# Application Research of Multi-Mode Relay in Future Heterogeneous Networks
## 3 Outage Probability Expression Derivation and Analysis  
The outage events in the communication system can be denoted as: if the data rate falls below the threshold rate, the outage event of communication happens. The data rates between A and B are defined as B\(\sim\)A: \(F_{BA}=0.5\text{log}_{2}\left(1+f_{BA}\right)\), A\(\sim\)B: \(F_{AB}=0.5\text{log}_{2}\left(1+f_{AB}\right)\), respectively. The size of SINR is related to MMR powers (\(p_{r1}\) and \(p_{r2}\)) and channel conditions (\(H_{a}\) and \(H_{b}\)). If there is not any data rate which is lower than the threshold rate \(F_{th}\), it means both rates are greater than the threshold rate \(F_{th}\), at this time, the communication system does not have an outage event. Motivated by this, the system outage probability can be expressed as in [18]. \[ \begin{array}{l}P_{out}=\text{Pr}\left\{\min\left(F_{AB},F_{BA}\right)<F_{th} \right\}=1\\ -\text{Pr}\left(f_{BA}>f_{th},f_{AB}>f_{th}\right)\end{array}  \tag{14}\]  
where \(f_{th}=2^{2F_{th}}-1\). Because of the SINR \(f\) is related to MMR power \(p_{r1}\) (\(i=1,2\)), and the two \(p_{r}\) are independent to each other, so Formula (14) can be rewritten as Equation (15). For convenience in the further derivation, we define a factor \(k\), let \(k=\frac{f_{BA}}{f_{AB}}\), substituting Equations (12) and (13) into \(k\), then have Equation (16). \[ \begin{array}{l}P_{out}=1-\text{Pr}\left(f_{BA}>f_{th}|f_{BA}<f_{AB}\right)- \text{Pr}\left(f_{AB}>f_{th}|f_{BA}>f_{AB}\right)\\ =1-\text{Pr}_{1}\ \ -\text{Pr}_{2}\end{array}  \tag{15}\]  
\[\begin{array}{l}k=\frac{p_{s}\sigma_{E}^{2}\left(H_{a}+H_{b}\right)+4p_{s} \sigma_{E}^{2}H_{b}+\left(H_{b}+\sigma_{E}^{2}\right)\mathbf{M}+2p_{s}\sigma_{ E}^{4}+\frac{p_{s}\left(H_{a}+H_{b}\right)+M+2p_{s}\sigma_{E}^{2}}{p_{r2}}}\\ \frac{p_{s}\sigma_{E}^where  
\(\Theta=p_{r1}p_{r2}\sigma_{E}^{2}\left[p_{s}\left(H_{a}+H_{b}\right)+\text{M}+2p_ {s}\sigma_{E}^{2}\right]\)  
\(\omega=\left(4p_{s}\sigma_{E}^{2}+\text{M}\right)\)\(p_{r1}p_{r2}\)  
\(\zeta=p_{s}\left(H_{a}+H_{b}\right)+\text{M}+2p_{s}\sigma_{E}^{2}\)  
Obviously, \(k\) is relevant with \(p_{r1}\), \(p_{r2}\) and \(H.\) In the following research derivation on MMR, we let \(p_{r1}>p_{r2}\),  
(1) When \(H_{a}<H_{b}\), it is obvious that \(k>1\), then \(f_{BA}>f_{AB}.\) So  
\[{Pr_{2}}^{\prime}=\text{Pr}\left(f_{AB}>f_{th}|H_{b}>H_{a}\right) \tag{18}\]  
(2) When \(H_{a}>H_{b}\), we divide two situations to further study:  
(a) If \(H_{a}\gg H_{b}\), overcomes the influence of \(p_{r1}>p_{r2}\), then \(k<1\),\(f_{BA}<f_{AB}\). What is the relationship between \(H_{a}\) and \(H_{b}\)? We let \(k<1\), then get Equation (19)  
\[\left(1-\frac{p_{s}}{\omega}\left(p_{r1}-p_{r2}\right)\right)H_{a}>\left(1+ \frac{p_{s}}{\omega}\left(p_{r1}-p_{r2}\right)\right)H_{b}+\frac{\text{M}+2p_{ s}\sigma_{E}^{2}}{\omega}\left(p_{r1}-p_{r2}\right) \tag{19}\]  
Making the simplification by substituting \(k_{1}=\frac{p_{s}}{\omega}\left(p_{r1}-p_{r2}\right)\) and \(k_{2}=\frac{\text{M}+2p_{s}\sigma_{E}^{2}}{\omega}\left(p_{r1}-p_{r2}\right)\) into Equation (15), and if \(1-k_{1}>0\), we get \(H_{a}>\frac{1+k_{1}}{1-k_{1}}H_{b}+\frac{k_{2}}{1-k_{1}}\)  
Now we achieve  
\[{Pr_{1}}^{\prime}=\text{Pr}\left(f_{BA}>f_{th}|H_{a}>\frac{1+k_{1}}{1-k_{1}}H _{b}+\frac{k_{2}}{1-k_{1}}\right) \tag{20}\]  
(b) If \(H_{a}>H_{b}\), not overcome the influence of \(p_{r1}>p_{r2}\), then \(k>1\),\(f_{BA}>f_{AB}\)  
\[{Pr_{2}}^{\prime\prime}=\text{Pr}\left(f_{AB}>f_{th}|H_{a}>H_{b}>\frac{1-k_{1}} {1+k_{1}}H_{a}-\frac{k_{2}}{1+k_{1}}\right) \tag{21}\]  
In conclusion, the outage probability is  
\[{Pr_{1}}={Pr_{1}}^{\prime} \tag{22}\]  
\[{Pr_{2}}={Pr_{2}}^{\prime}+{Pr_{2}}^{\prime\prime}\]  
where \(\Omega\) represents the diagonal matrix with the elements \(v_{n}\), \(\phi\left(\Omega\right)\) is the number of different diagonal elements of matrix \(\Omega\).