# Cellular-V2X Communications for Platooning: Design and Evaluation
## 5 Theoretical Bounds  
This Section provides a numerical comparison between the sequential and simultaneous transmission schemes in terms of the time required to complete a CAM cycle when varying the composition of the platoons, CAM size and network parameters. The analysis is intended to provide theoretical bounds for such metrics, while neglecting the specific channel dynamics and spatial dimension, affecting interference and propagation conditions, i.e., all platoons are assumed to be served by the same eNodeB and all platoon vehicles are under each other coverage. **Sequential transmission. ** The selection of the CAM cycle duration must take into account that the PM-to-PM step in a platoon with \(L-1\) PMs is \(T_{PM-to-PM}^{seq}=(L-1)\cdot\Delta\) in the best case, where \(\Delta=T_{req}+T_{sched}+T_{grant}+T_{tx}\) is the time required to perform one CAM transmission considering the handshake between the PM and the eNodeB for obtaining the frequency resources, as shown in Figure 3. Let us consider \(T_{req}\), \(T_{grant}\) and \(T_{tx}\) as constant values. In this case, \(\Delta\) is minimized if \(T_{sched}=1\) TTI for each PM \(i\) belonging to the platoon, i.e., there are enough RBs to schedule the transmission of PM \(i\) in one TTI. This can be achieved depending on the total number of PMs (considering other platoons, too) that are simultaneously active; this is because we assume dedicated resources for V2X platooning communications. Assuming that one TTI has \(M\) available RBs and that the transmission of one CAM needs \(D<M\) RBs, the maximum number of PMs that can transmit in the same TTI is \(K_{max}<\lceil M/D\rceil\). This means that \(\Delta\) is minimized if there are \(K\leq K_{max}\) synchronized platoons. Otherwise, the eNodeB cannot schedule all the PMs in one TTI and some transmission grants must be allocated during subsequent TTI(s), hence increasing \(T_{sched}\). The overall CAM completion time is computed by adding a \(\Delta\) contribution to \(T_{PM-to-PM}^{seq}\) to account for the PL CAM transmission. Results in Figure 4 show that only platoons with few vehicles (around 7) can be supported while ensuring that the CAM completion time is below 100 ms. The maximum number of synchronized platoons \(K_{max}\) as a function of \(D\) is reported in Figure 4b. In order to provide exemplary values for \(D\), let us consider a 300-byte CAM and a CQI equal to 7. According to tables in [17] and after some straightforward computations, we obtain that the eNodeB has to allocate 14 RBs, hence \(D=14\). Instead, if we consider perfect channel quality, i.e., \(CQI=15\), we obtain \(D=4\). Under such range of \(D\) settings, \(K_{max}\) is in-between 5 and 10. **Simultaneous transmission. ** In such a case, all the PMs of a platoon start the CAM transmission at the same time. Assuming that \(T_{req}\), \(T_{grant}\) and \(T_{tx}\) stay constant for all PMs, the duration of the PM-to-PM phase for one platoon depends on \(T_{sched}\), i.e., the time required to complete the scheduling for all the PMs, which in turn depends on both their number and the CAM size. In more details, the number of RBs required to schedule the whole platoon is \((L-1)\cdot D\). Since the eNodeB can allocate at most \(M\) RBs in one TTI, \(T_{sched}=\left\lceil\frac{(L-1)\cdot D}{M}\right\rceil\). As a result, we obtain \(T_{PM-to-PM}^{sim}=T_{req}+\left\lceil\frac{(L-1)\cdot D}{M}\right\rceil+T_{ grant}+T_{tx}\). Again, this is the ideal case in which the eNodeB is serving one platoon only. In the worst case, \(N\) platoons are synchronized, i.e., their PMs transmit at the same instant. In this case, the time required to accommodate all transmissions becomes \(T_{PM-to-PM}^{sim}=T_{req}+\left\lceil\frac{N\cdot(L-1)\cdot D}{M}\right\rceil+T_{grant }+T_{tx}\). In general, if the platoons have different length \(L_{j}\), then \(T_{PM-to-PM}^{sim}=T_{req}+\left\lceil\frac{\sum_{j=1}^{N}(L_{j}-1)\cdot D}{M} \right\rceil+T_{grant}+T_{tx}\).