## 5 Performance Evaluation  
### Analysis  
Here, we perform an average-case analysis of SPS to understand its dynamics in terms of the many parameters it uses. In particular, we focus on the collision probability upon (re)selection because under the sensing and/or the lookahead, it is the only occasion when the collision can take place. Although the collision upon the (re)selection leads to a multiple packet collision because of the semi-persistency, the number of such ensued packet collisions will be identical for both schemes. Thus, we focus on the collision probability on (re)selection. To facilitate analysis, we apply some simplifying assumptions. First, we set \(probResourceKeep=0\), namely, we do not keep the current resource upon reselection [35]. Second, we assume that the vehicle population is static, namely no churn in the membership. Suppose the vehicular UE \(V\) reached \(RC=0\) at subframe \(n\). Let \(N_{subCH}\) is the number of subchannels in the given carrier, and \(T=T2-T1+1\) is the selection window width. In the reselection for a vehicle \(V\), there are \(R_{tot}=N_{subCH}\cdot T\) single-subframe resources in the selection window. Among them, \(R=R_{tot}(1-CBR)\) are available for reselection. Here, CBR measured in subframe \(n\) is defined to be the portion of subchannels in the resource pool whose RSSI measured by the UE exceed a (pre-)configured threshold sensed over subframes \([n-100,n-1]\)[39]. Essentially, it is the resource use. Although the UEs that reselect in \([n-T+1,n]\) will not use their current resource locations because \(probResourceKeep=0\), other UEs do not use these locations because of their sensing result in the previous RRI (see discussion in Section 4), so \(R\) accounts for this inefficiency as well. In each subframe, there are \(N_{sf}=N_{subCH}\cdot CBR\) UEs transmitting on average. In subframe \(n\), there are \(N_{sf}^{\prime}=N_{sf}/C\) nodes on average whose RC counter reached 0  
Figure 5: Number of additional bits to include the lookahead in the SCI Format 1, including the ten offset bits to the subframe in the lookahead. including \(V\), where \(C=(C1+C2)/2\). For simplicity, we assume that every UE has the same RRI of 100 ms so that every UE can reselect only a single resource in the selection window. Any one of these contending UEs with \(RC=0\) reselects a resource location in \(R\) with probability \(p=1/R\). In subframe \(n\), the probability \(P_{col,n}\) that a UE \(V\) collides with other UE(s) on the same subframe \(n\) is given by  
\[P_{col,n} = 1-(1-p)^{N_{sf}^{\prime}-1} \tag{1}\]  
In subframe \(n-k\)\((0<k\leq T-1)\), there are \(N_{sf}^{\prime}\) UEs that expired their RCs that can be contending with \(V\). With \(V\), they share \(T-k\) subframes in the selection window. The probability that a UE there selects one resource in the overlapped window is \(p^{(n-k)}=(T-k)/T\cdot p\). As above, the probability \(P_{col,n-k}\) that a UE \(V\) collides with other UE(s) on this subframe \(n-k\) is given by  
\[P_{col,n-k} = 1-\left\{1-p^{(n-k)}\right\}^{N_{sf}^{\prime}} \tag{2}\]  
Then, the total collision probability for \(V\) can be computed as  
\[P_{col} = 1-\prod_{k=0}^{T-1}(1-P_{col,n-k}) \tag{3}\] \[\approx 1-\prod_{k=0}^{T-1}\left\{1-p^{(n-k)}\right\}^{N_{sf}^{\prime}}\] \[= 1-\prod_{k=0}^{T-1}\left\{1-\frac{T-k}{T}\times\frac{1}{N_{sub CH}\cdot T\cdot(1-CBR)}\right\}^{N_{subCH}\cdot CBR/C}\]  
Notice that \(P_{col}\) is a function of the system parameters \(T1\), \(T2\), \(N_{subCH}\), \(C1\), \(C2\), and the traffic load (CBR). In contrast, the SPS/LA scheme removes the collisions caused by the UEs transmitting in the past subframes (Equation (2)), hence \(P_{col}=P_{col,n}\). This is because these UEs make their plans known, \(V\) can avoid the locations they plan to use in its sensing window. The only collisions are caused by the UEs in the same subframe as \(V\) with their RC expiring, whose transmissions are mutually hidden (Figure 4). Figure 6 shows the collision probabilities upon (re)selection for the two schemes as a function of CBR.