5.2 Time-Space Diagrams and Bandwidth Optimization

1.2 Time-Space Diagrams & Bandwidth Optimization

A time-space diagram is a two-dimensional distance-time graph used by traffic engineers and signal technicians to plot, analyze, and optimize signal timing plans along a coordinated arterial corridor. By visually displaying signal phase indications and vehicle trajectories over time, the time-space diagram enables technicians to evaluate platoon progression, identify progression bottlenecks, and calculate coordination efficiency.

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Fundamentals & Graphic Conventions

In standard North American traffic engineering practice, a time-space diagram uses the following coordinate axes:

• Horizontal Axis ($x$-axis): Represents elapsed time in seconds, advancing from left to right across one or more background cycle lengths ($C$).
• Vertical Axis ($y$-axis): Represents distance along the arterial corridor in feet or meters, plotting successive signalized intersections according to their physical geographic spacing.

At each intersection baseline along the vertical axis, the green, yellow change, and red clearance intervals for the main-street coordinated phases are displayed horizontally. Green intervals are drawn as open or green bars, while red intervals are shaded solid red.

Vehicle platoons moving along the corridor are depicted as progression bands—sloped parallel lines that pass through green signal displays at successive intersections.

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Progression Speed ($V$) & Trajectory Slope

The slope of a progression band on a time-space diagram represents the progression speed ($V$) of the vehicle platoon traveling along the arterial:

$\text{Slope} = \frac{\Delta \text{Distance}}{\Delta \text{Time}} = V$

• Steeper Slope: Represents higher progression speeds (e.g., 45 mph or 66 ft/s).
• Flatter Slope: Represents lower progression speeds (e.g., 25 mph or 36.7 ft/s).

Design Speed vs. Posted Speed Limit

The design progression speed selected for timing calculations is typically set 2 to 5 mph lower than the posted speed limit. This accounts for real-world factors such as heavy traffic friction, commercial vehicles, platoon dispersion, and driver reaction delays. Setting the design progression speed too high forces platoons to arrive before green indications begin, causing abrupt stops.

Queue Clearance Time ($g_q$)

At downstream intersections, vehicles arriving during the red interval form a standing queue at the stop bar. When the downstream signal turns green, these queued vehicles require time to react, accelerate, and clear the intersection before the arriving upstream platoon reaches the stop bar. To prevent the arriving platoon from slowing down or stopping behind this standing queue, the lead edge of the green bandwidth at the downstream signal must be adjusted by adding queue clearance green time ($g_q$):

$g_q = t_{start} + (n_{queue} \times h)$

Where $t_{start}$ is driver start-up lost time (typically 2.0 seconds), $n_{queue}$ is the number of queued vehicles per lane, and $h$ is the average queue discharge headway (typically 2.0 to 2.2 seconds per vehicle).

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Green Bandwidth Mechanics ($B$)

Green bandwidth ($B$) is the width of the progression band measured in seconds along the horizontal time axis. It represents the maximum continuous time window available for a platoon of vehicles to pass through all coordinated signals along the corridor without encountering a red light.

Inbound vs. Outbound Bandwidth

• Outbound Bandwidth ($B_{out}$): Measures progression bandwidth for traffic moving away from the central city core or master reference intersection.
• Inbound Bandwidth ($B_{in}$): Measures progression bandwidth for traffic moving toward the central city core.

Progression Band Boundaries

A progression band is bounded by two key vehicle trajectories:

1. Lead Vehicle Trajectory: The trajectory of the first vehicle in the platoon departing the upstream intersection at the onset of green (adjusted for queue clearance).
2. Lag Vehicle Trajectory: The trajectory of the last vehicle in the platoon that successfully passes through all downstream signals before main-street green terminates.

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Quantitative Metrics & Bandwidth Calculations

Traffic signal technicians use three primary mathematical metrics to quantify coordination quality:

1. Coordination Efficiency ($E$)

Coordination efficiency ($E$) expresses green bandwidth as a percentage of the total background cycle length:

$E = \left( \frac{B}{C} \right) \times 100\%$

• Target Range: A well-designed arterial coordination plan typically achieves an efficiency of 40% to 55%.
• Interpretation: An efficiency of 45% on a 100-second cycle indicates a 45-second continuous green window for platoon progression.

2. Maximum Attainable Bandwidth Capacity ($B_{max}$)

The maximum achievable bandwidth along a multi-intersection corridor can never exceed the shortest coordinated green split along that corridor:

$B_{max} \le \min(g_1, g_2, g_3, \dots, g_n)$

Where $g_n$ is the main-street effective green split in seconds at intersection $n$.

3. Attainability Ratio ($A$)

Attainability ($A$) measures how effectively the signal timing plan utilizes the available green time at the tightest intersection (the bottleneck):

$A = \left( \frac{B}{g_{min}} \right) \times 100\%$

An attainability of 100% signifies that the maximum theoretical bandwidth permitted by the bottleneck split has been fully achieved by the offset design.

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Bottleneck Identification & Capacity Constraints

A bottleneck intersection is the signal along an arterial corridor with the most restricted main-street green split. Bottlenecks are typically caused by heavy cross-street arterial traffic, complex multi-phase left-turn requirements, or long pedestrian clearance times on side streets.

Impact of Bottlenecks

If Intersection A has 60 seconds of main-street green, Intersection B has 35 seconds, and Intersection C has 55 seconds, Intersection B is the bottleneck ($g_{min} = 35\text{ seconds}$). Regardless of how perfectly offsets are optimized, the maximum green bandwidth across the entire three-signal corridor cannot exceed 35 seconds.

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Summary of Time-Space Variables