10.2. Link cost function

As the flow increases towards the capacity of the stream, the average stream speed reduces from the free flow speed to the speed corresponding to the maximum flow. This can be seen in the graph shown below.

This segment explains how as traffic flow (the number of vehicles on the road) nears the maximum capacity of a road (where it can no longer handle additional traffic efficiently), the average speed of vehicles decreases. Initially, under low traffic conditions, cars can move freely and travel at high speeds. However, as more cars enter the stream, they start to slow down due to congestion, leading to longer travel times.

Imagine a funnel: when you pour water slowly, it flows out easily. However, if you pour too much water at once, it starts to overflow and flow slows down. Similarly, vehicles on a road will flow smoothly when there are fewer cars, but as more cars are added, they impede one another, causing slower speeds.

The interzonal flows are assigned to the minimum paths computed on the basis of free-flow link impedances (usually travel time). But if the link flows were at the levels dictated by the assignment, the link speeds would be lower and the link travel time would be higher than those corresponding to the free-flow conditions.

This part discusses how traffic flows between zones are initially calculated based on the fastest or 'free-flow' conditions. However, once the traffic is assigned, the actual speeds and travel times experienced during peak flow conditions differ from these initial estimates. This discrepancy necessitates a reevaluation of the paths taken as the traffic conditions change.

Think of planning a trip with no traffic: you estimate your travel time based on open roads. However, when you actually drive during rush hour, the traffic slows you down, and your travel time increases. This means your original estimation needs adjustment based on the actual conditions you encounter.

The relation between the link flow and link impedance is called the link cost function and is given by the equation: t=t0[1+αxβ] where t and x is the travel time and flow, respectively on the link, t0 is the free flow travel time, and k is the practical capacity. α and β are the model parameters, for which the value of α = 0.15 minimum and β = 4.0 are typically used.

This section introduces the mathematical model used to describe how travel time increases with higher traffic flow. The equation relates the travel time (t) with flow (x) and free-flow conditions (t0). It shows that as traffic increases, travel times grow, indicating that there's a practical capacity (k) beyond which flow increases lead to significantly higher travel times. The parameters α and β determine how sensitive travel time is to changes in traffic flow.

Consider how a water pipeline functions: at low flow rates, water moves smoothly. As you increase the flow, you notice resistance and pressure build-up, causing delays (akin to increased travel time). The equation models this pressure versus flow relationship, helping engineers understand how to manage road capacities effectively.

The types of traffic assignment models are all-or-nothing assignment (AON), incremental assignment, capacity restraint assignment, user equilibrium assignment (UE), stochastic user equilibrium assignment (SUE), and system optimum assignment (SO). The frequently used models all-or-nothing, user equilibrium, and system optimum will be discussed in detail here.

This chunk outlines the various methodologies employed in traffic assignments. Each model has distinct characteristics for how trips are assigned to different routes. For example, AON assigns all trips to the shortest path without considering congestion, while UE focuses on conditions where no driver can reduce their travel time by switching routes. Understanding these different models can help planners select the most suitable approach based on traffic conditions and objectives.

When choosing a route for a journey, you can either take the fastest route without checking for traffic (similar to AON), or check traffic apps for the path where delays are minimal (similar to UE). Different models reflect these decision-making processes, helping understand how traffic behaves under various conditions.

--