## 3 Grant-Free Scheduling  
This paper uses grant-free scheduling with \(K\)-repetitions and shared resources to evaluate the reliability and latency levels that can be achieved in the presence of aperiodic traffic. Following [20], reliability for URLLC services is defined as the percentage of data packets that are successfully delivered before the latency deadline \(L\) established by the service or application. Following 3 GPP standards [11], UEs transmit the same data packet in \(K\) consecutive transmission slots with a duration \(T_{slot}\). The UE randomly selects an RB (Resource Block) for each transmission from the \(U\) RBs available per \(T_{slot}\). This is illustrated in Figure 1 that represents the time/frequency resource grid map in 5G NR, where the unit is an RB. In 5G NR, a wideband channel is divided into sub-frames, slots and RBs. An RB is the smallest unit of frequency resources that can be allocated to a UE. Without loss of generality, this study considers a numerology \(\mu\) equal to 3 with a subcarrier spacing of 120 kHz [21]. An RB is then 1440 kHz (\(\Delta f\)) wide in frequency (12 sub-carriers of 120 kHz) and lasts for one time slot with the duration \(T_{slot}\) equal to 0.125 ms. The reliability at the medium access level that can be achieved with grant-free scheduling with \(K\)-repetitions and shared resources depends upon two main factors. The first factor is the possibility that a packet is not correctly received due to the collision of all its \(K\) replicas with other transmissions; this is due to the random selection of the RB for the transmission of each replica. The study in [15] showed that the possibility to successfully deliver a packet increases with the number \(K\) of replicas. The second factor is the effect of self-collisions. A self-collision occurs when a UE has to transmit a new packet, and the transmission of the \(K\) replicas of the previous packet has not finished. If this happens, the new packet must be stored, and its transmission is delayed until all the replicas of the previous packet have been transmitted. This delay can result in the case that the new packet cannot be delivered within the latency limit, and hence self-collisions can impact the reliability of URLLC services. It is important then 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. In this case, the reliability or probability \(P_{rel}\) that a packet is correctly received by the BS must consider the probability \(P_{sc}\) that the transmission of the \(K\) replicas of a packet is not completed before the latency deadline \(L\) due to the effect of self-collisions. For the packets that are not affected by the effect of self-collisions, it must be considered the probability \(P_{c}\) that a packet is not correctly received due to the collision of all its \(K\) replicas with other transmissions. Hence, \(P_{rel}\) can be expressed as:  
\[P_{rel}=1-(P_{sc}+(1-P_{sc})\cdot P_{c}) \tag{1}\]  
In [15], its authors presented an expression to approximate the probability \(P_{c}\) of the collision of the \(K\) replicas of a packet with the transmission of other UEs. The expression was derived in scenarios where \(N\) UEs share the same pool of RBs. However, [15] did not analyze the impact of self-collisions, since the study only considered low values of \(K\) (equal to or lower than 4). For these low values, self-collisions might not have an impact upon the reliability, as will be later shown. In this paper, we analytically derive the exact probability of any collision of the \(K\) replicas of a packet with packets transmitted by other UEs (\(P_{c}\)). We also quantify the impact of self-collisions (\(P_{sc}\)), and analytically compute the reliability that can be achieved by grant-free scheduling with \(K\)-repetitions and shared resources (\(P_{rel}\)).