# A Throughput and Energy Efficiency Scheme for Unlicensed Massive Machine Type Communications +
## 2 System Model
Hence, a Poisson point process \(\Phi\) is used to model the interfering nodes distributed over an infinite two-dimensional plane where \(\lambda>0\) (average number of nodes per m\({}^{2}\)) denotes the spatial density. This model is also elaborated in detail in [20; 44]. For modeling the wireless channel in this paper, we consider distance dependent path-loss and quasi-static fading model. Here, \(r_{i}\) represents the distance between the reference receiver and the \(i\)-th interfering node. Based on Slivnyak theorem [44], the reference receiver being located arbitrarily at the origin means that the receiver's position is at the center of the Euclidean space. Having this location as a fixed point makes locating the surrounding elements easier [27; 44]. More detailed explanation about using this tool in modeling mMTC networks and why it is useful can be found in [45; 46]. The channel gain between the reference receiver and the \(i\)-th interfering node is shown as \(S_{i}\). If we consider \(P_{T_{x}}\) to be the transmit power and \(\alpha>2\), the path-loss exponent, then the power received at the reference receiver is equivalent to \(P_{T_{x}}g_{i}r_{i}^{-\alpha}\). With these in hand, the signal to interference ratio at the reference receiver \(\text{SIR}_{0}\) Signal-to-interference-plus-noise ratio (SINR) is defined as \(\text{SINR}=\frac{S}{N+I}\) where S denotes the desired signal power and N and I the noise power and interference power, respectively; however, in interference limited scenarios such as the one studied in this paper, where noise can be neglected, the SINR is reduced to signal-to-interference ratio (SIR) [44].) is then defined as:  
\[\text{SIR}_{0}=\frac{P_{\text{S}}g_{0}r_{0}^{-\alpha}}{P_{\text{P}}\sum_{i\in \Phi}g_{i}r_{i}^{-\alpha}}, \tag{1}\]  
in which, \(P_{\text{P}}\) denotes the licensed nodes transmit power and \(P_{\text{S}}\) is the transmit power used by the unlicensed users for their transmissions. (Note that noise is not considered in this analysis, even if it is considered, adding noise would not have much effect on the output as also stated in [47].) The data rate in this model is defined as \(\log(1+\beta)\) in \(\text{bits}/\text{s}/\text{Hz}\). This is justifiable since the reference link takes advantage of point-to-point Gaussian codes and interference-as-noise decoding rules [20; 48; 49]. This spectral efficiency can only be achieved if the SIR is greater than a given threshold, which here is defined as \(\beta\), i.e., \(\text{SIR}>\beta\). An outage event happens if a transmitted message is not successfully decoded at the receiver side, meaning that \(\text{SIR}\leq\beta\). The probability of the system being in outage is \(P_{\text{out}}=\Pr[\text{SIR}<\beta]\). In this model, there is possibility of retransmission in case an outage event occurs. There can be up to \(m\) retransmissions in the network; hence, if the message is not successfully decoded by the receiver after \(1+m\) transmission attempts (first transmission plus \(m\) retransmissions), it is then dropped [21], which will result in packet loss. Here, \(P_{\text{suc}}\) is used to refer to the probability of having a successfully decoded message and is \(P_{\text{suc}}=1-P_{\text{out}}^{1+m}\). In order to compute \(P_{\text{out}}\), the previously mentioned channel gains \(g\) are considered to be quasi-static (squared envelopes), which are also independent and identically distributed exponential random variables (Rayleigh fading) with mean 1 [41]. In this model, the licensed nodes, which are also the source of interference, are not static but rather highly dynamic. Therefore, we consider that their locations with respect to the reference receiver are constantly changing during each transmission. Considering a Poisson point process \(\Phi\), it is possible to characterize the signal-to-interference ratio at the reference link \(\text{SIR}_{0}\) as in [20].