# Asynchronous Flipped Grant-Free SCMA for Satellite-Based Internet of Things Communication Networks
## 4 AF-SCMA Analytical Performance Derivation  
This section derives the performance of AF-SCMA in terms of PLR and network throughput \(T\). Firstly, we need first to analyze the probability of the number of packets interfering with a specific one. Assuming the system traffic load follows a Poisson distribution with parameter \(\lambda\) within one packet transmission duration \(\tau\). Thus, the probability of \(k\) packets arrived within \(\tau\) can be expressed as  
\[f\left(k;\lambda\tau\right)=\frac{\left(\lambda\tau\right)^{k}exp\left(- \lambda\tau\right)}{k!}, \tag{1}\]  
where \(k\) is a Poisson random variable with intensity \(\lambda\). In asynchronous system, collisions occur when two or more packets arrive within \(\pm 1\) packet duration \(\tau\). Therefore, \(\lambda\) is given by  
\[\lambda=2 GG_{c}N_{rep}, \tag{2}\]where \(G\) is the normalized logical traffic load of medium access control (MAC) layer, which is measured in bits/symbol/carrier. AF-SCMA coding gain \(G_{c}\) is defined as \(G_{c}=1/\left({{{rlog}_{2}}M}\right)\), where \(r\) is the FEC coding rate and is the cardinality of SCMA codebooks. \(N_{rep}\) is the number of transmitted replicas, which is a constant of 2 in this paper. To derive the PLR of AF-SCMA, we also need to analyze the power of the received packets which influences the decoding performance directly. Since MTDs are widely distributed in a satellite beam, uplink LMS channel fading conditions of each MTD are different. Additionally, the antenna gain of MTDs or the satellite also fluctuates in a small range due to the environment variations. Therefore, power control mechanisms cannot work perfectly, and the power of each received packet at the gateway station will fluctuate around its desired value \(\left[{{{E_{b}}\mathord{\left/{\vphantom{{E_{b}}{N_{0}}}}\right.\kern-1.2 pt}{N_ {0}}}}\right]_{D}\). As a result, the real received power of individual packet can be expressed as \(\left[{{{E_{b}}\mathord{\left/{\vphantom{{E_{b}}{N_{0}}}}\right.\kern-1.2 pt}{N_ {0}}}}\right]\left({a}\right)={a^{2}}\left[{{{E_{b}}\mathord{\left/{\vphantom{{E_{b}}{N_ {0}}}}\right.\kern-1.2 pt}{N_{0}}}}\right]_{D}\), where \(a\) is the power amplitude and \(a\) is assumed to follow a lognormal distribution \(L(a)\) with parameters of a mean \(\mu\) (in dB) and a standard deviation \(\sigma\) (in dB). Specifically, \(L(a)\) is given by [35]:  
\[L\left({a}\right)=\frac{{20}}{{\sqrt{2\pi}ln10\sigma a}}exp\left[{-\frac{{{ \left({20 log_{10}a-\mu}\right)}^{2}}}{{2\sigma^{2}}}}\right],\left({a>0} \right). \tag{3}\]  
Furthermore, we exploit the discrete lognormal probability density function (pdf) mentioned in [10] to characterize the discrete power amplitude of the received packets, and the lognormal pdf is represented as:  
\[{{p_{{{E_{b}}}\mathord{\left/{\vphantom{{E_{b}}{N_{0}}}}}\right.\kern-1.2 pt}{N _{0}}}}\left({i}\right)=\left[{\frac{{{E_{b}}}}{{{N_{0}}}}}\right]_{D}{{ \int_{{{E_{b}}}\mathord{\left/{\vphantom{{E_{b}}{N_{0}}}}\right.\kern-1.2 pt}{N _{0}}}}}{{{{E_{b}}\mathord{\left/{\vphantom{{E_{b}}{N_{0}}}}\right.\kern-1.2 pt}{N _{0}}}}}}{{{{E_{b}}\mathord{\left/{\vphantom{{E_{b}}{N_{0}}}}\right.\kern-1.2 pt}{N _{0}}}}}}{{{a^{2}}L\left({a}\right)da},\quad\text{for }i=1,2,...,\infty, \tag{4}\]  
where \({{E_{b}}\mathord{\left/{\vphantom{{E_{b}}{N_{0}}}}\right.\kern-1.2 pt}{N_{0}}} \left({i}\right)\) is a possible received power value, and it increases in an equally small step in logarithmic domain (e.g., 0.1 dB). In AF-SCMA, the decoding process is based on a unit of a chunk, and the detection of different chunks is independent of each other. Therefore, we derive the chunk loss ratio (\(CLR\)) first and then obtain PLR according to \(CLR\). However, due to the adoption of recursive IC process in the sliding window, we should derive an iterative model for \(CLR\).