# Service Level Agreement Violations in Cloud Storage: Insurance and Compensation Sustainability
## 4 Loss Statistics
### The Pareto-Lognormal Model
#### Number of Long Outages  
For the expected number of long outages, we go back to the general expression in Equation (11). Since the number of outages is a random variable, we can again exploit the product of averages. After recalling Equation (30) for the expected number of outages and the tail of the lognormal distribution, we have  
\[\mathbb{E}[N_{\text{lf}}] =\mathbb{E}[N_{\text{f}}]\mathbb{E}[\mathbb{I}_{[D_{1}>w]}]= \mathbb{E}[N_{\text{f}}]\mathbb{P}(D_{i}>w)\] \[=\mathbb{E}[N_{\text{f}}]\left[1-\text{G}\left(\frac{\ln(w)-\mu} {\sqrt{2}\sigma}\right)\right]\] \[=\left[1-\left(\frac{h}{T}\right)^{\alpha}+\sum_{i=1}^{\infty} \text{G}\left(\frac{T-\mathbb{E}[Z_{i}]}{\sqrt{\mathbb{V}[Z_{i}]}}\right)\right] \tag{34}\] \[\times\left[1-\text{G}\left(\frac{\ln(w)-\mu}{\sqrt{2}\sigma} \right)\right].\]  
As to the variance, we adopt the same approach as in Section 4.2:  
\[\mathbb{V}[N_{\text{lf}}] =\mathbb{E}^{2}[\mathbb{I}_{[D_{i}>w]}]\mathbb{V}ar[N_{\text{f}}] +\mathbb{V}[\mathbb{I}_{[D_{i}>w]}]\mathbb{E}[N_{\text{f}}]\] \[=\mathbb{P}^{2}(D_{i}>w)\mathbb{V}[N_{\text{f}}]+(1-\mathbb{P}(D_ {i}>w))\mathbb{P}(D_{i}>w)\mathbb{E}[N_{\text{f}}]\] \[=\left[1-\text{G}\left(\frac{\ln(w)-\mu}{\sqrt{2}\sigma}\right) \right]^{2}\] \[\times\left[1-\left(\frac{h}{T}\right)^{\alpha}+\sum_{i=1}^{ \infty}(2i+1)\text{G}\left(\frac{T-\mathbb{E}[Z_{i}]}{\sqrt{\mathbb{V}[Z_{i}] }}\right)-\mathbb{E}[N_{\text{f}}]^{2}\right] \tag{35}\] \[+\left[1-\text{G}\left(\frac{\ln(w)-\mu}{\sqrt{2}\sigma}\right) \right]\left[\text{G}\left(\frac{\ln(w)-\mu}{\sqrt{2}\sigma}\right)\right]\] \[\times\left[1-\left(\frac{h}{T}\right)^{\alpha}+\sum_{i=1}^{ \infty}\text{G}\left(\frac{T-\mathbb{E}[Z_{i}]}{\sqrt{\mathbb{V}[Z_{i}]}} \right)\right].\]  
Again, we can now easily get the expected value and the variance of the economic loss \(X\):  
\[=\mathbb{E}[N_{\text{f}}]\left[1-\text{G}\left(\frac{\ln(w)-\mu} {\sqrt{2}\sigma}\right)\right]\]0  
The equilibrium equation may not be amenable to a closed form solution, depending on the nonlinear nature of the utility function. We can find an approximate solution by expanding both terms through a Taylor series in the neighbourhood of \(\omega-\mathbb{E}[X]\):  
\[ \begin{split} u(\omega-P)&\simeq u(\omega- \mathbb{E}[X])+(\mathbb{E}[X]-P)u^{\prime}(\omega-\mathbb{E}[X]),\\ u(\omega-X)&\simeq u(\omega-\mathbb{E}[X])+( \mathbb{E}[X]-X)u^{\prime}(\omega-\mathbb{E}[X]),\\ &+\frac{(\mathbb{E}[X]-X)^{2}}{2}u^{\prime\prime}(\omega- \mathbb{E}[X]).\end{split}  \tag{43}\]  
By employing the second of these Taylor expansions, we get  
\[ \begin{split}\mathbb{E}[u(\omega-X)]&\simeq u( \omega-\mathbb{E}[X])+(\mathbb{E}[X]-\mathbb{E}[X])u^{\prime}(\omega-\mathbb{E }[X])\\ &+\frac{\mathbb{V}[X]}{2}u^{\prime\prime}(\omega-\mathbb{E}[X])\\ &=u(\omega-\mathbb{E}[X])+\frac{\mathbb{V}[X]}{2}u^{\prime \prime}(\omega-\mathbb{E}[X]).\end{split}  \tag{44}\]  
Finally, by replacing those expressions in the equilibrium Equation (42), we obtain the maximum tolerable premium  
\[P\simeq\mathbb{E}[X]+\frac{\mathbb{V}[X]}{2}r(\omega-\mathbb{E}[X]), \tag{45}\]  
where we have introduced the risk aversion function \(r\), which takes into account the effect of the utility function:  
\[r(x)=-\frac{u^{\prime\prime}(x)}{u^{\prime}(x)}. \tag{46}\]  
Though several options are possible for the utility function (see, e.g., Section 1.3 in [23]), a typical assumption is to consider a constant risk aversion coefficient  
\[r(x)=\delta>0. \tag{47}\]  
In that case, the utility function exhibits the Constant Absolute Risk Aversion (CARA) property [24]; the only function that satisfies the CARA property is the exponential function \(u(x)=1-e^{-\delta x}\), which leads to the maximum tolerable premium  
\[P\simeq\mathbb{E}[X]+\delta\frac{\mathbb{V}[X]}{2}. \tag{48}\]  
The CARA property has been assumed, e.g., in [13; 25].