## 3 Methods
### Stochastic Queuing Model Based on Markov Chain  
In Figure 3, within the context of the IoT-SBL strategy, the vehicle queue in front of the signalized intersection stop line is segmented into two categories: private vehicles in the regular lane and private vehicles in the dynamic bus lane. We denote \(L_{t}\) and \(K_{t}\) as the numbers of queued private vehicles in the regular lane and the dynamic bus lane at time \(t\), respectively. Additionally, \(S_{t}\) represents the number of time intervals that have elapsed since the last green light phase at the intersection. This framework characterizes the traffic queuing system at that moment under the IoT-SBL strategy as \(\{L_{t},K_{t},S_{t},t>0\}\), where \(L_{t}\geq 0,0\leq K_{t}\leq V,1\leq S_{t}\leq G+R\). In the proposed model, V denotes the maximum capacity of the dynamic bus lane (i.e., the maximum number of private vehicles permitted in the IoT-SBL). For numerical experiments, \(V\) is set to 15 vehicles, based on typical urban road configurations (e.g., a 200 m lane length with an average vehicle spacing of 13 m). Consequently, \(K_{t}\) operates within \(0\leq K_{t}\leq 15\). While \(L_{t}\) (regular lane queue) has no theoretical upper limit, its practical range is constrained by traffic demand, signal cycle duration, and intersection discharge capacity. Figure 3: Model specification diagram for the IoT-SBL strategy. The signaling system, denoted as \(\{S_{t},t>0\}\), is governed by the signal intersection and is characterized as a deterministic process within a state space defined by \(\{1,2\ldots m,m+1\ldots m+n\}\). In this state space, each signal state corresponds to consecutive time intervals \(T\) during the signaling cycle of the intersection. Private vehicles arrive randomly and independently, with access to the IoT-SBL or the ability to depart from the PTSI stop line only at the boundaries of these time intervals. Consequently, the established traffic queuing mechanism can be effectively represented as a semi-regenerative process \(\{L_{t},K_{t},S_{t},t>0\}\), which can be analyzed through traditional methods of traffic queuing dynamics. We define \((t_{n}=nT,n\geq 0)\) as the termination time of the \(n\)th time slot. Here, \(L_{n}=L(t_{n})\) denotes the number of private vehicles in the regular lane, while \(K_{n}=K(t_{n})\) represents the count of private vehicles in the IoT-SBL. Additionally, \(S_{n}=S(t_{n})\) indicates the number of time intervals that have elapsed since the start of the most recent green light phase at the signal intersection, beginning from time \(n\). The overall signaling state throughout the complete cycle can be represented as \(S_{n}\). Given that private vehicles can only enter the IoT-SBL or exit the signal intersection stop line at the boundaries of the time intervals, the system state can be characterized by \(\{L_{n},K_{n},S_{n},n>0\}\). In the semi-regenerative process \(\{L_{n},K_{n},S_{n},t_{n}>0\}\), the associated Markovian update process can be illustrated by \(\{L_{t},K_{t},S_{t},t>0\}\) Consequently, the embedded Markov chain is further defined as \(\{L_{n},K_{n},S_{n},n>0\}\). Let \(l\) and \(k\) represent the counts of private vehicles in the social lane and the IoT-SBL, respectively, while \(s\) indicates the signal state at time \(t\). The state space \(\varnothing\) of the process \(\{L_{n},K_{n},S_{n},n>0\}\) is defined as follows:  
\[\varnothing=\{(l,0,s),l\geq 0,1\leq s\leq g+r\} \tag{1}\] \[\cup\{(l,k,s),0\leq l\leq V,1\leq k\leq\min(k,V),V-l\leq s\leq V-k\}\] \[\cup\{(l,k,s),l\geq V,1\leq k\leq\min(k,V),1\leq s\leq V-k\}\] \[\cup\{(l,k,s),l\geq V,1\leq k\leq\min(k,V),g+l\leq s\leq c\}\]  
Queuing the state space in lexicographic order, the one-step probability transition matrix of an embedded Markov chain \(\{L_{n},K_{n},S_{n},n>0\}\) can be derived as follows:  
\[\mathbb{Q}=