## 3 Outage Capacity  
A very critical performance metric for each OWC system is its outage capacity, \(C_{out}\), which stands for the capacity guaranteed for a percentage rate of \((1-r)\) of the channel realizations [44; 45].  
\[\text{Pr}[C<C_{out}]=r, \tag{12}\]where Pr[.] denotes probability, and \(C\) represents the instantaneous channel capacity which is obtained as follows [43]:  
\[C=B\log_{2}(1+\gamma)=\frac{1}{\ln 2}B\ln(1+\gamma), \tag{13}\]  
where \(B\) is the channel's bandwidth.  
It is appropriate to clarify here the differences between outage probability, outage capacity, and capacity metrics. Capacity denotes the achievable maximum data rate just only for a specific moment, while the outage capacity represents the capacity guaranteed for a specific probability of the channel realizations. Additionally, the outage probability represents the probability that the instantaneous SNR at the receiver side falls below a specific SNR threshold that corresponds to the receiver's sensitivity, while the outage capacity could represent the capacity guaranteed for a specific outage probability. Moreover, higher outage probability values correspond to a degraded outage performance for the system, while higher outage capacity values correspond to an upgraded outage performance for the system [36; 44]. Indeed, considering Equation (12), the outage capacity refers to the minimum capacity which is required for a specificpercentage rate of channel realizations, while outage probability refers to the required instantaneous SNR threshold.  
Since, in a real wireless channel, the instantaneous electrical SNR, \(\gamma\), is a random variable, it becomes evident from Equation (13) that \(C\) is also a random variable. Therefore, the probability of Equation (12) is evaluated asfollows [42; 45]:  
\[r=\int\limits_{0}^{C_{out}}f_{C}(C)\;dC, \tag{14}\]  
where \(f_{C}(C)\) is the PDF of the random variable \(C\).  
Using Equations (11) and (13), along with standard technique of transforming random variables, \(f_{C}(C)=\frac{f_{j}(\gamma)}{|\partial C/\partial\gamma|}\Bigg{|}\)\(\gamma=2^{\frac{C}{8}}-1\)[47], the latter PDF is obtained asfollows:  
\[f_{C}(C)=\frac{\psi^{2}\ln 2}{2B}\bigg{(}\frac{\psi^{2}}{\psi^{2}+2}\bigg{)}^{ \frac{\psi^{2}}{2}}\mu^{\frac{-\psi^{2}}{2}}2^{C/B}\bigg{(}2^{C/B}-1\bigg{)}^ {\frac{\psi^{2}-2}{2}}. \tag{15}\]  
Thus, by substituting Equation (15) into Equation (14), we get  
\[r=\frac{\psi^{2}\ln 2}{2B}\bigg{(}\frac{\psi^{2}}{\psi^{2}+2}\bigg{)}^{ \frac{\psi^{2}}{2}}\mu^{\frac{-\psi^{2}}{2}}\int\limits_{0}^{C_{out}}2^{C/B} \Big{(}2^{C/B}-1\Big{)}^{\frac{\psi^{2}-2}{2}}\;dC. \tag{16}\]  
In order to calculate the integral of Equation (16), we set \(u=\sqrt{2^{C/B}-1}\), \(du=\frac{2^{C/B}\ln 2}{2B\sqrt{2^{C/B}-1}}dC\). Thus, the above integral is calculated as  
\[\int\limits_{0}^{2^{C_{out}/B}-1}u^{\psi^{2}-2}u(2B/\ln 2)\;du=\frac{2B}{ \psi^{2}\ln 2}\Big{(}\sqrt{2^{C_{out}/B}-1}\Big{)}^{\psi^{2}}. \tag{17}\]  
Therefore, by substituting Equation (17) into Equation (16), we obtain  
\[r=\bigg{(}\frac{\psi^{2}}{\psi^{2}+2}\bigg{)}^{\frac{\psi^{2}}{2}}\mu^{\frac{ -\psi^{2}}{2}}\Big{(}\sqrt{2^{C_{out}/B}-1}\Big{)}^{\psi^{2}}, \tag{18}\]which can also be expressed by substituting Equation (9) into Equation (18) asfollows:  
\[C_{out}/B=\log_{2}\left\{10^{\frac{2}{\
u^{2}}\log{(r\frac{\varphi^{2}}{\varphi^{2 }+2})}^{-\frac{\
u^{2}}{2}}(\frac{\
u^{2}\eta^{2}\,\lambda_{0}^{2}B\,\exp{[- \alpha(\lambda)\delta]}}{(\varphi^{2}+2)N_{0}})^{\frac{\varphi^{2}}{2}}}\right]+ 1\right\}. \tag{19}\]  
The derived analytical expressions in Equations (18) and (19) reveal that, in addition to the available bandwidth, the outage capacity for TOW links depends strongly on the pointing error's strength, average electrical SNR at the receiver's side, skin-induced attenuation, and specific link characteristics such as skin channel thickness, power and noise spectral densities, and operational wavelength.