## 3 Methodology
### The Dynamic Adjusting Model of Traffic Queuing Time
#### 3.1.2 The Dynamic Adjusting Model  
Based on the traffic flow pattern in a single intersection [9], this study applies the dynamic adjusting model in three intersections. The three formulae, including the average vehicle length (\(ALC_{t,i}\)), total road length (\(RT_{t,i}\)),and the real-time-accumulated vehicles of each road length (\(AR_{t,i}\)), are considered in [9].  
The average length of vehicle \(i\), \(ALC_{t,i}\), is measured by the average of the sum of maximum length and minimum length of reaching vehicle \(j\)th as follows.  
\[ALC_{t,i}=\frac{(MAX_{t,j}+MIN_{t,j})}{2} \tag{1}\]  
We can obtain the vehicles capacity of the intersection roads, \(RT_{t,i}\), via the ratio of the length intersection roads \(R_{t,i}\) to the average length of each vehicle, \(ALC_{t,i}\) as show in (2).  
\[RT_{t,i}=\frac{R_{t,i}}{ALC_{t,i}} \tag{2}\]  
In a traffic queuing time model, the real-time-accumulated vehicles of each intersection road (e.g., \(AR_{t,i}\)) can be calculated in (3). In (3), we can add the initial queued vehicles and the difference of the reaching vehicles and their reducing amount of vehicle on each intersection road. Therefore, we obtain the real-time-accumulated vehicles of each intersection road \(AR_{t,i}\), where total accumulated vehicles must less equal than the vehicles capacity of its intersection road in (3).  
\[AR_{t,i}=\begin{array}{l}(SCW_{t,i}+CI_{t,i}-CO_{t,i})\\ AR_{t,i}\leq RT_{t,i}\end{array} \tag{3}\]  
We assume that the initial time of the traffic light is a constant value \(C\). We set up the initial time of each traffic light of intersection roads as \(C\) equal to \(S_{t,i}\) in (4).  
\[S_{t,i}=C\]The initial time of each traffic light, \(S_{t,i}\), plus the time of the traffic light changed, \(PT\), is denoted the fixed time of green light. The real-time-accumulated vehicles of each intersection road, \(AR_{t,i}\), times the average passing time of the vehicles, \(N\), is denoted the actual times of green light. Finally, we obtain the dynamic adjusting time of green light on each intersection road, \(\Delta T_{t,i}\), by the difference of the fixed time of green light and the actual times of green light in (5). Note that the negative value of \(\Delta T_{t,i}\) represents the protect times of the initial fixed time of green light which is not sufficient and is required to be increased into the next period adjusting time of the traffic light immediately.  
\[\Delta T_{t,i}=\left(S_{t,i}+PT\right)\ -\ \left(AR_{t,i}\times N\right) \tag{5}\]  
In other words, the \(AR_{t,i}\) adjusts the green light times to \(S_{t+1,i}\) through the \(\Delta T_{t,i}\). Then we derive the total next period adjusting time of the traffic light on each intersection road, \(S_{t+1,i}\), that can be presented by the initial time of each traffic light minus its dynamic adjusting time of green light previous period in (6).  
\[S_{t+1,i}=S_{t,i}-\Delta T_{t,i} \tag{6}\]  
In the next period, queued vehicle \(SCW_{t+1,i}\) is obtained by inheritance from current period real-time-accumulated vehicles of each intersection road \(AR_{t,i}\), as described in (7).  
\[SCW_{t+1,i}=AR_{t,i} \tag{7}\]  
Indeed, we set the period \(t+2\) adjusting time of the green light on each intersection road, \(S_{t+2,i}\) obtained by inheritance from the total period \(t+1\) adjusting time of the green light on each intersection road \(S_{t+1,i}\) minus its dynamic adjusting time \(\Delta T_{t,i}\), as described in (8).  
\[S_{t+2,i}=S_{t+1,i}-\Delta T_{t+1,i} \tag{8}\]