## 2 Summary of the Cell Transmission Model  
It is well known that the CTM is a macroscopic traffic flow model proposed by [PERSON] [29]. As far as we know, [PERSON] [29; 30] first introduced the CTM as an improvement to the Lighthill-Whitham-Richards model [31; 32; 33]. It is widely used in traffic management due to its strong mathematical framework, which allows for the efficient simulation of traffic dynamics. The CTM divides a road segment into several cells, numbering them as \(i=1\), \(2\ldots n\) starting from the upstream end, where each cell represents a portion of the roadway with uniform traffic characteristics. The flow of vehicles between cells follows pre-defined traffic flow principles based on vehicle conservation and the fundamental diagram of traffic flow. In this article, for simplicity, we consider a two-way grid network \(G=(C,\,A)\) to illustrate simulation results based on our traffic congestion control strategies, adapted from our previous work [17]. In this model, \(C\) represents the set of nodes, and \(A\) denotes the set of links. A link \(a=(l,\,m)\) connects nodes \(l\) and \(m\), where \(A_{l}\) represents the set of links directed toward node \(l\) and \(B_{m}\) refers to the set of links departing from node \(m\). Each link is divided into \(\lambda\) discrete cells, with time segmented into intervals such that the length of a cell corresponds to the distance traveled by free-flow traffic within one time interval \(\delta\). As shown in Figure 1, each link \(a\) consists of two distinct sections [16]: (1) a downstream queue storage area, where vehicles are sorted based on their turning movements, and (2) an upstream reservoir, where turning movements mix. In a specific case, such as the ninth cell, the downstream queue storage area is subdivided into three sections, i.e., \(\varphi_{ac}\), \(\varphi_{ab}\), and \(\varphi_{ad}\), which serve as separate queuing zones. Table 1 outlines the key notations, primarily sourced from [PERSON] et al. [17]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Meaning** \\ \hline \(n^{i}_{a}(t)\) & The count of vehicles present in cell \(i\) of link \(a\) at the beginning of time interval \(t\). \\ \(y^{i}_{a}\left(t\right)\) & The quantity of vehicles entering cell \(i\) of link \(a\) during time interval \(t\). \\ \(N^{i}_{a}(t)\) & The highest possible number of vehicles that can occupy cell \(i\) of link \(a\) during time interval \(t\). \\ \(Q^{i}_{a}(t)\) & The greatest possible number of vehicles that can enter cell \(i\) of link \(a\) during time interval \(t\). \\ \(n^{i}_{ab}(t)\) & The count of vehicles present in cell \(i\) of link \(a\) at the beginning of time interval \(t\) that have link \(b\) as their next destination. \\ \(y^{i}_{ab}(t)\) & The quantity of vehicles entering cell \(i\) of link \(a\) during time interval \(t\) that will proceed to link \(b\) next. \\ \(d^{i}_{a}(t)\) & The total congestion delay experienced by vehicles present in cell \(i\) of link \(a\) during time interval \(t\). \\ \(\phi_{ax}\) & The percentage of vehicles moving from link \(a\) to links \(b\), \(c\), and \(d\). \\ \(d(t)\) & The overall congestion delay experienced by all vehicles within the entire network during time interval \(t\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definitions and corresponding meanings of notations. Figure 1: Elements of link \(a\). Below, we briefly introduce its key parameters and mathematical formulas. The key parameters in the CTM are as follows:  
1. Cell Capacity (\(N_{i}\))--The maximum number of vehicles a cell can hold. 2. Free Flow Speed (\(v_{f}\))--The speed at which vehicles move in uncongested conditions. 3. Jam Density (\(\lambda\))--The maximum vehicle density a cell can sustain before complete congestion occurs. 4. Flow Rate (\(q_{i}\))--The number of vehicles moving from one cell to the next per time step. The CTM follows two main equations, as follows:  
1.