1.2 Model-Based vs. Rule-Based ASCT Algorithms

1.2 Model-Based vs. Rule-Based ASCT Algorithms

The fundamental distinction among Adaptive Signal Control Technology (ASCT) platforms lies in the mathematical logic of their optimization engines. Adaptive systems fall into two overarching algorithmic paradigms: Model-Based Optimization Systems (which simulate traffic dynamics through explicit physical and mathematical equations) and Rule-Based / Expert Systems (which govern phase transitions through finite-state logic, heuristic rules, and dynamic queue thresholds).

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1. Model-Based Optimization Paradigms

Model-based systems—exemplified by SCOOT, OPAC (Optimized Policies for Adaptive Control), and RHODES (Real-Time Hierarchical Optimized Distributed Effective System)—construct a real-time mathematical representation of traffic behavior within the roadway network.

The Objective Function: Performance Index (PI)

Model-based systems optimize signal timing by minimizing an objective function known as the Performance Index ($PI$). The $PI$ represents the total operational cost of traffic within the network over a defined planning horizon: $PI = \sum_{i=1}^{N} \left( d_i + K \cdot S_i \right)$ Where:

• $N$ = total number of links or approaches in the controlled network.
• $d_i$ = total vehicle delay incurred on link $i$ (expressed in vehicle-hours or vehicle-seconds).
• $S_i$ = total number of vehicle stops incurred on link $i$.
• $K$ = user-configurable stop penalty weighting factor, expressed in equivalent seconds of delay per vehicle stop (typically calibrated between 4 and 20 seconds/stop).

Engineering Significance of Stop Weighting Factor ($K$)

• Setting $K = 0$ configures the optimizer to purely minimize total system delay, even if doing so forces vehicles in progression platoons to undergo frequent stop-and-start maneuvers.
• Increasing $K$ ($K = 16 - 20$) penalizes stops heavily, instructing the algorithm to preserve unbroken arterial green bands. This reduces fuel consumption, heavy vehicle wear, and rear-end crash potential on high-speed arterials ($v \ge 45\text{ mph}$).

Robertson's Platoon Dispersion Model

To project how a dense platoon of vehicles discharging from an upstream intersection spreads out as it travels down an arterial link, SCOOT and classic traffic models apply Robertson's Platoon Dispersion Model: $q_2(t + t_a) = F \cdot q_1(t) + (1 - F) \cdot q_2(t + t_a - 1)$ Where:

• $q_1(t)$ = vehicle departure flow rate from the upstream intersection during discrete time step $t$.
• $q_2(t + t_a)$ = predicted vehicle arrival flow rate at the downstream intersection stop bar during time step $t + t_a$.
• $t_a$ = minimum travel time over the link, defined as: $t_a = \beta \cdot T$ where $T$ is the average free-flow cruise travel time and $\beta$ is an empirical travel time factor (typically calibrated to 0.8).
• $F$ = platoon dispersion smoothing factor, defined mathematically as: $F = \frac{1}{1 + \alpha \cdot \beta \cdot T}$
• $\alpha$ = empirical platoon dispersion factor (dimensionless):
  • $\alpha \approx 0.20 - 0.25$: High-type access-controlled suburban arterials with wide lane geometry, few driveways, and minimal roadside friction (platoons stay tightly clustered).
  • $\alpha \approx 0.35$: Typical urban divided arterial with moderate commercial driveway access.
  • $\alpha \approx 0.50$: Dense downtown central business district (CBD) streets with frequent transit bus stops, pedestrian friction, on-street parking maneuvers, and heavy turning turbulence (platoons disperse rapidly).

Cyclic Flow Profiles (CFPs)

Model-based systems continuously measure vehicle flows across each cycle using advance system detectors located immediately downstream of the upstream intersection. This continuous stream of count and occupancy measurements forms a Cyclic Flow Profile (CFP):

1. The upstream detector samples vehicle arrivals in 1-second bins, recording the exact shape, density, and duration of the discharging platoon relative to the upstream signal cycle.
2. The upstream CFP is processed through Robertson's dispersion equation to generate the Predicted Downstream Arrival Profile.
3. The offset optimizer slides the downstream green interval across this predicted arrival profile, evaluating the exact time when the leading edge of the platoon reaches the downstream stop line.
4. The optimizer adjusts the coordination offset to ensure the green interval opens precisely before the platoon arrives, preventing the front of the platoon from encountering a red indication while minimizing wasted green time prior to platoon arrival.

Rolling Horizon Dynamic Programming (OPAC & RHODES)

Unlike SCOOT's incremental step approach, algorithms like OPAC and RHODES apply dynamic programming over a rolling horizon:

• The planning horizon (typically 50 to 120 seconds) is divided into stages (e.g., 5-second intervals).
• The algorithm models actual detected platoons within an immediate stage and uses statistical forecasting for future stages.
• It evaluates all feasible phase switching sequences to minimize queue accumulation, executing phase transitions for the immediate stage, rolling the horizon forward by one time step, and repeating the optimization continuously.

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2. Rule-Based & Expert Systems (Finite-State Machines)

Rule-based systems—most notably InSync—abandon the concept of fixed background cycle lengths, predefined splits, and static offset relationships. Instead, they model traffic control as a Finite-State Machine (FSM) governed by heuristic rules, queue thresholds, and dynamic cost/urgency functions.

