31.5. Relation between time mean speed and space mean speed

The relation between time mean speed and space mean speed can be derived as below. Consider a stream of vehicles with a set of substream flow q , q , ...q , ...q having speed v , v , ...v , ...v . The fundamental relation between flow(q), density(k) and mean speed v is, q = k v (31.6)

This chunk introduces the basic relationship among three critical variables in traffic flow: flow (q), density (k), and mean speed (v). Flow is defined as the number of vehicles passing a point in a specified time, density as the number of vehicles per unit length of road, and mean speed as the average speed of those vehicles. The equation q = k * v provides the fundamental link that expresses how these variables are interconnected during traffic flow.

Imagine a busy highway where the flow of traffic is like water flowing through a pipe. The number of cars (flow) is determined by how tightly packed the cars are on the road (density) and how fast they are moving (mean speed). If more cars are crammed into the same space, they will flow more slowly.

Therefore for any substream q , the following relationship will be valid. q = k v (31.7) The summation of all substream flows will give the total flow q. Σq = q (31.8)

In this chunk, we explore the concept of substreams, which represent smaller sections of traffic flow within the larger stream. Each substream has its own flow (q_i), density (k_i), and speed (v_i). By applying the same fundamental relationship, we can derive that the total flow is simply the summation of the flows of these substreams. This emphasizes how individual segments of traffic combine to create the overall flow seen on the road.

Think of a large river that is made up of many small streams. Each small stream represents a substream of traffic. Just as the total flow of the river is the sum of all these smaller streams, the total traffic flow on a busy highway can be analyzed by adding the flows of all the smaller segments of traffic.

Similarly, the summation of all substream density will give the total density k. Σk = k (31.9)

Parallel to the way flow is summarized for substreams, the chunk emphasizes that the total density of traffic is determined by summing up the densities of the individual substreams. This relationship allows for an understanding of how the average concentration of vehicles across different portions of a road contributes to the overall traffic situation.

Imagine packing a suitcase with clothes from different drawers. Each drawer's clothes can be thought of as individual substreams. When you combine clothes from all drawers (substream densities), you get the total amount of clothing packed into the suitcase (total density).

Space mean speed averages the speed over space. Therefore, if k vehicles has v speed, then space mean speed is given by, Σk v_i / k (31.11)

This chunk defines how space mean speed is calculated. Unlike time mean speed, which considers how fast vehicles are over a time period, space mean speed focuses on the distribution of speeds across a road segment. The equation provided calculates the mean speed considering the number of vehicles at different speeds, providing a more spatially reflective speed measure.

Think of a classroom where students are competing in a race. While some students run quickly, others run slowly. The space mean speed would depict an average that considers how many students (vehicles) are running at certain speeds across the entire classroom (road space), similar to assessing the overall speed of the classroom instead of just the fastest runner.

Time mean speed averages the speed over time. Therefore, Σq v_i / q (31.12)

In this section, the concept of time mean speed is introduced as it calculates the average speed considering the number of vehicles passing a specific point over time. This helps illustrate how speed can vary over different periods while highlighting the total flow of traffic during that time.

Imagine how long it takes to bake cookies. Depending on how many you bake at once (which corresponds to flow), you measure the average time taken to bake them over cooking periods. Just like baking multiple batches at different intervals provides the complete picture of baking speed, time mean speed reflects the overall speed pattern of vehicles over time.

By adding and subtracting v and doing algebraic manipulations, v can be written as, σ² = v² + (31.18) Hence, time mean speed is space mean speed plus standard deviation of the spot speed divided by the space mean speed.

This concluding mathematical manipulation reveals the final relationship between time mean speed and space mean speed, indicating that the time mean speed will always be equal to or greater than the space mean speed due to the standard deviation being a non-negative value. It shows how the variability in vehicle speeds (standard deviation) influences the average speed calculated over time versus space.

Consider a basketball game's performance. If all players score consistently, the average scoring (time mean speed) will be high. However, if one player has variable performance (standard deviation), it could bring the team's average performance down (space mean speed). Thus, even if one player excels consistently, group dynamics will still lead to an overall lower average due to variability.

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