## 4 The Proposed Communication Strategy
### The Problem of DL Priority  
In standard LoRaWAN, the GWs work in half duplex mode only, i.e., they cannot receive and transmit simultaneously. Moreover, in commercial GWs, if there is the need to send a DL message, the reception of any incoming signal is interrupted, i.e., the concurrent UL packet is lost. Accordingly, the mechanism proposed in this paper for coordinating  
Figure 9: A schematic diagram of transmission of RP, UPs and DCP for 5 EDs belonging to the same cluster where each ED transmits UPs at the same time using different channels from sub-band g.  
simultaneous UL UPs, which is based on periodic delivery of DCPs, could dramatically affect the PLR of UPs. It is then of paramount importance to evaluate the PLR due to GW transmissions. To this aim, it is worth noting that the UL packet is lost by the concurrent DL transmission either when the DL packet is ongoing at the UL packet arrival time, or if it is started during the reception of the UL packet, since the GW gives priority to transmission anyway. Accordingly, denoting by \(\tau\) and \(D\) the durations of a DCP and of an UP, respectively, the PLR of UPs is equal to the probability that at least one DCP is generated in the interval \(\Delta=\tau+D\). To elaborate, in the considered setting DCPs are created as a response of RPs transmitted in the UL by each node. Accordingly, the DCPs arrival process statistics is equivalent to that of RPs generation process. Let then denote by \(T\) the rescheduling period set by each ED. Owing to inevitable clock drifts, the actual rescheduling time can be modeled as a random variable (rv) \(r=T+\delta\), where \(\delta\) is the clock error. In the considered scenario we deal with internal clocks which are natively embedded inside the ED microcontroller. This choice allows to save cost, energy, and complexity with respect to external clocks. In this case, it is shown in [30] that the clock errors are unbiased and that they can be reasonably modeled as independent and identically distributed (IID) Gaussian rvs, i.e., \(\delta\)-\(\mathcal{N}(0,\sigma)\). Accordingly, also the interarrival times \(r\) are IID rvs, i.e., the arrival process of DCPs belong to the class of renewal processes [31]. More specifically, we have \(r\)-\(\mathcal{N}(T,\sigma)\). From the theory of renewal processes, it is possible to evaluate the time asymptotic density \(w(x)\) for the the time elapsed from a generic time till the next arrival, i.e.,  
\[w(x)=\tfrac{1}{T}(1-F_{r}(x)) \tag{1}\]  
where \(F_{r}(x)\) is the cdf of \(r\). In the following, we are interested in evaluating the probability that an UP does not experience any collision with DCPs. To this aim, it is reasonable to assume that different nodes are characterized by independent clocks and independent time delays, and, hence, the probability \(P_{\text{C}}\) that the an UP does not experience any collision can be evaluated as the product of individual probabilities of all independent events. To elaborate, let us denote by \(N\) the number of EDs and by \(\boldsymbol{\tau}=\{\tau_{1},\tau_{2},\ldots,\tau_{N}\}\) the DCP time duration of each node. Such terms depends on the SF used by the correspondent nodes to transmit DCPs, i.e., the higher SF the longer \(\tau\). Since the adopted SFs in the UL depend on the channel conditions of each ED, e.g., the distance from the GW, it is reasonable to consider \(\boldsymbol{\tau}\) as a set of i.i.d. rvs with individual pdf \(f_{\tau}(t)\). Similarly, also \(D\) depends on the SF adopted by the ED to transmit an UP and, hence, it can be characterized by a given pdf \(f_{D}(y)\).