Local Urgency & Cost Optimization Functions

In a rule-based architecture, every phase $j$ in the controller accumulates an urgency score ($U_j$) that grows dynamically based on real-time detection data: $U_j = w_1 \cdot Q_j + w_2 \cdot \left( t_{w,j} \right)^m + w_3 \cdot P_j$ Where:

• $Q_j$ = estimated queue length on approach $j$ (derived from stop-bar and advance video/radar detection).
• $t_{w,j}$ = wait time of the longest-waiting vehicle on approach $j$ since red display.
• $m$ = exponential weighting exponent ($m \ge 1.5$), which accelerates urgency non-linearly to prevent side-street starvation.
• $P_j$ = pedestrian actuation call status (with high weighting to ensure pedestrian clearance guarantees).
• $w_1, w_2, w_3$ = calibrated scaling parameters.

The Dynamic "Green Tunnel"

Rather than establishing progression through a repeating common cycle timer, rule-based systems create Green Tunnels along arterial corridors:

1. Advance sensors (radar or video) track vehicle platoons approaching the intersection, calculating vehicle counts, approach speeds, and platoon density.
2. When an arterial platoon approaches, the local edge processor requests a Phase Hold on the coordinated through movement, opening a dynamic green tunnel sized specifically to accommodate the detected platoon's length.
3. Platoon Clearance & Immediate Service Transition: As soon as the last vehicle in the platoon passes the advance detector—or when headways exceed a critical passage threshold (gap-out)—the green tunnel collapses immediately.
4. Rather than dwelling in main-street green until a fixed cycle timer expires, the state machine instantly switches to service conflicting side-street or left-turn movements with the highest accumulated urgency score ($U_j$).
5. Once side-street queues are served, the system immediately returns to arterial readiness, awaiting the next detected platoon.

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3. Dynamic Modulation: Cycle-by-Cycle vs. Interval-by-Interval

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4. Degree of Saturation Formulations & Split Allocation Logic

In both model-based and saturation-driven systems, allocating green splits requires calculating the Capacity ($c_i$) and Degree of Saturation ($DS_i$) for every critical phase $i$: $c_i = s_i \cdot \left( \frac{g_i}{C} \right)$ $DS_i = \frac{V_i}{c_i} = \frac{V_i}{s_i \cdot \left( \frac{g_i}{C} \right)} = \frac{V_i \cdot C}{s_i \cdot g_i}$ Where:

• $V_i$ = approach arrival volume (vehicles per hour, vph).
• $s_i$ = saturation flow rate (vehicles per hour of green, vphg).
• $g_i$ = effective green time allocated to phase $i$ (seconds).
• $C$ = total cycle length (seconds).

Equisaturation Rebalancing Principle

To achieve optimal network efficiency, ASCT algorithms attempt to equalize the Degree of Saturation across competing critical conflicting phases ($DS_1 = DS_2 = \dots = DS_k$). For two critical conflicting phases ($1$ and $2$), the green split ratio must equal the flow-ratio proportion: $\frac{g_1}{g_2} = \frac{V_1 / s_1}{V_2 / s_2}$

Worked Engineering Example: Split Rebalancing Calculation

Consider a two-phase critical intersection operating with cycle length $C = 100\text{ s}$ and total clearance time $L = 10\text{ s}$ (total available green $G = C - L = 90\text{ s}$):

• Phase 2 (Main Arterial): Volume $V_2 = 1,400\text{ vph}$, Saturation Flow Rate $s_2 = 3,500\text{ vphg}$ (2 through lanes), Current Green $g_2 = 60\text{ s}$. $c_2 = 3,500 \cdot \left( \frac{60}{100} \right) = 2,100\text{ vph}$ $DS_2 = \frac{1,400}{2,100} = 0.67$
• Phase 4 (Cross Street): Volume $V_4 = 600\text{ vph}$, Saturation Flow Rate $s_4 = 1,750\text{ vphg}$ (1 through lane), Current Green $g_4 = 30\text{ s}$. $c_4 = 1,750 \cdot \left( \frac{30}{100} \right) = 525\text{ vph}$ $DS_4 = \frac{600}{525} = 1.14$

Analysis: Phase 4 is severely oversaturated ($DS_4 = 1.14 > 1.0$), resulting in failing cycles and growing residual queues, while Phase 2 operates with excess capacity ($DS_2 = 0.67$).

ASCT Rebalancing: Calculate flow ratios $y_i = V_i / s_i$: $y_2 = \frac{1,400}{3,500} = 0.40, \quad y_4 = \frac{600}{1,750} = 0.343, \quad Y = y_2 + y_4 = 0.743$ Allocate available green time ($G = 90\text{ s}$) proportionally: $g_{2,\text{new}} = G \cdot \left( \frac{y_2}{Y} \right) = 90 \cdot \left( \frac{0.40}{0.743} \right) = 48.5\text{ s} \approx 49\text{ s}$ $g_{4,\text{new}} = G \cdot \left( \frac{y_4}{Y} \right) = 90 \cdot \left( \frac{0.343}{0.743} \right) = 41.5\text{ s} \approx 41\text{ s}$ Recalculate Degrees of Saturation: $DS_{2,\text{new}} = \frac{1,400}{3,500 \cdot (49/100)} = 0.816$ $DS_{4,\text{new}} = \frac{600}{1,750 \cdot (41/100)} = 0.836$ Both phases now operate safely below capacity ($DS < 0.85$) with balanced saturation across the barrier.

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5. Comprehensive ASCT Algorithm Comparison Matrix