# Resolving Persistent Packet Collisions through Broadcast Feedback in Cellular V2X Communication +
## 4 Consecutive Error Prevention Performance: Evaluation
### Duration of Consecutive Collisions
For instance, if the resource reservation period (RRP) is 100 ms (see Figure 7a), \(5\leq X_{i}\leq 15\). It is decremented on every packet transmission in the run. At a given resource keep probability \(P_{k}\), the number 1 of packet runs that retain the same frequency resource at a vehicle is geometrically distributed, whose mean is given by the Equation (6). \[\mathbb{E}[l]=\frac{1}{1-P_{k}} \tag{6}\]  
Suppose \(l_{1}\) and \(l_{2}\) are the numbers of such consecutive packet runs from two different vehicles colliding on the same resource at the same periodicity. Because what matters most is the packet collisions within a safety-critical distance, one can assume that the vehicles are within a mutual communication range. If one vehicle sees that another is already using a frequency resource, it will not select it under SPS. Therefore, any colliding runs should start simultaneously. Thus, the colliding span will be determined by the shorter length of the two, _min_[\(l_{1}\), \(l_{2}\)]. Although it may happen that \(l_{1}<l_{2}\) with \(\sum_{i=1}^{l_{1}}X_{i}>\sum_{j=1}^{l_{2}}X_{j}\), the probability becomes increasingly smaller as 1 becomes larger. This is because \(\sum_{i=1}^{l_{1}}X_{i}\) tends  
Figure 8: Average consecutive packet collision duration; 95% confidence interval, y-axis in log scale. to a normal distribution centered at \(l\overline{X}\). For the two geometrically distributed random variables with parameter \(p=1-P_{k}\), the minimum and the maximum are given as in Equations (7) and (8), respectively. \[\mathbb{E}[min\{l_{1},\ l_{2}\}]=\frac{1}{2p-p^{2}} \tag{7}\]  
\[\mathbb{E}[max\{l_{1},\ l_{2}\}]=\mathbb{E}[l_{1}]+\mathbb{E}[l_{2}]-\mathbb{E }[min\{l_{1},\ l_{2}\}]=\frac{2}{p}-\frac{1}{2p-p^{2}} \tag{8}\]  
At \(p=0.2\) (i.e., \(P_{k}=0.8\)), the shorter of the two is averaged at \(\mathbb{E}[min\{l_{1},\ l_{2}\}]=2.778\) runs. The longer, on the other hand, is averaged at \(\mathbb{E}[max\{l_{1},\ l_{2}\}]=10\)-\(2.778=7.222\) runs. With such a difference, the Irwin-Hall distribution tells us that the probability of the smaller number of runs having a larger number of packets is small. So, one can approximately estimate the number of colliding packets as \(\mathbb{E}\ [min\{l_{1},\ l_{2}\}]\cdot\mathbb{E}\ [X]\), which is 27.78 packets. It is equivalent to 2.778 s in time span at 10 Hz beaconing, which to a first-order approximation agrees with the result in Figure 8. Note that this can be a critical time in V2X communication, as it can lead to serious problem where two or more cars become \"invisible\" [14] to the neighbors on the wireless channel. The performance of the two other schemes starkly contrasts with SPS. In the SPS + alternating scheme, which alternates between two resources, the chances of packet collisions in both frequency locations with another vehicle should be slim. Therefore, the vast majority of collision events are composed of single packet losses, as Figure 8 confirms. Although this scheme successfully fragments a possibly long consecutive packet loss event in SPS into interspersed single losses so that the invisibility problem is practically resolved, it does not terminate the persistently occurring collisions as the SPS + colFeedback scheme does. Namely, the loss-every-other-packet behavior will continue until the next resource reselection. Consequently, Section 4.4 will show that the PRR performance of the scheme is adversely affected. In comparison, the performance of the proposed SPS + colFeedback scheme has a slightly larger average collision duration than SPS + alternating. However, the difference tapers as the traffic density increases. It suggests that the larger duration under lower traffic densities in this scheme is due to the relatively low availability of the feedback providers. As the traffic density increases, so does the availability. The collision event converges to a single packet loss in this scheme as well.