## 4 Proposal
### Proposed Algorithm
#### 4.2.2 Modified Dijkstra Algorithm (MD)
* For \(average\ length=vehicle\ average\ length+min\ gap\), is taken as an average length for all vehicles 3.8 m and 2.4 m of vehicle separation, so that the total is 6.2 m.

Figure 18: Proposed vehicular network. (b) Origin/Destination Pairs for Each Vehicle and Approximate Flow

In this case the vehicles depart each of a different origin and destination. Since we defined the flow of each street in the network, it was necessary to establish the subset of possible origins and destinations (O/D) so that any vehicle can reach its destination node. The O/D of a vehicle is assigned randomly of that set when the vehicle enters the network. Additionally, it was established a number \(F\) of vehicle incorporating in the network at each step of the simulation, thus, allowing a random number of vehicles joining each time. This is done in order to establish the robustness of the proposed solution because under different trip conditions, overall state of the network could change and it is useful to examine how the solution behaves under uncontrolled parameters. ### Simulation Description

Since we didn't used a specialized vehicular traffic simulator, it was necessary to model the behavior of a vehicle while it is moving in the vehicular network under certain assumptions. The vehicles are going to have three possibilities when following a route: move, wait and exit the network. When the vehicle starts its trip, it moves along the initial street in such a way that its final position never exceede the front vehicle position minus the minimal separation. If the separation is exceeded, the vehicle does not move and waits. The position changes of the vehicle are totally discrete. The flow diagram for the vehicle movement is shown in the figure below:

### Comparison Metrics

#### 5.3.1 Route Efficiency

As the simulation developed acts in a discrete way, it was necessary to make some approximations in order to establish the route efficiency for a vehicle. We start from the fact that the number of times a

Figure 19: Vehicle movement simulation. vehicle must wait affects the quality of the route it is following. Efficiency in this case was defined in the following way:

\[efficiency_{v}=\frac{number\ of\ moves}{number\ of\ moves+waits} \tag{8}\]

where the number of moves for a vehicle is determined by the total \"slots\" in the street or its capacity, and the waits are generated each time the vehicle cannot advance. This metric is proposed to directly be compared with the total distance traveled so that although the car takes a more costly route according to his distance, if there are not generated waits in the trip the efficiency is 1. #### 5.3.2 Traveled Distance

It is the total distance traveled by the vehicle from the origin node till the destination node:

\[td=\sum_{(i,j)\in R_{v}}d_{(i,j)} \tag{9}\]

#### 5.3.3 Fairness Index

The fairness index is used as a compare metric to establish how the individual optimization for the vehicle's route efficiency is related with the way the road infrastructure is being used. Following the Equation [2] used in Section 3.2, the fairness is calculated as follows:

\[f_{\alpha}(k)=\frac{\left[\sum Use_{street_{i}}/Capacity_{street_{i}}\right]^{ 2}}{S*\sum\left(Use_{street_{i}}/Capacity_{street_{i}}\right)^{2}} \tag{10}\]

where S is the total number of streets in the vehicular network and \(k\) the current state. #### 5.3.4 Congestion

The average congestion is presented with the purpose of relate it with the efficiency and fairness obtained:

\[\mu(k)=\frac{\sum\left[\frac{Use_{(street_{i})}}{Capacity_{(street_{i})}} \right]}{S} \tag{11}\]

where S is the total number of streets in the vehicular network and \(k\) the current state. ## 6 Obtained Results

This section presents the results for each implementation in the set of evaluated scenarios. First, we present the control scenario results with the two different validation approaches described in Section 5.1.1: N vehicles and F vehicles. Later, we present the reference scenario results. The next section describes the result presentation.