## 3 Integrated Radio Coverage and Device Capacity Analysis Framework  
### Theoretical Approach  
The objective of this subsection is to present the mathematical models that were considered for the integrated radio coverage and device capacity dimensioning for LoRaWAN and NB-IoT. In order to assess the coverage and capacity of LoRAWAN, it is important to briefly study the mechanisms related to channelization, framing, and medium access in PHY and MAC layer, respectively. As discussed in [1,2], in LoRaWAN, there are eight PHY transmission modes that include seven modes using the LoRa modulation (with Spreading Factors (SFs) ranging from 7 to 12) and another mode assuming a Gaussian Frequency Shift Keying (GFSK) modulation. The first six modes utilize a channel bandwidth equal to 125 kHz, whereas the last two modes use 250 kHz. Each mode has its own coverage and throughput figures depending also on the link-layer procedures for transmitting and receiving packets. All LoRaWAN nodes transmit their packets using a pure Aloha access protocol. Based on the propagation model used in [17] and on the receiver sensitivity values for the different spreading factors reported in [15], Table 1 shows the maximum distance values from the LoRaWAN gateway:For the abovementioned calculations, the path loss was calculated by the expression  
\[PL=10n\times log_{10}(d)+PL_{0}+L_{h}\times log_{10}(h_{d})+M_{s} \tag{1}\]  
where d is the distance of the device from the LoRaWAN gateway in km, n is the path loss exponent, \(PL_{0}\) is the reference path loss, \(L_{h}\) is the loss related to the height of the device antenna \(h_{d}\), and \(M_{s}\) is the shadowing margin. For the urban scenario, the following values were considered: \(n=4.179\), \(PL_{0}=102.86\), \(L_{h}=-6.3\), \(h_{d}=50\) cm, and \(M_{s}=10\) dB. For each of the abovementioned distance values, a separate LoRaWAN channel is considered to be used, with increasing spreading factor, trading off throughput for additional coverage. The symbol rate in LoRa modulation is given by the following expression:  
\[R_{s}=\frac{BW}{2^{SF}} \tag{2}\]  
where \(BW\) is the channel bandwidth, and \(SF\) is the considered spreading factor. One important metric that captures the actual radio resources required by a device transmission is related to the Time over the Air (ToA) for a transmitted packet and is calculated with Expression (2) [1; 22; 2]. PL is the application layer payload size in bytes, SF is the spreading factor, DE is the data rate optimization overhead that applies for SFs higher than 10 at 125 kHz channelization, and CR is the coding rate. The ToA values computed from Expression (2) can lead to the corresponding derivations of the actual device throughput and capacity figures. \[ToA=\frac{1}{R_{s}}\left\{12.15+\left[1+max\bigg{[}ceil\bigg{(}\frac{8 PL+24-4 SF}{4(SF-2 DE)}\bigg{)}(CR+4)\bigg{]},0\bigg{]}\right\} \tag{3}\]  
The device throughput is given by the following expression:  
\[T_{D}=\frac{8 PL}{ToA} \tag{4}\]  
whereas the actual maximum number of devices that can be supported within the coverage area is equal to:  
\[N_{D}=\frac{T_{p}}{ToA}\frac{1}{N_{c}}\frac{1}{2e} \tag{5}\]  
where \(T_{p}\) is the packet transmission period per device, and \(N_{c}\) is the number of employed LoRaWAN channels. In Expression (4), the Aloha access scheme collisions are considered by dividing the estimated devices' number over 2e. For modeling the NB-IoT access network key, PHY and MAC choices in terms of framing and channelization were considered. NB-IoT has three operating modes that involve in-band operation (assuming a Physical Resource Block (PRB) within the standard LTE band); a guard-band mode, where NB-IoT uses an LTE guard band that is not used by LTE; and a standalone mode, where NB-IoT uses one legacy GSM carrier. One frame of 10 ms duration consists of 10 subframes of duration 1 ms each. Each subframe is further