## 5 Performance Evaluation
In particular, Figure 6 depicts the number of UEs that can be supported with a given reliability and latency \(L\) when \(U\) decreases and \(\lambda\) is set equal to 0.1 packets (the reliability is equal to \(P_{rel}=1-P_{c}\) when the effect of self-collisions is not taken into account, i.e., \(P_{sc}=0\)). Figure 6 shows that the number of UEs that grant-free scheduling with \(K\)-repetitions can support for a given set of requirements strongly depends upon the number of RBs available. UEs randomly select an RB for each transmission from the \(U\) RBs available per slot. The probability that several UEs select the same RB for their transmissions increases when the number of RBs per slot decreases. Consequently, the probability \(P_{c}\) that a packet is not correctly received due to packet collisions, increases. In addition, the number of UEs that can achieve a target reliability level also decreases when the number of RBs per slot decreases. For example, 443 UEs can be supported with \(L=0.5\) ms (and hence \(K=4\)) and \(P_{c}=10^{-5}\) when \(U\) is equal to 5 RBs. This number decreases to 69 UEs when \(U\) decreases to 3 RBs. This is a significant reduction of 84%. This reduction increases when the reliability demand increases. For example, 86 UEs can be supported with \(P_{c}=10^{-9}\) and \(L=1\) ms (and hence \(K=8\)) when \(U\) is equal to 6. However, only 6 UEs can achieve these values of \(P_{c}\) and \(L\) if \(U\) decreases to 4 (i.e., a 93% reduction). Figure 5: Number of UEs supported with different requirements (\(L\) and \(P_{rel}=1-P_{c}\), when \(P_{sc}=0\)): (**a**) \(\lambda=0.1\) packets; (**b**) \(\lambda=1\) packet. All previous results have been derived without considering the effect of self-collisions. Self-collisions were illustrated in Figure 2, and the probability of self-collision was derived in Section 3.2. As previously described, if a packet \(p_{2}\) is generated before the \(K\) replicas of the previous packet \(p_{1}\) have been transmitted, \(p_{2}\) will be stored and transmitted after completing the transmission of the \(K\) replicas of \(p_{1}\). Due to the time that \(p_{2}\) is stored, the transmission of its \(K\) replicas may finish after the latency deadline \(L\). As presented in Section 3.2, it is not possible to complete the transmission of the \(K\) replicas of \(p_{2}\) before the latency deadline \(L\) if \(p_{2}\) is generated before \(t_{2}-L\) (\(t_{2}\) is the time at which the transmission of the \(K\) replicas of \(p_{2}\) is finished as shown in Figure 2). This results in that the probability \(P_{sc}\) (the probability that the transmission of \(K\) replicas of a packet is not completed before \(L\) due to the effect of self-collisions) is equal to the probability that the time between the generation of two consecutive packets at a UE falls within the interval [0, \(\Delta t\)], where \(\Delta t\) represents the time difference between \(t_{2}-L\) and the time \(t_{p_{1}}\) at which \(p_{1}\) is generated (see (25) and (26)). We consider that packets are generated following a Poisson process with exponential inter-arrival time. As a result, \(\Delta t\) is homogeneously distributed between \(\Delta t_{1}\) and \(\Delta t_{2}\). For \(K\) = 4 in Figure 2, \(\Delta t_{1}\) is equal to 0 and \(\Delta t_{2}\) is equal to \(T_{slot}\), since \(p_{1}\) can be homogeneously generated between \(t_{0}\) and \(t_{0}-T_{slot}\). When \(K\) = 6, \(\Delta t_{1}\) is equal to 4 \(\cdot T_{slot}\), and \(\Delta t_{2}\) is equal to (4+1) \(\cdot T_{slot}\), since \(p_{1}\) can be homogeneously generated between \(t_{0}\) and \(t_{0}-T_{slot}\). Similarly, \(\Delta t_{1}\) and \(\Delta t_{2}\) are equal to 8 \(\cdot T_{slot}\) and (8+1) \(\cdot T_{slot}\) for \(K\) = 8. Table 1 shows the value of \(P_{sc}\) given in (26) when \(\Delta t\) is equal to \(\Delta t_{1}\) or \(\Delta t_{2}\) considering \(L\) = 1 ms and \(K\) = 4, 6 and 8.