# A Survey of LoRaWAN for IoT: From Technology to Application
## 6 MAC Layer Performance Evaluation
### LoRaWAN Network Models
#### 6.1.1 Mathematical Models  
Several analytical network models for LoRaWAN performance are described in the literature [25; 34; 69; 70; 71; 72; 73; 74]. They differ based on what type of SF interference is considered, if the channel fading is considered [25; 73] or if other technology interference is considered too [25]. On the other hand, the LoRa modulation mathematical model was presented in [4]. It was shown analytical that, in comparison to FSK modulation, LoRa modulation performs better in case of frequency selective fading. At the beginning of LoRaWAN research, the channel access model was assumed to be Aloha. In [34], authors modeled the channel access in LoRaWAN as Aloha based and assessed via the simulations the network capacity. However, their assumption that, if the transmission time of two packets overlaps, none of the packets will be received correctly is proven to be wrong by other empirical studies [47; 75]. Thus, the network capacity of LoRaWAN was under-determined by over-determining the collision ratio. A mathematical model based on Aloha is shown in [69]. Authors of [69] considered only MAC layer characteristics including acknowledgments and retransmissions. Frame generation by end nodes is modeled as Poisson distribution, while model considers uplink packet probability as well as downlink ACK probability for a successful transmission. The proposed model offers the possibility to find the PER under certain traffic load. It is shown that PER is 50% lower when retransmissions are considered compared to the case without retransmissions. In [70], the model has been extended to include capture effect with each co-SF interferer separately, while it misses the impact of channel fading. In [71], a system model for uplink communication in single gateway deployments is shown using stochastic geometry modeling. For the outage probability of a desired signal in uplink two conditions are considered: if the signal is above a certain SNR threshold and if the signal is at least 6 dB stronger than the dominant co-SF interferer. This study later was complemented by considering also the cumulative co-SF interference and inter-SF interference alone, as well as cumulative impact of both [73]. Compared to [70], studies in [71; 73] consider also impact of channel fading modeled as Rayleigh fading. Authors of [72] extended the model presented in [71] for the case when gateway is equipped with multiple antennas. Analytically, it is shown that by employing two and four antennas at gateway the coverage probability is increased by 1.5 and 1.97 times, respectively, compared to single antenna case [71]. In addition, the impact of message replication in coverage probability is shown, being utilized better for low density networks. In higher density networks, it causes more problems (higher collision probability) than that can increases the coverage probability. Another analytical model is presented in [76] that considers the impact of duty cycling in latency and collision performance. The SF orthogonality assumption makes it not realistic in terms of collision performance evaluation, however, it is one of the first models that analysis the latency behavior under duty cycling regulation constraints. In [74], the joint time-frequency interference is considered as a two-dimensional plane to account for the overlap of transmitted packets in both dimensions. The generated mathematical model considers the capture effect too, meaning that even under collision the packet with better received SINR can get through. In [77], a Markovian chain model to assess the performance of the join procedure in LoRaWAN was presented. Based on generated model, the delay performance and energy consumption of join procedure are assessed. It is shown that the joining delay is impacted by the number of channels and sub-bands used, being around 50 s when all channels are used [77]. The percentage of inactivated nodes has a negative impact in the activation delay by increasing the delay linearly.