## 4 Results
### Definition of the Lower-Level Problem  
The lower-level optimization problem targets optimization of the duration of the green light of the traffic lights \(u_{k}\), \(k=1,\ldots,16\), and superiority of the outgoing over the incoming flow of each junction of the network. The traffic light cycles are denoted by \(y_{j}\), \(j=1\),..., 8. The amber light is 1/10 th part of the traffic light cycle. For the first crossroad section, the duration of the green lights \(u_{1}\) and \(u_{2}\) are, respectively, for the horizontal and vertical direction. The sum \(u_{1}\) and \(u_{2}\) of the first traffic light represents 9/10 th part of the cycle duration \(y_{1}\). This condition can be formalized by the following equation:  
\[u_{1}+u_{2}=0.9y_{1}.\]  
The optimization goal is the maximization of the green light durations, satisfying a set of constraints. A part of the constraints represents a modification of the store-and-forward model, applied for each queue of the junction. In this model, the intention is the outgoing flow of each direction to be bigger than the incoming to the junction traffic flow. Because the traffic flows are evaluated for cycle duration, their values correspond to the number of vehicles which can pass or make queues in the network links. In that manner, we formalize the requirement for decreasing the queue lengths in front of the junction. Let us describe the constraints for the first crossroad. According to the store-and-forward model and Equation (5), the queue length \(x_{1}\) is determined by the initial number of vehicles \(x_{1_{0}}\) plus the number of incoming vehicles \(x_{1 in}\) decreased by the number of the outgoing vehicles \(x_{1 out}\). Equation (5) can be presented in the following form:  
\[x_{1}=x_{1_{0}}+x_{1 in}-x_{1 out}, \tag{6}\]  
where the notation \(x_{i_{0}}\) means vehicles for direction \(i=1\),\(30\) and \(``_{0}\"\) is the notation of the initial number of vehicles. The incoming flow \(x_{1 in}\) depends on the duration of the traffic light cycle \(y_{1}\), the road saturation \(s_{1}\), and a coefficient \(a_{3}\), which formalizes the proximity of the model to reality:  
\[x_{1 in}=a_{3}s_{1}y_{1}.\]  
The outgoing flow depends on the green light duration \(u_{1}\) and the road saturation \(s_{1}\). We have in mind that the vehicles move straight ahead and turn to the left and right in the vertical directions. The part of the turning cars is denoted by coefficient \(a_{2}\), and because  
Figure 3: Network topology. there are two turning directions, the turning vehicles are \(2a_{2}\). The outgoing traffic flow from the first crossroad in a horizontal direction from west to east is:  
\[x_{1 out}=(1+2a_{2})s_{1}u_{1}.\]  
The flows from the vertical direction are not considered, because they do not belong to the internal traffic in the network in Figure 3.  
Equation (6) becomes:  
\[x_{1}=x_{1_{0}}+a_{3}s_{1}y_{1}-(1+2a_{2})s_{1}u_{1}. \tag{7}\]  
Our intention is for the outgoing flow to be bigger than the incoming flow in order to decrease the queue length \(x_{1}\). This can be formalized by the inequality:  
\[a_{3}s_{1}y_{1}-(1+2a_{2})s_{1}u_{1}\leq-x_{1_{0}}.\]  
The outgoing flows of \(x_{2}\) and \(x_{4}\) are determined analogically. A little bit different is the expression of the connection with the second-junction traffic flow, \(x_{3}\). The incoming flow from the second to the first junction depends on the sum of the green light duration \(u_{3}\) of the second junction and the road saturation \(s_{3}\) and the turning vehicles from the vertical directions of the second junction:  
\[x_{3 in}=s_{3}u_{3}+2a_{2}s_{4}u_{4}.\]  
The outgoing flow is:  
\[x_{3 out}=(1+2a_{2})s_{1}u_{1}.\]  
The constraint formalizing the condition the outgoing flow of \(x_{3}\) to be bigger than the incoming flow is:  
\[x_{1}=x_{1_{0}}+x_{1 in}-x_{1 out}, \tag{6}\]0  
By the same way, the rest traffic flows of the network are formalized. They represent a part of the constraints of the lower-level optimization problem.