## 4 Consecutive Collision Problem Analysis of the JCS Sidelink Resource Allocation
### Markov Chain Model of SB-SPS  
In this section, a Markov chain analytical model [46] is presented, as shown in Figure 3. At any slot \(t\), the corresponding \(RC(t)\in[0,15]\). If \(RC(t)=0\), it is randomly re-initialized s.t. \(RC(t+1)\in[5,15]\). Thus, the probability that  
\[\Pr\{RC(t+1)=i|RC(t)=0\}=\frac{1}{11}. \tag{10}\]  
Denote \(\pi_{i}\) as the probability that \(RC(t)=i,0\leq i\leq 15\). According to Figure 3, \(\pi_{i}\) satisfies:  
\[\pi_{i}=\begin{cases}\pi_{0}&\text{for }0\leq i\leq 4,\\ \frac{1}{11}\pi_{0}+\pi_{i+1}&\text{for }5\leq i\leq 14,\\ \frac{1}{11}\pi_{0}&\text{for }i=15.\end{cases} \tag{11}\]  
Using the normalization condition \(\sum_{i=0}^{15}\pi_{i}=1\), and solving (10), we obtain \(\pi_{0}=\frac{1}{11}\).  
Since access collisions are caused by resource reselection, we first define the probability of a reselection resource collision. A collision will occur when multiple vehicles reselect the same resources within overlapping selection windows. Assume that at time \(t\), vehicle UE0 is in the state \(\text{RC}=0\), and it performs a reselection during the selection window \([t,t+\text{RRI}]\). During this time, other UEs also engage in reselection. UEs transition to \(\text{RC}=0\) with probability \(\pi_{0}\) and reselect with probability \(1-P_{k}\), moving to a new state. Since the RC states of each UE are independent, if \(n\) out of \(N_{\text{UE}}\) UEs are involved in reselection, the probability that other vehicles also trigger reselection when UE0 triggers reselection is given by:  
\[P_{s}(n)=\Pr\{n\;RC_{UEs}=0,\;\text{Reselect}|RC_{UEO}=0,\;\text{ Reselect}\} \tag{12}\] \[=\begin{pmatrix}N_{UE}\\ n\end{pmatrix}(\pi_{0}(1-P_{k}))^{n}(1-\pi_{0}(1-P_{k}))^{N_{UE}-n}\]  
Figure 3: Markov chain for state transition of SPS.  
where \(n\) represents the number of vehicles simultaneously reselecting within UEO's reselection window. Access collision may occur if at least one other vehicle selects the same available PRB as UE0.  
We define the collision involving \(n\) UEs reselecting within the overlapping selection window as an \(n\)-fold collision, given by the following formula:  
\[\begin{split}& P_{r}(n)=\Pr\{n\text{-fold Collision}|n\ RC_{UEs}=0,\text{ Reselect}\}\\ &=1-\left(1-\frac{1}{N_{a}}\right)^{n}\end{split} \tag{13}\]  
where \(\overline{N_{a}}\) represents the average number of available PRBs within the selection window. This number is influenced by the vehicle density and the packet size. Higher vehicle density and larger packet sizes result in fewer available resources.  
Thus, when UE0 performs reselection within the selection window, the access collision probability can be obtained as follows:  
\[\begin{split}& P_{c}=\Pr\{\text{Collision}|RC_{UE0}=0,\text{ Reselect}\}\\ &=\sum_{n=1}^{N_{UE}}P_{r}(n)P_{s}(n)\end{split} \tag{14}\]  
In SB-SPS, the same resources would be continuously colliding after an access collision. The collision will last for at least \(\min\{RC_{\text{UE0}},RC_{\text{UE1}}\}\) times between two UE0 and UE1. Therefore, it is useful to adjust \(\Delta RC\), which is the \(RC\) decreasing step size to mitigate consecutive collisions. Increasing the \(\Delta RC\) can reduce consecutive collisions. However, an overly aggressive increase in \(\Delta RC\) will increase the number of vehicles entering the selection window simultaneously, thereby increasing the probability of access collision, according to (14). Therefore, it is necessary to optimize the \(\Delta RC\).