## 3 Collision Probability Analysis  
One of the key challenges in C-V2X communication is understanding how distance, speed and resource block allocation affect collision probabilities [24]. Candidate resource blocks (RBs) are first determined during reselection using the SB-SPS scheme [25; 26]. After reselection is initiated, the SPS system takes into account all physical resource blocks (PRBs) between \(T_{a}\) and \(T_{g}\) slots and disqualifies all PRBs that match either of the two criteria [27]:  
1. All RBs with an average time over the SPS are excluded. 2. All RBs that UE knows to be reserved (due to prior packet receptions) are excluded. After completing the exclusions, UE randomly chooses one of the remaining PRBs to broadcast together with a new (random) value for counter. We refer to the period of time between the reselection trigger and the new transmission as \(\Delta T\). Similar to LTE Proximity Services, UEs are presumed to be half-duplex [28], which prevents them from decoding additional packets while they are transmitting. Resource Reservation Interval (RRI) is denoted as \(1/\tau\), where \(\tau\) is the resource reservation rate, and UE can periodically reserve slots. Any value from 1 ms to 100 ms, or multiples of 100 ms from 200 ms to 1000 ms, can be used for the RRI. UE will produce a counter \(T_{c}\) at random when it reserves an RB, use that RB \(T_{c}\) times and then decrease the number \(T_{c}\) by 1 after each transmission. In Figure 1, considering that two or more UEs can choose the same RB, the SNR at UE1 is low. This scenario typically happens as a result of UE2 being outside of the UE1 limited transmission range, but it can also occur if another UE1 performs reselection concurrently with UE2 [4]. There are three types probability of interferences in this scenario:  
1. If the interference occurs when UE1 has already released the packet and UE2 is releasing the packet in the same RRI duration; \[I_{U_{u}}=P(T_{U_{u}})\] (1)
2. If the packet delay time is interfering with any UEs (given that the first event has been taken into consideration); \[I_{tr}(d_{1-2})=P(T_{tr}|T_{U_{tr}})\] (2)3. If the first collision has already occurred and interference of the new released packet occurred due to the same UE (given that the first two events have been taken into consideration); \[I_{col}(d_{1-2})=P(T_{col}|T_{tr},T_{U_{u}})\] (3) where \(I_{U_{u}}=\) interference caused by Uplink Unicast; \(T_{U_{u}}=\) time required for Uplink Unicast; \(I_{tr}=\) interference caused by transmission range; \(T_{tr}=\) time required for the packet to cover the transmission range; \((d_{1-2})=\) distance between two UEs; \(I_{col}=\) interference caused by 1 st collision; \(T_{col}=\) time required for the packet caused by collision. Considering these three interferences, probability of the packet reception rate (PRR) for three different cases can be defined by  
\[P_{PRR}(I_{U_{u}}) = (1-I_{U_{u}}) \tag{4}\] \[P_{PRR}(I_{tr}(d_{1-2})) = (1-I_{tr}(d_{1-2}))\] (5) \[P_{PRR}(I_{col}(d_{1-2})) = (1-I_{col}(d_{1-2}))\] (6) Considering the three cases as independent events, the probability of total PRR is \[P_{PRR}(d_{1-2})=(1-I_{U_{u}})(1-I_{tr}(d_{1-2})(1-I_{col}(d_{1-2})) \tag{7}\]
* \(I_{U_{u}}\) calculation: This interference illustrates the possibility that UE1 and UE2 choose the same slot for transmission but that, because of the Uplink Unicast nature of UE operation, UE1 is unable to decode UE2's packet [4; 30]. This interference depends on the number of packets generated per second by vehicles, \(N\). If the number of packets is 100 per second, we can assume that the RBs of UE are distributed equally in time. \[I_{U_{u}}=\frac{N}{100}\] (8)
* \(I_{tr}(d_{1-2})\) calculation:  
Figure 1: Resource selection and reselection between two UEs based on different interference [29]. The likelihood that the receive power of a packet delivered from UE1 to UE2 at a distance of \(d_{1}\), \(d_{2}\) falls below the sensing threshold transmission range.