## 5 Performance Evaluation  
This section evaluates the capacity of 5G NR grant-free scheduling with \(K\)-repetitions and shared resources to meet the reliability and latency requirements of URLLC services. To this aim, we use the analytical expressions that are derived in Section 3 and were validated in the previous section. Reliability for URLLC services is defined as the percentage \(P_{rel}\) of data packets that are successfully received by the BS before the latency deadline established by the service or application. In this study, we analyze first the reliability, considering only the effect of collisions from other UEs. This study analyzes then the impact of self-collisions on the capacity of 5G NR grant-free scheduling with \(K\)-repetitions and shared resources to achieve the reliability levels demanded by URLLC services. This is particularly relevant, as this study extends the state of the art by evaluating the capacity of 5G NR grant-free scheduling to sustain reliability levels even higher than 1-10\({}^{-9}\). This study also evaluates the performance of 5G NR grant-free scheduling as a function of the number of UEs, the number of reserved radio resources, and the number \(K\) of replicas. The performance of 5G NR grant-free scheduling is evaluated considering a single cell with \(N\) UEs. Packets are generated by each UE following a Poisson process with exponentially inter-arrival time. The average packet inter-arrival time is equal to \(1/\lambda\), where \(\lambda\) is the average number of packets generated per second. UEs transmit small packets with a size of 32 bytes [22]. Radio resources are divided in \(6\times 12\) subcarriers (i.e., \(U=6\)) with a subcarrier spacing of 120 kHz (i.e., \(T_{slot}=0.125\) ms). Figure 4 shows the probability \(P_{c}\) that a packet is not correctly received at the BS due to the collisions from other UEs experienced by all of the replicas of a packet (This would correspond to the reliability achieved with 5G NR grant-free scheduling if there were no self-collisions, i.e., \(P_{sc}=0\) and \(P_{rel}=1-P_{c}\)). The figure shows the value of \(P_{c}\) that can be achieved as a function of the number of UEs for latency requirements (\(L\)) of 0.25, 0.5, 0.75 and 1 ms. We focus on services with the most stringent latency requirements, given the challenge to satisfy high reliability levels when latency decreases [23]. For each value of \(L\), the grant-free scheduling scheme is executed with the maximum possible number of replicas  
Figure 3: Comparison of analytical and simulation results for different latency requirements \(L\) and number of repetitions \(K\) (\(U=6\), \(\lambda=0.1\) packets). \(K\) that can be transmitted within the required latency. For example, if the maximum latency \(L\) that can be tolerated is equal to 1 ms, the maximum number of replicas \(K\) that can be transmitted within 1 ms is equal to 8 (\(L\) = 1 ms corresponds to 8-\(T_{slot}\) when \(T_{slot}\) = 0.125 ms). Figure 4 also shows the performance achieved for two values of \(\lambda\) (0.1 and 1 packet(s)). The results depicted in Figure 4 clearly show that reducing the probability \(P_{c}\) of not receiving a packet to values as low as 10-9, (and hence reaching reliability levels of 1-10-9 when the effect of self-collisions is not considered), can only be achieved with high values of \(K\) and values of \(L\) equal to 0.75 or 1 ms. Figure 4 also shows that the probability \(P_{c}\) increases with the number of UEs, since the risk of collision is higher. As a result, the capacity of 5G NR grant-free scheduling to support high reliability levels is significantly decreased as the number of UEs to be supported increases. Figure 4 also shows that the difficulty in supporting high reliability levels increases with \(\lambda\), since the probability \(P_{c}\) increases as a result of a higher risk of collision between UEs. Figure 5 depicts the number of UEs that can be supported with a given latency requirement (\(L\)) and a reliability of \(P_{rel}=1-P_{c}\) when \(P_{sc}=0\).