## 3 Method
### Cycle Time  
As a second step in the method, the cycle time is calculated. To this end, the red-red time or RR (in seconds) must be calculated according to Equation (1). Similarly, the yellow time or AA (in seconds) is obtained according to Equation (2). For safety reasons, the minimum and maximum values of AA were considered to be 3 s and 6 s, respectively. Regarding RR, a value greater than 3 s could confuse vehicles into using it as part of the AA phase, causing them to start moving before the vehicle green light.  
\[RR=\frac{(n\cdot W)+L_{a}}{v} \tag{1}\]  
\[AA=t_{r}+\frac{v}{2\cdot\overline{f}} \tag{2}\]  
The losses (L) are obtained from the sum of RR and AA. Since the initial loss (\(\uplambda\)1) and final gain (\(\uplambda\)2) vary between 2 and 2.5 s, they are considered equal (and thus cancel each other out).  
Additionally, the minimum pedestrian green time (G\({}_{\text{min}}\)) is calculated using Equation (3), where D represents the maximum walking distance for pedestrians to cross. In the case without a Shibuya-type crossing, D is obtained by multiplying the number of lanes (n) by the lane width (W) for each branch of the intersection. However, in the case with a Shibuya-type crossing, D is calculated using the expression in Equation (4), where n\({}_{1}\) and n\({}_{2}\) refer to the number of lanes per branch of the intersection.  
\[G_{min}=max\{8\ ;\ 5+\frac{D}{v_{p}}\} \tag{3}\]  
\[D_{diag}=\sqrt{(n_{1}\cdot W)^{2}+(n_{2}\cdot W)^{2}} \tag{4}\]  
The load factors (y\({}_{1}\)) for each branch of the intersection are obtained using the expression in Equation (5), where q\({}_{i}\) represents the vehicular flow and S\({}_{i}\) is the saturation flow for each movement i. The vehicular flow refers to the total flow per movement on each branch of the intersection, and S is the sum of the saturation flows for each lane. Then, Y will be the sum of the y\({}_{i}\) values, considering only the critical movements i (those with the highest required green time).  
In the case of two branches in the intersection with bidirectional flow, the critical movement will be the right-turn movement for each branch.  
\[y_{i}=\frac{q_{i}}{S_{i}} \tag{5}\]  
To obtain the cycle time (c), the minimum cycle time (C\({}_{\text{min}}\)), optimal cycle time (C\({}_{\text{o}}\)), and practical cycle time (C\({}_{\text{p}}\)) must be calculated. The minimum value between C\({}_{\text{min}}\), C\({}_{\text{o}}\), and C\({}_{\text{p}}\) is then chosen. The maximum value for the cycle time (C) is defined as 120 s, while the minimum value for C is set to 40 s. However, in the case of Shibuya-type crossing, a cycle time C\({}^{\prime}\) is considered as the sum between C and G\({}_{\text{min}}\), considering D\({}_{\text{diag}}\).  
\[C_{min}=\frac{L}{1-Y} \tag{6}\]  
\[C_{o\_webster}=\frac{1.5\cdot L+5}{1-Y} \tag{7}\]  
\[C_{o\_ackclik}=\frac{(1.4+k)\cdot L+6}{1-Y} \tag{8}\]  
\[C_{p}=\frac{L}{1-11\cdot\overline{Y}} \tag{9}\]In Equation (8), k is Akcelik's constant, which can be assigned a value based on its objective function. If k = 0, then the delays are minimized.  
To calculate the effective green time distribution (\(\mathrm{v_{ei}}\)) for each vehicular movement i, the expression in Equation (10) is used, where \(\mathrm{v_{ei}}\) must be greater than or equal to \(\mathrm{G_{min}}\). In the case where \(\mathrm{v_{ei}}<\mathrm{G_{min}}\), it must be established that \(\mathrm{v_{ei}}\) is equal to \(\mathrm{G_{min}}\).  
\[v_{ei}=(C-L).\frac{y_{i}}{Y} \tag{10}\]  
When there is a vehicle-pedestrian conflict, vehicles can only make the turn during the yellow phase, so \(\mathrm{v_{ei}}\) = AA. On the other hand, for a Shibuya-style crossing, the exclusive pedestrian phase (phase 3) allows vehicles to make the turn using the entire effective green time from the previous phases (phase 1 and phase 2) (see Figure 3).