## 5 Performance Evaluation
\(\Delta t=\Delta t_{1}\) corresponds to the scenario where self-collisions are less probable, while \(\Delta t=\Delta t_{2}\) corresponds to the case in which they are more probable. The results in Table 1 show that the probability of self-collision is non-negligible. For example, \(P_{sc}\) can reach values equal to \(1.25\times 10^{-4}\) and \(9.99\times 10^{-4}\) when \(K\) is equal to 4 and 8, respectively, and \(\lambda\) = 1 packet. It is also important to highlight that a comparison of results in Figure 4 and Table 1 shows that \(P_{sc}\) can be actually higher than \(P_{c}\). This is for example the case when \(K\) = 8: \(P_{c}\) is lower than \(10^{-7}\) and \(10^{-5}\) for \(\lambda\) equal to 0.1 and 1 packet(s), respectively (Figure 4), while \(P_{sc}\) is approximately equal to \(10^{-4}\) and \(10^{-3}\) (Table 1). Grant-free scheduling can hence be limited by the effect of self-collisions, in particular when \(K\) increases. It is then important that the reliability (or probability that a packet is correctly received before the latency deadline) of grant-free scheduling with \(K\)-repetitions and shared radio resources is computed considering both the effect of collisions from other UEs and the effect of self-collisions following (1). Figure 7 plots \(1-P_{rel}\) for different values of \(K\) and \(L\) when considering both \(P_{c}\) and \(P_{sc}\). The results are plotted considering \(\Delta t=\Delta t_{1}\) for computing \(P_{sc}\). \(\Delta t=\Delta t_{1}\) corresponds to the case where self-collisions are less probable. Figure 4 shows that it is necessary to transmit a high number of replicas \(K\) within \(L\) to combat collisions from other UEs and correctly receive a packet at the BS. For example, Figure 4 shows that \(K\) must be equal to 8 in order to achieve \(P_{rel}=1\)-\(10^{-9}\) when \(P_{sc}=0\) and \(\lambda\) is equal to 1 packet. However, Table 1 showed that the effect of self-collisions increases with \(K\) even to the point that self-collisions limit the reliability that can be achieved. This is actually shown in Figure 7 when we consider \(L=1\) ms. In principle, it could be possible to satisfy a 1 ms latency requirement if we transmit 4, 6 or 8 replicas of a packet. Figure 7 shows that if \(K=4\) and \(\Delta t=\Delta t_{1}\) (for computing \(P_{sc}\) in (26)), the impact of self-collisions is not relevant, and the reliability levels of 1-\(10^{-5}\) can be satisfied for more than 500 UEs and 80 UEs when \(\lambda\) is equal to 0.1 and 1 packet(s), respectively; these results are in line with those observed in Figure 4 for \(K=4\). However, when \(K\) is equal to 6 or 8, the effect of self-collisions becomes more relevant (Table 1), and Figure 7 shows that it can actually limit the maximum reliability that can be achieved independently of the number of UEs. In fact, the maximum reliability that can be achieved is approximately equal to \(1-P_{sc}\). In this case, for \(K=8\) and \(\lambda=1\) packet/s, the maximum reliability (when \(P_{sc}\) is computed considering \(\Delta t=\Delta t_{1}\)) that can be achieved is 1-\(10^{-3}\) when the latency requirement \(L\) is equal to 1 ms. It should be noted that reliability levels even higher than \(1-P_{c}=1\)-\(10^{-9}\) were achieved when the effect of self-collisions was not considered (Figure 4). The results discussed so far correspond to the scenario where \(P_{sc}\) has been computed considering \(\Delta t=\Delta t_{1}\). This corresponds to the scenario where self-collisions are less probable. Figure 7 also shows the reliability that can be achieved with \(L=1\) ms and \(K=4\) when \(\Delta t=\Delta t_{avg}\). This \(\Delta t_{avg}\) is the average value of \(\Delta t\). \(\Delta t_{avg}=(\Delta t_{1}+\Delta t_{2})/2\), since \(\Delta t\) is homogeneously distributed between \(\Delta t_{1}\) and \(\Delta t_{2}\). Figure 7 shows that in this case it is not possible to achieve a reliability higher than 1-\(6.3\times 10^{-5}\) and 1-\(6.3\times 10^{-4}\) when \(\lambda\) is equal to 0.1 and 1 packet(s).