# Radio Resource Dimensioning for Low Delay Access in Licensed OFDMA IoT Networks +
## 2 IoT Network Model  
We consider a sensor network in which a random number of active nodes sensors and collectors are distributed in a given area \(\mathcal{A}\) according to two independent homogeneous PPP with intensities\(\lambda_{a}\) and \(\lambda_{b}\). We assume that the sensors are equipped with a single antenna and that the collector is equipped with \(n_{r}\) receiver antennas. We assume that a sensor is active \(n_{a}\) times per day, the mean service time is \(\
u^{-1}\) (s) and the interarrival rate is \(\frac{n_{a}}{2\lambda<60\times 60}\)\(\rho\) per second and per km\({}^{2}\). The active sensors nodes form then a spatial PPP \(\Phi_{a}\) with intensity  
\[\lambda_{a}=\frac{n_{a}}{24\times 60\times 60}\rho\
u^{-1}. \tag{1}\]  
The frequency reuse pattern in the network is equal to 1. At a given collector situated at \(y_{0}\in\Phi_{b}\), the received power on a given RR from a sensor \(x\in\Phi_{a}\) transmitting with power \(P_{\text{RR}}\) (mW per RR) is computed as, \(P_{r}(y_{0},x)=P_{\text{RR}}\)\(\alpha\)\(|y_{0}-x|^{-\beta}\)\(A_{f}\) where \(\alpha\) and \(\beta\) are respectively the attenuation factor (that includes the average shadowing) and the path-loss exponent that are computed from the Okumura-Hata model, \(A_{f}\) is the fading coefficient with distribution depending on the antenna configuration and the used transmission mode (single-user or multiuser). The sensor nodes are considered as static and the fading channel is considered as flat during the transmission. The Channel State Information (CSI) is only available at the receiver side and not at the transmitter side. Each sensor is connected to the collector on which the average received power (by averaging over the fading and the shadowing) is the highest. This is equivalent to connect the sensor to the nearest collector. Assuming a collector \(y_{0}\), the set of the sensors connected to this latter is defined as \(\Phi_{c}(y_{0})=\{x\in\Phi:\forall y\in\Phi_{b}-\{y_{0}\}:|y_{0}-x|<|y-x|\}\). Nodes transmitting in the same frequency band generate additive interference with power  
\[I\left(y_{0}\right)=\sum_{x_{i}\in\Phi_{I}(y_{0})}P_{\text{RR}}\alpha\left|y_{ 0}-x_{i}\right|^{-\beta}A_{f,i}, \tag{2}\]  
with  
\[\Phi_{I}\left(y_{0}\right)=\cup_{y\in\Phi_{b}y\
eq y_{0}}\left\{x_{i}:x_{i} \text{ randomly selected in }\Phi_{c}(y)\right\}. \tag{3}\]  
The received SINR at the given collector \(y_{0}\) is,  
\[\text{SINR}(y_{0},x)=\frac{P_{\text{RR}}\ \alpha\ |y_{0}-x|^{-\beta}\ A_{f}}{P_{ n}+I(y_{0})}, \tag{4}\]  
with \(P_{n}\) being the random exponential noise power with mean power of \(\bar{P}_{n}=KTB\) where \(K\) is the Boltzmann's constant \(K=1.379\times 10^{-23}\) W Hz\({}^{-1}\) K\({}^{-1}\), \(T\) the absolute temperature in kelvins \(T=290\) K and \(B\) the bandwidth. The power of the random exponential noise power is characterized by its Laplace transform as,  
\[\mathcal{L}_{P_{n}}(s)=\mathbb{E}\left[e^{-sN}\right]=\frac{1}{s\bar{P}_{n}+1}. \tag{5}\]  
Due to Slivnyak-Mecke Theorem in [8], the statistical behavior in the PPP remains unchanged when adding a collector at the center of this region. This defines the typical cell centered at the origin \(\Phi_{c}(0)\) as illustrated in Figure 1. The typical cell properties are reviewed in Proposition 1.  
**Proposition 1** (Typical cell properties).: _The typical cell average number of sensors is,_  
\[N_{s}\triangleq\mathbb{E}\left[\sum_{x\in\Phi_{a}}\mathds{1}_{\{x\in\Phi_{c} (0)\}}\right]=\frac{\lambda_{a}}{\lambda_{b}} \tag{6}\]  
_The probability distribution function of \(r=|x|\) the distances between \(x\in\Phi_{c}(0)\) and the \(o\) collector is,_  
\[f(r)=2\pi\lambda_{b}\exp(-\lambda_{b}\pi r^{2}). \tag{7}\]  
_The average radius of the typical cell is \(1/(2\sqrt{\lambda_{b}})\)._Proof.: We provide for completeness the proof in Appendix A.