4.1 Unsignalized Intersections (TWSC, AWSC) & Modern Roundabouts Capacity

Unsignalized Intersections & Modern Roundabouts Capacity

Unsignalized intersections—encompassing Two-Way Stop-Controlled (TWSC) intersections, All-Way Stop-Controlled (AWSC) intersections, and Modern Roundabouts—rely on driver gap acceptance, spatial yielding behavior, and right-of-way rules rather than fixed or actuated signal phases. Analyzing their operational performance under the Highway Capacity Manual (HCM 7th Edition) requires understanding conflicting traffic streams, critical headways, follow-up times, and geometric deflection parameters.

1. Two-Way Stop-Controlled (TWSC) Intersections

At a TWSC intersection, major-street through and right-turning vehicles proceed unimpeded, while major-street left turns and all minor-street movements must yield right-of-way to conflicting traffic streams. The capacity of each subordinate movement depends strictly on the availability of acceptable time gaps in the conflicting traffic stream ($v_c$).

+-----------------------------------------------------------------------------+
|                   TWSC GAP ACCEPTANCE PARAMETERS (HCM)                      |
+-----------------------------------------------------------------------------+
| Parameter                  | Definition                                     |
+----------------------------+------------------------------------------------+
| Critical Headway (t_c)     | Minimum time interval in conflicting stream    |
|                            | required for one minor vehicle to enter/cross. |
| Follow-Up Headway (t_f)    | Time headway between departure of successive   |
|                            | queued minor vehicles entering the SAME gap.   |
| Conflicting Flow (v_c)     | Total hourly equivalent flow rate of higher-   |
|                            | priority vehicular streams crossing the path.  |
| Potential Capacity (c_p)   | Maximum theoretical capacity of minor movement |
|                            | assuming zero blockage from other minor queues.|
+-----------------------------------------------------------------------------+

Potential Capacity Formulation ($c_p$)

Under HCM gap acceptance theory assuming Poisson-distributed arrivals in the conflicting stream, the potential capacity ($c_{p,x}$) in passenger cars per hour (pc/h) for subject movement $x$ is computed as:

$c_{p,x} = v_{c,x} \cdot \frac{\exp\left(-\frac{v_{c,x} \cdot t_{c,x}}{3600}\right)}{1 - \exp\left(-\frac{v_{c,x} \cdot t_{f,x}}{3600}\right)}$

Where:

• $v_{c,x}$ = conflicting flow rate for movement $x$ (veh/h or pc/h).
• $t_{c,x}$ = critical headway for movement $x$ (seconds).
• $t_{f,x}$ = follow-up headway for movement $x$ (seconds).

Base Critical ($t_{c,base}$) and Follow-Up ($t_{f,base}$) Headways (HCM Values)

Critical Headway Adjustments

The base critical headway is adjusted for site-specific conditions:

$t_{c,x} = t_{c,base} + t_{c,HV} \cdot P_{HV} + t_{c,G} \cdot G - t_{c,T} - t_{3,LT}$

Where:

• $t_{c,HV}$ = adjustment factor for heavy vehicles (typically $1.0\text{ s}$ for multilane, $2.0\text{ s}$ for 2-lane).
• $P_{HV}$ = proportion of heavy vehicles in subject movement.
• $t_{c,G}$ = adjustment factor for minor-street approach grade ($0.1\text{ s}$ per $1%$ upgrade for minor left/through; $0.2\text{ s}$ per $1%$ upgrade for minor right).
• $G$ = percent grade divided by 100.
• $t_{c,T}$ = adjustment factor for T-intersection geometry ($0.7\text{ s}$ reduction for minor-street left turn at 3-leg intersection).
• $t_{3,LT}$ = adjustment for two-stage gap acceptance if a wide raised median is present ($0.7\text{ s}$ reduction).

Movement Hierarchy, Conflicting Streams, & Impedance

Right-of-way priority at a TWSC intersection follows a strict four-tier hierarchical ranking system. Movements with higher numerical rank must yield right-of-way to all movements of lower numerical rank.

+-----------------------------------------------------------------------------+
|                     TWSC MOVEMENT HIERARCHICAL RANKS                        |
+-----------------------------------------------------------------------------+
| Rank 1 (Highest Priority)  | Major-Street Through Movements                 |
|                            | Major-Street Right-Turn Movements              |
+----------------------------+------------------------------------------------+
| Rank 2                     | Major-Street Left-Turn Movements               |
|                            | Minor-Street Right-Turn Movements              |
+----------------------------+------------------------------------------------+
| Rank 3                     | Minor-Street Through Movements                 |
+----------------------------+------------------------------------------------+
| Rank 4 (Lowest Priority)   | Minor-Street Left-Turn Movements               |
+----------------------------+------------------------------------------------+

Conflicting Flow Rates ($v_{c,x}$)

Each movement encounters a distinct set of conflicting movements:

• Major Left Turn (Rank 2): Conflicts with opposing major through flow ($v_{T,opp}$) plus opposing major right-turn flow ($v_{R,opp}$, if right turns merge into same lane).
• Minor Right Turn (Rank 2): Conflicts with major through flow from the left ($v_{T,left}$) and major right turns from the left.
• Minor Through (Rank 3): Conflicts with both major through streams ($v_{T1} + v_{T2}$), major right turns from the left, major left turns from both directions, and opposing minor right turns.
• Minor Left Turn (Rank 4): Conflicts with all major through, right, and left movements, plus opposing minor through and minor right movements!

