$ for 3 lanes).
• $f_{LT}$ = Left-turn adjustment factor: $0.95$ for protected exclusive left-turn lanes; complex empirical equations for permitted/shared phases.
• $f_{RT}$ = Right-turn adjustment factor: $0.85$ for protected exclusive right-turn lanes; $1.00 - 0.15 P_{RT}$ for shared lanes.
• $f_{Lpb}, f_{Rpb}$ = Pedestrian/bicycle conflict adjustment factors for turning movements across crosswalks.

Lane Group Capacity, Green Ratios, and Degree of Saturation

Effective Green Time ($g$)

Effective green time represents the actual duration of service provided to a movement during each signal cycle:

$$g = G + Y + R_c - t_L$$

Where:

• $G$ = displayed green interval (s).
• $Y$ = yellow change interval (s).
• $R_c$ = red clearance (all-red) interval (s).
• $t_L$ = total lost time per phase $= l_1 + l_2$ (typically $l_1 = 2.0\text{ s}$ start-up lost time and $l_2 = 2.0\text{ s}$ clearance lost time; if $Y + R_c = t_L$, then $g = G$).

Lane Group Capacity ($c$)

Capacity ($c$) is the maximum hourly volume the lane group can service:

$$c = s \times \left( \frac{g}{C} \right)$$

Where $C$ is the total cycle length in seconds, and $g/C$ is the green ratio.

Degree of Saturation / Volume-to-Capacity Ratio ($X$)

$$X = \frac{v}{c} = \frac{v}{s \times (g/C)} = \frac{v \times C}{s \times g}$$

Where $v$ is the peak 15-minute demand flow rate in veh/h ($v = V / PHF$).

Critical Movement Analysis ($X_c$)

For an entire intersection operating under multi-phase control, the critical degree of saturation ($X_c$) determines overall system adequacy:

$$X_c = \frac{\sum (v/s)_{ci} \times C}{C - L} = \frac{\sum y_{ci}}{1 - (L/C)}$$

Where:

• $\sum (v/s)_{ci} = \sum y_{ci}$ = sum of flow ratios for critical lane groups across all signal phases.
• $L = \sum t_{Li}$ = total lost time per cycle across all critical phases.
• If $X_c > 1.00$, the intersection is geometrically and temporally deficient regardless of how green time is allocated.

Control Delay Modeling ($d = d_1 \times PF + d_2 + d_3$)

The primary measure of effectiveness for signalized intersections is average control delay ($d$) in seconds per vehicle ($\text{s/veh}$):

$$d = d_1 \times PF + d_2 + d_3$$

1. Uniform Delay ($d_1$)

Uniform delay assumes perfectly uniform, deterministic vehicle arrivals throughout the cycle:

$$d_1 = \frac{0.5 \, C \left( 1 - \frac{g}{C} \right)^2}{1 - \left[ \min(1, X) \times \frac{g}{C} \right]}$$

2. Progression Adjustment Factor ($PF$)

Progression modifies uniform delay based on arrival platoon quality:

$$PF = \frac{(1 - P) f_{PA}}{1 - (g/C)}$$

Where $P$ is the proportion of vehicles arriving on green, classified into six Arrival Types (AT 1 to AT 6):

• AT 1: Dense platoon arriving during red (worst progression; $PF > 1.0$).
• AT 3: Random, uncoordinated isolated arrivals ($P = g/C \implies PF = 1.00$).
• AT 5: Dense platoon arriving at the onset of green (favorable progression; $PF \approx 0.40 - 0.70$).
• AT 6: Exceptional progression with advanced ITS / dynamic green band ($P \approx 1.0$).

3. Incremental Delay ($d_2$)

Accounts for random Poisson arrival fluctuations, individual cycle failures, and sustained oversaturation queues:

$$d_2 = 900 T \left[ (X - 1) + \sqrt{(X - 1)^2 + \frac{8 k I X}{c T}} \right]$$

Where:

• $T$ = analysis period duration in hours ($0.25\text{ h}$ for a 15-minute analysis).
• $k$ = controller actuation parameter ($k = 0.50$ for fixed-time pretimed signals; $0.04$ to $0.50$ for fully-actuated controllers).
• $I$ = upstream filtering/metering adjustment factor ($I = 1.00$ for isolated signals; $< 1.0$ for coordinated arterials with upstream signals).
• $c$ = lane group capacity (veh/h).
• $X$ = volume-to-capacity ratio ($v/c$).

4. Initial Queue Delay ($d_3$)

Accounts for pre-existing queues ($Q_b$) unmet from the prior analysis period carried over into the current period.

Level of Service (LOS) Criteria for Signalized Intersections

Unlike unsignalized intersections (where LOS F begins at $> 50.0\text{ s/veh}$), signalized intersections have a higher tolerance threshold due to expected cycle lengths. LOS F begins at $> 80.0\text{ s/veh}$ or when $v/c > 1.00$.

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Numerical Step-by-Step Problem: Signalized Capacity & Delay Calculation

Given Data

• Single isolated through lane group ($N = 1$)
• Cycle length $C = 90.0\text{ s}$, Effective green $g = 45.0\text{ s}$ ($g/C = 0.50$)
• Demand volume $V = 720\text{ veh/h}$, $PHF = 0.90 \implies v = 800\text{ veh/h}$
• Pretimed controller ($k = 0.50$, $I = 1.00$, $T = 0.25\text{ h}$, $PF = 1.00$, $d_3 = 0.0$)
• Adjusted saturation flow rate $s = 1,800\text{ veh/h/ln}$

Step 1: Compute Capacity ($c$)

$$c = s \times \left( \frac{g}{C} \right) = 1,800 \times 0.50 = 900\text{ veh/h}$$

Step 2: Compute Volume-to-Capacity Ratio ($X$)

$$X = \frac{v}{c} = \frac{800}{900} = 0.8889$$

Step 3: Compute Uniform Delay ($d_1$)

$$d_1 = \frac{0.5 \times 90 \times (1 - 0.50)^2}{1 - (0.8889 \times 0.50)} = \frac{45 \times 0.25}{1 - 0.4444} = \frac{11.25}{0.5556} = 20.25\text{ s/veh}$$

Step 4: Compute Incremental Delay ($d_2$)

$$d_2 = 900 \times 0.25 \left[ (0.8889 - 1) + \sqrt{(0.8889 - 1)^2 + \frac{8 \times 0.50 \times 1.00 \times 0.8889}{900 \times 0.25}} \right]$$ $$d_2 = 225 \left[ -0.1111 + \sqrt{(-0.1111)^2 + \frac{3.5556}{225}} \right] = 225 \left[ -0.1111 + \sqrt{0.01234 + 0.01580} \right]$$ $$d_2 = 225 \left[ -0.1111 + \sqrt{0.02814} \right] = 225 \left[ -0.1111 + 0.1678 \right] = 225 \times 0.0567 = 12.76\text{ s/veh}$$

Step 5: Total Delay and Level of Service

$$d = d_1(PF) + d_2 + d_3 = 20.25(1.00) + 12.76 + 0 = 33.01\text{ s/veh}$$

Since $20.0 < d \le 35.0\text{ s/veh}$, the lane group operates at LOS C.