31. Fundamental relations of traffic flow

Speed is one of the basic parameters of traffic flow and time mean speed and space mean speed are the two representations of speed. Time mean speed and space mean speed and the relationship between them will be discussed in detail in this chapter. The relationship between the fundamental parameters of traffic flow will also be derived. In addition, this relationship can be represented in graphical form resulting in the fundamental diagrams of traffic flow.

This chunk introduces us to two crucial concepts in traffic flow analysis: time mean speed and space mean speed. Time mean speed refers to the average speed of vehicles at a specific location over a set duration, whereas space mean speed takes into account the variations in speed across a specific stretch of road. The chapter also indicates that these concepts will be illustrated graphically, helping us visualize how they relate to each other and the fundamental parameters of traffic flow such as volume and density.

Imagine a group of runners on a track. Time mean speed is akin to measuring how fast each runner completes a lap over a set period, while space mean speed can be thought of as looking at the average speed of every runner at any given time along the track. Hence, understanding both speeds provides a fuller picture of the runners' performance.

As noted earlier, time mean speed is the average of all vehicles passing a point over a duration of time. It is the simple average of spot speed. Time mean speed v is given by, v = \frac{1}{n} \sum_{i=1}^{n} v_{i}, (31.1) where v is the spot speed of ith vehicle, and n is the number of observations.

In this section, we learn how to calculate time mean speed, which is simply an average of the speeds of all vehicles passing a given point during a specified timeframe. The formula provided helps us compute this average using the speeds of individual vehicles (spot speeds) and the total number of vehicles observed. This gives us a clear understanding of the average speed as experienced by vehicles on the road over that time span.

Consider a classroom where five students take a test one after the other. If their test scores are 80, 90, 70, 85, and 95, the time mean speed would be comparable to the average score of all students, calculated simply by adding all scores and dividing by the number of students. This average gives us an overall view of the students’ performance in that particular test duration.

The space mean speed also averages the spot speed, but spatial weightage is given instead of temporal. This is derived as below. Consider unit length of a road, and let v_{i} be the spot speed of ith vehicle. Let t be the time the vehicle takes to complete unit distance and is given by 1/v_{i}. Therefore from the above equation, average travel time t_s is given by, t_s = \frac{1}{n} \sum_{i=1}^{n} \frac{1}{v_{i}}. (31.3)

Space mean speed is different from time mean speed in that it considers the time each vehicle takes to travel a certain distance, effectively weighing speeds according to how long vehicles are on the road. This gives a more realistic view when analyzing traffic flow, particularly because slower vehicles influence the overall traffic conditions more than faster ones. The formula provided calculates the average travel time for vehicles, which can then be inverted to obtain the space mean speed.

Think of a parade where some floats move slowly while others go fast. The space mean speed focuses on how long each float takes to cover a distance. If a float crawls at 5 mph while another zooms at 50 mph, the overall pace of the parade will reflect the slower float’s performance much more than the fast one, similar to how space mean speed highlights the effect of slower vehicles in traffic.

The relation between time mean speed and space mean speed can be derived as below. ... Therefore, time mean speed will always be greater than space mean speed since standard deviation cannot be negative.

This section derives the relationship between time mean speed and space mean speed mathematically. It shows that the time mean speed is always greater than or equal to the space mean speed because of the standard deviation involved in the calculations. The intuition behind this is that if all vehicles travel at the same speed, then both measures would be equal. However, in real-world traffic, there are usually variations in speeds, leading to a higher time mean speed.

Imagine a group of friends driving toward a concert. If one friend is stuck in traffic while the others are cruising along, the average speed of the group (time mean speed) will be higher than the speed at which they would all travel if they stuck together (space mean speed). The slower friend effectively brings down the overall 'group speed' when considering space, while the others accelerate the average when time is counted.

The relationship between the fundamental variables of traffic flow, namely speed, volume, and density, is called the fundamental relations of traffic flow... This is the fundamental equation of traffic flow. Please note that, v in the above equation refers to the space mean speed.

This section encapsulates the core principles of traffic flow, linking speed, volume, and density into a fundamental equation. It establishes that the flow of traffic (how many vehicles pass a point in a given time) is directly related to the density of vehicles on the road and their average speed over a distance. This equation serves as a critical tool for understanding and analyzing traffic patterns and behaviors.

Think of a water pipe. The flow of water through the pipe can be compared to traffic flow: if the pipe is narrow (high density), the water's movement will be slower (lower speed), while a wider pipe (lower density) allows water to flow quickly (higher speed). Just as the dimensions of the pipe affect water flow, road conditions and density dictate how smoothly traffic moves.

--