## 2 Architecture and Operational Analysis
### Queue-Length Models
#### 2.3.2 Model 2: Queue Length of Vehicles Arriving Ahead of Schedule in Current Cycle  
Upon the activation of the green phase for the merging direction at the upstream intersection in the current operational cycle, if the vehicles merging from this direction arrive before the green phase for the through movement at the downstream intersection starts, they must come to a stop and wait behind the queue of vehicles delayed from the previous cycle. This action creates a new queue of vehicles, and the queuing time for subsequently joining vehicles is determined by the time of arrival of the first vehicle at the end of the queue and the moment when the queuing wave and stopping wave intersect, as shown in Figure 4.  
When the leading vehicle merges from the upstream intersection during the current cycle's green phase duration and reaches the end of the queue of delayed vehicles at the downstream intersection, the relationship equation for the initiation of queuing can be expressed as: \(t_{tc,k}^{arr}(t)=t_{tc,k}^{s}+g_{t_{u},k,slr}^{b}(t)\). In the current cycle, when the leading vehicle from the merging traffic reaches the end of the queue of delayed vehicles and comes to a stop, it generates a stopping wave propagating backward at a speed of \(V_{pw}\). This causes subsequent arriving vehicles to closely follow the preceding queue, coming to a stop and waiting. After the green for the through movement at the downstream intersection is activated, the queued vehicles in front of the stop line will start to move one by one, creating a start-up wave propagating backward at a speed \(V_{sw}\) toward the end of the queue. The maximum queue length at the intersection is reached when the start-up wave propagating backward meets the stopping wave generated by the arriving vehicles. Simultaneously, one can establish the relationship between the queue length of the arriving vehicles in the current cycle and offset \(T_{os,k}\), as follows  
\[T_{os,k}+\frac{L_{p,k-1}^{det}+L_{p,k}^{arr}}{V_{sw}}=\frac{L_{g}-L_{p,k-1}^{ det}}{V_{g}}+\frac{L_{p,k}^{arr}}{V_{pw}} \tag{4}\]  
Finally, the queue length of the arriving vehicles in the current cycle \(L_{p,k}^{arr}\) is obtained as  
\[L_{p,k}^{arr}=\frac{V_{pw}\times V_{sw}}{V_{pw}-V_{sw}}\times\left(\frac{L_{ g}-L_{p,k-1}^{det}}{V_{g}}-\frac{L_{p,k-1}^{det}}{V_{sw}}-T_{os,k}\right) \tag{5}\]  
Figure 3: Queue of vehicles delayed from the previous cycle.  
To summarize, for an L-S (S-L) cycle transition, as the offset decreases (increases), the queue length of the delayed vehicles from the previous cycle gradually increases (decreases). There is a varying amount of change such that:  
\[\Delta\left(\frac{L_{p,k-1}^{det}}{V_{sw}}-\frac{L_{p,k-2}^{det}}{V_{sw}} \right)>\Delta\left(\left|T_{os,k}-T_{os,k-1}\right|\right)\ \text{(L-S cycle transition)} \tag{6}\]  
\[\Delta\left(\left|\frac{L_{p,k-1}^{det}}{V_{sw}}-\frac{L_{p,k-2}^{det}}{V_{sw }}\right|\right)>\Delta\left(T_{os,k}-T_{os,k-1}\right)\ \text{(S-L cycle transition)} \tag{7}\]  
The change in the L-S (S-L) cycle transition results in the queue length of vehicles arriving ahead of the schedule in the current cycle gradually decreasing (increasing) over several consecutive cycles.