Impedance & Movement Capacity ($c_{m,x}$)

When a higher-priority minor movement (such as a major left turn or minor through) develops a queue, it obstructs lower-priority movements (such as minor left turns), even if gaps are available in the major through stream. This is called impedance.

The unblocked probability ($p_{0,j}$) for movement $j$ is:

$p_{0,j} = 1 - \frac{v_j}{c_{m,j}}$

Where $v_j$ is the demand volume and $c_{m,j}$ is movement capacity. The movement capacity for Rank 4 (minor left turn) is calculated by multiplying potential capacity by the product of the unblocked probabilities of all higher-ranking conflicting minor movements:

$c_{m,Rank4} = c_{p,Rank4} \cdot p_{0,MajLT1} \cdot p_{0,MajLT2} \cdot p_{0,MinTH}$

Shared Lane Capacity (Harders Formulation)

When multiple minor movements (e.g., left, through, right) share a single approach lane, the shared-lane capacity ($c_{SH}$) is governed by the harmonic mean:

$c_{SH} = \frac{\sum v_i}{\sum \left( \frac{v_i}{c_{m,i}} \right)}$

Where $v_i$ is the volume of movement $i$ and $c_{m,i}$ is the movement capacity of movement $i$.

2. All-Way Stop-Controlled (AWSC) Intersections

At an AWSC intersection, every entering vehicle must come to a complete stop. Right-of-way is governed by arrival order (first-to-stop, first-to-go) and clockwise priority rules for simultaneous arrivals.

AWSC Departure Headway & The 5 Service Regimes

Under HCM 7th Edition, departure headway ($h_d$) is modeled using an iterative 5-regime probability model that evaluates the interaction of vehicles queued on conflicting, opposing, and subject approaches:

+-----------------------------------------------------------------------------+
|                   AWSC FIVE SERVICE OPERATIONAL REGIMES                     |
+-----------------------------------------------------------------------------+
| Regime 1 | Isolated departure: No vehicles present on opposing or cross     |
|          | approaches (minimum departure headway, highest service rate).    |
| Regime 2 | Vehicle present on other approaches, but not queued.             |
| Regime 3 | Vehicles queued on minor/conflicting cross approaches.           |
| Regime 4 | Vehicles queued on opposing major approach.                      |
| Regime 5 | All four approaches simultaneously queued (full saturation /     |
|          | maximum departure headway ~ 7.0 to 9.0 s/veh).                   |
+-----------------------------------------------------------------------------+

The service time ($t_s$) for an approach is calculated as:

$t_s = h_d - t_m$

Where $h_d$ is the weighted departure headway across all 5 regimes and $t_m$ is the move-up time in the queue (typically $2.0\text{ to }2.3\text{ s}$). Approach capacity is $c = 3600 / h_d$.

3. Modern Roundabouts Capacity & Operational Formulations

Modern roundabouts operate on three fundamental principles:

1. Yield-at-Entry (Off-side priority): Entering vehicles must yield right-of-way to circulating traffic already in the roundabout.
2. Geometric Deflection: Vehicle paths are deflected around a central island to constrain entry and circulating speeds to $15\text{–}25\text{ mph}$.
3. Flared / Designated Entry Lanes: Multi-lane entries channel movements to appropriate circulatory lanes without weaving.

HCM 7th Edition / NCHRP Report 572 Entry Capacity Model

Roundabout entry capacity ($c_e$) is an exponential function of the circulating flow rate ($v_c$) conflicting with the entry lane:

$c_{e,pce} = A \cdot \exp\left( -B \cdot v_{c,pce} \right)$

Where:

• $c_{e,pce}$ = entry lane capacity in passenger car equivalents (pc/h).
• $v_{c,pce}$ = conflicting circulating flow rate passing in front of the entry lane (pc/h).
• $A$ = capacity intercept (maximum entry capacity at zero circulating flow).
• $B$ = slope parameter (sensitivity coefficient to circulating traffic).

HCM 7th Edition Roundabout Capacity Regression Parameters

Critical Note on Flared Entries: A flared entry or bypass lane increases entry width and local radius, effectively increasing $A$ and reducing critical headway $t_c$.

Control Delay & Level of Service (LOS) Criteria

Unsignalized Intersection Control Delay Formula

For TWSC, AWSC, and modern roundabouts, average control delay ($d$) in seconds per vehicle is computed from the HCM queuing equation:

$d = \frac{3600}{c_m} + 900 T \left[ (x - 1) + \sqrt{(x - 1)^2 + \frac{\left(\frac{3600}{c_m}\right) \cdot x}{c_m \cdot T}} \right] + 5$

Where:

• $d$ = average control delay (s/veh).
• $c_m$ = movement or lane capacity (pc/h).
• $x$ = volume-to-capacity ratio $= v / c_m$.
• $T$ = analysis period duration (hours; typically $0.25\text{ h}$ for a 15-minute peak period).
• $+5$ = constant acceleration/deceleration delay term ($5.0\text{ s/veh}$). For roundabouts with bypass lanes or high geometry, geometric delay is evaluated separately.

Level of Service (LOS) Thresholds Comparison

Unsignalized intersections use lower delay thresholds for Level of Service than signalized intersections. Motorists perceive stop-controlled and roundabout delays as more frustrating because they must actively search for gaps rather than passively waiting for a green signal display.