# Application Research of Multi-Mode Relay in Future Heterogeneous Networks
## 3 Outage Probability Expression Derivation and Analysis
In addition, \(\
u\left[i\right]\) denotes the ordered values of \(\phi\left(\Omega\right)\) different elements which is \(\
u\left[1\right]>\
u\left[2\right]>\cdots>\
u\left[\phi\left(\Omega\right)\right]\); \(\phi_{i}\left(\Omega\right)\) is the number of replications of \(i_{th}\) element; \(\eta_{i,j}\left(\Omega\right)\) is the character coefficient of matrix \(\Omega\). Then applied Equation (22) into Equation (21), get Equations (27) and (28). \[ \begin{array}{l}\Pr_{1}=F_{M}\left(m\right)\approx\sum\limits_{i=1}^{\phi \left(\Omega\right)}\sum\limits_{j=1}^{\phi_{i}\left(\Omega\right)}\eta_{i,j} \left(\Omega\right)\frac{\left(\
u\left[i\right]\right)^{-j}}{\Gamma\left(j \right)}*\exp\left(-\frac{1}{2\mu^{2}}\left(10f_{th}\sigma_{E}^{2}+\frac{1}{p _{r2}}-\frac{1}{p_{r1}}+\frac{2f_{th}}{p_{r2}}\right)\right)\\ *\int_{0}^{\infty}m^{j-1}e^{-\frac{m}{\
u\left[i\right]}}*\left(\frac{1}{2}- \frac{1}{2}\frac{p_{s}\left(\frac{1}{p_{r2}}-\frac{1}{p_{r1}}\right)}{m+4p_{r }\sigma_{E}^{2}}\right)*\exp\left(-\frac{1}{2\mu^{2}}\left(\frac{2p_{s}\sigma_ {E}^{2}\left(f_{th}-1\right)\left(\frac{1}{p_{r2}}-\frac{1}{p_{r1}}\right)+ \frac{p_{r}}{p_{r1}}+\frac{p_{r}}{p_{r2}}\left(\frac{1}{p_{r2}}-\frac{2}{p_{r1 }}\right)}{m+4p_{r}\sigma_{E}^{2}-p_{r}\left(\frac{1}{p_{r2}}-\frac{1}{p_{r1}} \right)}+\frac{2f_{th}}{p_{s}}m\right)\right)dm\end{array}  \tag{27}\]  
\[ \begin{array}{l}\Pr_{2}=F_{M}\left(m\right)\approx\sum\limits_{i=1}^{\phi \left(\Omega\right)}\sum\limits_{j=1}^{\phi_{i}\left(\Omega\right)}\eta_{i,j} \left(\Omega\right)\frac{\left(\
u\left[i\right]\right)^{-j}}{\Gamma\left(j \right)}*\exp\left(-\frac{1}{2\mu^{2}}\left(10f_{th}\sigma_{E}^{2}+\frac{1}{p _{r1}}-\frac{1}{p_{r2}}+\frac{2f_{th}}{p_{r1}}\right)\right)\\ *\int_{0}^{\infty}m^{j-1}e^{-\frac{m}{\
u\left[i\right]}}*\left(\frac{1}{2}+ \frac{1}{2}\frac{p_{s}\left(\frac{1}{p_{r2}}-\frac{1}{p_{r1}}\right)}{m+4p_{r }\sigma_{E}^{2}}\right)*\exp\left(-\frac{1}{2\mu^{2}}\left(\frac{2p_{s}\sigma_ {E}^{2}\left(1-f_{th}\right)\left(\frac{1}{p_{r2}}-\frac{1}{p_{r1}}\right)+ \frac{p_{s}}{p_{r1}}+\frac{p_{s}}{p_{r1}}\left(\frac{1}{p_{r1}}-\frac{2}{p_{r 2}}\right)}{m+4p_{s}\sigma_{E}^{2}+p_{r}\left(\frac{1}{p_{r2}}-\frac{1}{p_{r 1}}\right)}+\frac{2f_{th}}{p_{s}}m\right)\right)dm\end{array}  \tag{28}\]  
Above all, outage probability of the network can be expressed in Equation (29):  
\[P_{out}=1-Pr_{1}-Pr_{2} \tag{29}\]  
\[\Pr_{1}=\Pr_{2}=\frac{1}{2}\exp\left(-\frac{1}{\mu^{2}}\left(5f_{th}\sigma_{E }^{2}+\frac{f_{th}}{p_{r}}\right)\right)*\sum\limits_{i=1}^{\phi\left(\Omega \right)}\sum\limits_{j=1}^{\phi_{i}\left(\Omega\right)}\eta_{i,j}\left(\Omega \right)*\left(1+\frac{f_{th}*\
u\left[i\right]}{\mu^{2}p_{s}}\right)\]  
First, it should be noted that if \(p_{r1}=p_{r2}=p_{r}\), Equations (27) and (28) can be simplified as equal to the consequence of reference [13] (Equation (18)). Second, note that \(p_{r1}\) and \(p_{r2}\) are very similar in most cases, but not equal. They are all affected by target rate \(f_{th}\) and the difference of the reciprocals of two relay emission powers. When the signal power \(p_{s}\) is strong enough, the difference of the reciprocals of two relay emission powers will not be the main influence factor.