# Review of Traffic Assignment and Future Challenges
## 3 Traffic Assignment: Theoretical Background
### Dynamic Traffic Assignment (DTA)
#### 3.2.1 Time-Dependent User Equilibrium  
In the context of \(DTA\), the definition of the first principle of [PERSON]'s equilibrium (see Definition 2) is adapted to consider users' departure schedules along with the travel times they experience. In this context, we refer to it as dynamic user equilibrium, defined as follows:  
**Definition 3** (Dynamic User Equilibrium (\(DUE\))). : _The DUE refers to a situation in a transportation network consisting of multiple origin-destination (o-d) pairs over a specific time period, where the following conditions are satisfied:_  
* _For each_ \(o\)_-d pair and for each departure time interval, the routes taken by users exhibit a generalized cost (experienced travel time) that is both equal and minimized to the extent possible._
* _No user can unilaterally reduce their experienced generalized cost (travel time)._  
Definition (3) outlines the concept of \(DUE\) in a transportation network, considering departure time intervals and the minimization of experienced travel times for users across various \(o\)-\(d\) pairs. Like [PERSON]'s equilibrium defined for static assignment, \(DUE\) implies that different routes or paths available to connect \(o\)-\(d\) zones are used so that the perceived travel time for users is balanced and minimized. Moreover, dynamic equilibrium requires that no user can unilaterally reduce their experienced travel time. This means that users cannot individually choose to change their route to reduce their personal travel time without considering other users and the consequences on the overall system equilibrium. However, this equilibrium applies to each departure time interval, where no route should be significantly faster or slower than others for a given \(o\)-\(d\) pair. Indeed, it is difficult to measure the travel time of all alternative roads at a given instant (at the departure time of a given vehicle). The problem of estimating travel time makes the search for \(DUE\) mathematically intractable [75; 76]. To overcome this problem, other definitions of the equilibrium, such as Boston traffic equilibrium [60] and integral equilibrium [77] of the trajectory, have been proposed. However, these approaches require verifying the generally accepted assumption of monotonic increasing delay as a flow function. Even if this assumption is widely admitted in macroscopic and mesoscopic traffic models, experience and microscopic models can sometimes invalidate this assumption, mainly because intersection management is not accurately considered in models with low resolution. Let us take the example of adaptive traffic lights that manage the intersection of two alternative paths. Reducing the number of vehicles on one path will increase the travel time of the remaining vehicles (see Figure 3B1,B2). The relative gap measure is commonly employed for each \(o\)-\(d\) pair at a given departure time interval (\(o\)-\(d\), \(T\)) to assess solution quality. This gap quantifies the difference between the total cost of paths used by vehicles and the total cost of the shortest path used by  
Figure 3: Examples of microscopic phenomena: (**A**) Deadlock situation; (**B**) Comparison of scenarios (**B1,B2**): Increased travel time due to the extension of green light time. all vehicles [78]. The relative gap can be defined and expressed as follows to gauge the proximity of an equilibrium solution [79]:  
\[Rgap=\frac{\sum_{T}\sum_{(o,d)\in\delta}\sum_{k\in K_{ad}(T)}r_{k}^{ad}(T)\Big{(} c(k^{ad},T)-\mu^{od}(T)\Big{)}}{\sum_{t}\sum_{od}q^{ad}(T)\mu^{od}(T)}, \tag{13}\]  
where \(\mu^{od}(T)\) represents the generalized cost of the shortest path connecting origin-destination (\(o\)-\(d\)) pair for time interval, \(T\). Note that, in Equation (13), the expression \(\Big{(}c(k^{ad},T)-\mu^{od}(T)\Big{)}\) quantifies the difference between the cost experienced by the \(r_{k}^{od}(T)\) vehicles following path \(k^{ad}\) and the minimal cost, \(\mu^{od}(T)\). If the conditions of dynamic equilibrium were met at each time interval \(T\), this difference would be null.