## 3 Grant-Free Scheduling
### Self-Collisions  
The effect of self-collisions is illustrated in Figure 2. We may suppose that a UE starts transmitting a packet \(p_{1}\) that was generated before \(t_{0}\). Let us then suppose then that a second packet \(p_{2}\) is generated before the \(K\) replicas of the previous packet \(p_{1}\) have been transmitted. This is a self-collision. If a self-collision happens, \(p_{2}\) can be stored, and its transmission will start after the UE has transmitted the \(K^{\text{th}}\) replica of \(p_{1}\) (i.e., at \(t_{1}\) in Figure 2). The transmission of the \(K\) replicas of \(p_{2}\) will finish at \(t_{2}\) that is equal to:  
\[t_{2}=2\cdot K\cdot T_{slot}+t_{0} \tag{24}\]  
The transmission of the \(K\) replicas of \(p_{2}\) may finish after the latency deadline \(L\), due to the time \(p_{2}\) being stored as the \(K\) replicas of \(p_{1}\) are being transmitted. We then analyze the probability \(P_{sc}\) that the transmission of \(K\) replicas of a packet is not completed before \(L\) due to the effect of self-collisions. This probability depends upon the number of replicas \(K\) and on the time instant at which \(p_{2}\) was generated. Figure 2 illustrates how self-collisions affect the probability of completing the transmission of \(p_{2}\) before \(L\), with \(L\) equal to \(1\) ms. \(L=1\) ms implies that the maximum number of replicas \(K\) that can be transmitted per packet is \(8\). However, it is possible to transmit less than \(8\) replicas, and Figure 2 represents the case in which \(K\) is set equal to \(4\), \(6\) or \(8\). \(p_{2}\) can be transmitted before the deadline \(L\) if it is generated at any time instant after \(t_{2}-L\), where \(t_{2}\) is the time at which the transmission of the \(K\) replicas of \(p_{2}\) is finished (the transmission of \(p_{2}\) starts when the transmission of the \(K\) replicas of \(p_{1}\) has finished at \(t_{1}\)). If \(p_{2}\) is generated before \(t_{2}-L\), it is not possible to complete the transmission of the \(K\) replicas of \(p_{2}\) before the latency deadline \(L\). \(P_{sc}\) can then be computed as 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 (26)). \(P_{sc}\) can then be expressed as:  
\[P_{sc}(\Delta t)=\int_{0}^{\Delta t}\lambda\cdot e^{-t\cdot\lambda}\cdot\mathrm{ d}t \tag{25}\]  
\[\Delta t=t_{2}-L-t_{p_{1}}=2\cdot K\cdot T_{slot}-L-t_{p_{1}} \tag{26}\]  
As shown in (25) and (26), the negative effect of self-collisions increases with the value of \(K\), since \(K\) influences the time a packet might be stored until the transmission of the previous packet is finished. However, increasing the number \(K\) of replicas transmitted for each packet is preferred, in order to combat possible collisions with other UEs sharing the same pool of radio resources. The next section will analyze both the effect of collisions from other UEs and the effect of self-collisions to analyze the reliability achievable with the grant-free scheduling with \(K\)-repetitions and shared radio resources.