## 5 LMS Channel Models
### Two-State Markov Channel Model  
Figure 3 shows a two-state Markov model for a LMS channel [35; 36; 37; 38; 39; 57]. The LMS channel can be represented by a Gilbert-Elliott (GE) model, in which it is possible to distinguish between time intervals with high received power corresponding to a so-called _good_ channel state, and time intervals with low power level corresponding to a so-called _bad_ channel state. In particular, the good channel state represents the condition of LoS in the satellite communication characterized by a low packet loss probability, whereas the bad channel state is responsible for representing areas in which there are obstacles between the satellite and user terminal, characterized by an extremely high probability of packet loss [58]. The _good_ channel state, called \(G\) and numbered as 1, corresponds to LoS situations; the _bad_ channel state, called \(B\) and numbered 2, represents situations where the signal is blocked by an obstacle in the propagation path. The test courses were carefully selected to represent different types of environments (city, suburbs, rural roads, highways), and to comprise a mixture of cruising directions.  
In this GE (two-state Markov model), the assumption is made that the system remains in a state for a certain number of seconds before deciding to make a transition. The transition can happen towards the other state or the same (current) state. The transition probabilities are labeled with \(p_{11}\), \(p_{12}\), \(p_{22}\), and \(p_{21}\), where \(p_{nm}\) represents the probability of transitioning from state \(n\) to state \(m\). Only two of these transition probabilities are independent because of the relationships \(p_{11}+p_{12}=1\) and \(p_{22}+p_{21}=1\). Clearly, more complex satellite channel models have been proposed in the literature, which will be provided in the next section; however, the GE model represents the simplest yet reasonably realistic model of satellite fading.  
Considering a mobile user with speed \(v\), and indicating with \(B\) the _bad_ state and with \(G\) the _good_ state, the average extensions (in meters) of the shadowed and non-shadowed areas, \(Area_{B}\) and \(Area_{G}\), correspond to average time intervals \(Int_{B}\) and \(Int_{G}\), in which the channel remains in _bad_ or _good_ condition, respectively. Considering a transmission rate \(R\)  
Figure 3: Two-state [PERSON] channel model.  
the mean state durations normalized to the symbol duration and the transition probabilities \(p_{21}=G\) and \(p_{12}=B\) result in:  
\[\begin{cases}Int_{B}=\frac{1}{p_{21}}=\frac{1}{\mathrm{C}}=\frac{R}{\mathrm{D}} \cdot Area_{B}\\ Int_{\mathrm{G}}=\frac{1}{p_{12}}=\frac{1}{B}=\frac{R}{\mathrm{D}}\cdot Area_{G }\end{cases} \tag{14}\]  
The more representative model which first used the two-state model was that of [PERSON] [35]. This model uses a Rician distribution for describing the good channel behaviour representative of the LoS component, and a mixed Rayleigh and lognormal distribution incorporated in the Suzuki model for describing the NLoS condition representative of the bad channel state. The PDF of the [PERSON] model [59], describing the bad channel in which obstacles can block the direct signal component, and of the [PERSON] model are as follows:  
\[p_{Suzuki}(a)=\int_{0}^{+\infty}p_{Rayleigh}(a|\sigma^{2})p_{Legnormal}( \sigma^{2})d\sigma, \tag{15}\]  
\[p_{Lutz}(a)=C\cdot p_{Rice}(a)+(1-C)\cdot p_{Suzuki}(a), \tag{16}\]  
where C and \((1-\mathrm{C})\) represent the percentage of shadowing and unshadowing, respectively.