10. Traffic Assignment

The process of allocating a given set of trip interchanges to the specified transportation system is usually referred to as traffic assignment. The fundamental aim of the traffic assignment process is to reproduce on the transportation system, the pattern of vehicular movements which would be observed when the travel demand represented by trip matrices to be assigned is satisfied. The major aims of traffic assignment procedures are:

1. To estimate the volume of traffic on the links of the network and obtain aggregate network measures.
2. To estimate interzonal travel cost.
3. To analyze the travel pattern of each origin to destination (O-D) pair.
4. To identify congested links and to collect traffic data useful for the design of future junctions.

Traffic assignment is the method used to decide how traffic flows through a transportation system based on given trips. The goal is to mimic actual vehicle movements when travel demands are met. The procedures help estimate how much traffic will be on different parts of a network, costs of traveling between zones, travel patterns between different locations, and areas of congestion which can inform future designs. For example, if many people travel between two points, understanding how they move can help planners design roads that better accommodate this need.

Think of traffic assignment like planning a big event, like a concert. You need to predict how many people will come and how they will get there. You might analyze the routes they would likely take to optimize parking and entrances. Just like traffic planners aim to see how cars travel from one place to another.

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. That means traffic conditions worsen and congestion starts developing. 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. So the minimum path computed prior to the trip assignment will not be the minimum after the trips are assigned. The relation between the link flow and link impedance is called the link cost function and is given by the equation as shown below: t = t0[1 + αx^β] where t is the travel time and x is the 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, usually with α = 0.15 minimum and β = 4.0.

When traffic flow approaches the maximum capacity of a road, the speed of vehicles decreases. This means that as more cars enter a road, not only do they move slower, but it also takes longer to travel any distance. Traffic models try to calculate this using a link cost function. The equation provided shows how to calculate the travel time based on the current flow of traffic and free-flow scenarios, teaching planners how traffic conditions change as they get congested.

Imagine a water pipe: When there is little water, it flows freely. But as you try to push more water through without making the pipe larger, the flow slows down, and sometimes it even backs up. The link cost function helps planners understand the 'bottlenecks' in traffic flow, similar to how you would identify where water gets stuck in pipes.

In this method the trips from any origin zone to any destination zone are loaded onto a single, minimum cost, path between them. This model is unrealistic as only one path between every O-D pair is utilized even if there is another path with the same or nearly same travel cost. Also, traffic on links is assigned without consideration of whether or not there is adequate capacity or heavy congestion; travel time is a fixed input and does not vary depending on the congestion on a link. However, this model may be reasonable in sparse and uncongested networks where there are few alternative routes and they have a large difference in travel cost. This model may also be used to identify the desired path: the path which the drivers would like to travel in the absence of congestion. In fact, this model’s most important practical application is that it acts as a building block for other types of assignment techniques. It has a limitation that it ignores the fact that link travel time is a function of link volume and when there is congestion or that multiple paths are used to carry traffic.

The all-or-nothing assignment method simplifies traffic predictions by assuming every trip is taken via the fastest route, without considering other routes. This approach makes assumptions that may not hold true during busy times when other paths might be used. While helpful for understanding how drivers might prefer to travel, this method is limited since real traffic conditions are often complex with multiple factors affecting flow.

Imagine a group of friends deciding to drive somewhere. If they all follow the same GPS route, expecting it to be the fastest, they might get stuck in traffic together, unable to see that an alternative route could be faster. The all-or-nothing assignment works like this GPS assumption, ignoring other potential paths that could also be beneficial.

The user equilibrium assignment is based on Wardrop’s first principle, which states that no driver can unilaterally reduce his/her travel costs by shifting to another route. User Equilibrium (UE) conditions can be written for a given O-D pair as: f(cu)=0 for all k; cu >= 0 for all k. If c = 0, from f k ⇒ 0 implies that all used paths will have same travel time. If cu > 0, then ensures that all unused paths will have travel time greater than the minimum cost path. The assumptions in User Equilibrium Assignment are: 1. The user has perfect knowledge of the path cost. 2. Travel time on a given link is a function of the flow on that link only. 3. Travel time functions are positive and increasing.

User equilibrium assignment means that the route taken by drivers balances out in such a way that no one can find a faster alternative route on their own. The connections between flows and costs must be known to drivers, and it factors in how flow impacts travel time. This model helps ensure that any adjustments in route don’t create more delays, sticking to a predictable yet complex pattern.

Imagine a town where everyone knows the best routes to avoid traffic. If everyone takes the same route to a festival since it's the fastest, it may become congested. The user equilibrium assignment is when all drivers have chosen the best paths available based on current traffic, leading to a balance where no one benefits by switching routes. It's similar to how a community knows the best roads to take, but sometimes they just have to accept busy routes.

The system optimum assignment is based on Wardrop’s second principle, which states that drivers cooperate with one another in order to minimize total system travel time. This assignment can be thought of as a model in which congestion is minimized when drivers are told which routes to use. Obviously, this is not a behaviorally realistic model, but it can be useful to transport planners and engineers trying to manage traffic to minimize travel costs and therefore achieve an optimum social equilibrium.

System optimum assignment looks at the entire traffic system and seeks the best routes all drivers should take collectively to minimize travel time. While this isn't practical in real life—since people do not always follow directions—transport planners can still use it to evaluate and adjust traffic patterns effectively.

Think of this like a coach directing a football team on the best plays to run. While not every player may follow the direction perfectly in a game, understanding the best strategies helps optimize the team's chances for success. Similarly, transport planners can set guidelines and make adjustments based on how traffic flows should ideally work together.

Let us discuss briefly some other assignments also. 10.6.1 Incremental assignment is a process in which fractions of traffic volumes are assigned in steps. In each step, a fixed proportion of total demand is assigned, based on all-or-nothing assignment. After each step, link travel times are recalculated based on link volumes. When there are many increments used, the flows may resemble an equilibrium assignment; however, this method does not yield an equilibrium solution. Consequently, there will be inconsistencies between link volumes and travel times that can lead to errors in evaluation measures. Also, incremental assignment is influenced by the order in which volumes for O-D pairs are assigned, raising the possibility of additional bias in results.

Incremental assignment breaks down total traffic into smaller parts to be evaluated step-by-step, recalculating travel time after each iteration. While this approach might seem like it accurately represents an evolving traffic situation, it doesn't truly provide a balanced outcome. Depending on the sequence of traffic assignments, the results can differ, leading to potential biases that complicate the analysis.

Imagine trying to fill a large aquarium with water by pouring in buckets one at a time without knowing how much water is already inside. The resulting level will continuously change based on each addition, and if you're not careful about when and how you pour, you might spill or misjudge the final water level. Incremental assignment works similarly—adding traffic in pieces without achieving a final balanced outcome.

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