# Exploiting Outage and Error Probability
## 3 System Performance Analysis and Our Proposed Cooperative Routing Protocols
### Outage Probability  
Outage is defined as non-availability of signal at D. In the proposed model, outage occurs when direct transmission along with all the retransmissions fail to achieve the desired SNR threshold at D.  
The expression for outage probability (\(P_{out}\)) can be written as:  
\[ \begin{array}{lcl}P_{out}&=&Pr(\gamma_{SD}\leq\gamma_{0})Pr(\gamma_{SR_{1}D}+ \gamma_{SD}\leq\gamma_{0}|\gamma_{SD}\leq\gamma_{0})\\ &&Pr(\gamma_{SR_{1}D}+\gamma_{SR_{2}D}+\gamma_{SD}\leq\gamma_{0}|\gamma_{SR_{1}D }+\gamma_{SD}\leq\gamma_{0})\cdots\\ &&Pr(\sum_{i=1}^{m}\gamma_{SR_{i}D}+\gamma_{SD}\leq\gamma_{0}|\sum_{i=1}^{m-1} \gamma_{SR_{i}D}+\gamma_{SD}\leq\gamma_{0})\end{array}  \tag{11}\]  
In Equation (11), the first term \"\(Pr(\gamma_{SD}\leq\gamma_{0})\)\" represents the failure probability of direct link which requires the first relay to retransmit the signal. Therefore, first relay is needed to retransmit the signal. The second term \"\(Pr(\gamma_{SR_{1}D}+\gamma_{SD}\leq\gamma_{0}|\gamma_{SD}\leq\gamma_{0})\)\" represents the probability that combined signal at the destination is below \(\gamma_{0}\) when direct transmission has already suffered outage. Similarly, the third term \"\(Pr(\gamma_{SR_{1}D}+\gamma_{SR_{2}D}+\gamma_{SD}\leq\gamma_{0}|\gamma_{SR_{1}D }+\gamma_{SD}\leq\gamma_{0})\)\" shows that second retransmission is also failed to achieve SNR above \(\gamma_{0}\), provided that the first retransmission is also in outage. In third term, second retransmission is combined with the first retransmission along with directly transmitted signal by using MRC. These retransmissions continue till SNR above \(\gamma_{0}\) is achieved at destination or all available relays are utilized. When \(m\)th retransmission fails to achieve SNR greater than \(\gamma_{0}\), outage is considered to be occurred. Here \(\gamma_{SR_{i}D}\) is the equivalent SNR of \(S\to R_{i}\to D\). It is defined in Equation (12) below. \[\gamma_{SR_{i}D} = min(\gamma_{SR_{i}},\gamma_{R_{i}D}) \tag{12}\] \[= \frac{\gamma_{SR_{i}}\gamma_{R_{i}D}}{\gamma_{SR_{i}}+\gamma_{R_{i }D}+1} \tag{13}\]  
By using law of conditional probability, Equation (11) can be reduced to:  
\[P_{out}=Pr(\sum_{i=1}^{m}\gamma_{SR_{i}D}+\gamma_{SD}\leq\gamma_{0}) \tag{14}\]  
In order to calculate a closed-form expression, \(m\) is limited to 3 for the sake of simplicity. Hence, \(P_{out}\) can be expressed as:  
\[P_{out}=Pr(\gamma_{SR_{1}D}+\gamma_{SR_{2}D}+\gamma_{SR_{3}D}+\gamma_{SD}\leq \gamma_{0}) \tag{15}\]  
See Appendix C for derivation of Equation (15). To calculate \(P_{out}\), it is required to know the output SNR at the destination. Since MRC is used at destination, the SNR at D, \(\gamma_{D}\), is the sum of direct signal, \(\gamma_{SD}\), and relayed signals, \(\sum_{i=1}^{3}\gamma_{SR_{i}D}\). Where \(\gamma_{SR_{i}D}\) is the equivalent SNR of \(S\to R_{i}\to D\)[15; 29]. A tight upper bound for \(\gamma_{SR_{i}D}\) is given in [30]. Since fading is Rayleigh distributed, therefore, \(\gamma\) follows exponential distribution with mean \(\tilde{\gamma}\)[15]. The Probability Distribution Function (PDF) of \(min(\gamma_{SR_{i}},\gamma_{R_{i}D})\) is also exponential with a mean \(\tilde{\gamma}_{R_{i}}=min(\tilde{\gamma}_{SR_{i}},\tilde{\gamma}_{R_{i}D})= \frac{\gamma_{SR_{i}}\gamma_{R_{i}D}}{\tilde{\gamma}_{SR_{i}}+\tilde{\gamma}_{ R_{i}D}+1}\). For sum of exponentially distributed independent random variables, their PDF is the convolution of these variables. The mean of output SNR at D, \(\tilde{\gamma}_{d}\) is given as \(\tilde{\gamma}_{D}=\tilde{\gamma}_{SD}+\tilde{\gamma}_{R_{1}}+\tilde{\gamma }_{R_{2}}+\tilde{\gamma}_{R_{3}}\). PDF of \(gamma_{d}\), \(\tilde{\gamma}\) is given as:\[