# Service Level Agreement Violations in Cloud Storage: Insurance and Compensation Sustainability
## 6 Refund Sustainability
### Unit Refund Limit  
As hinted above, the compensation that cloud providers may pay their customers should be bounded, so that it does not pose a significant threat to the overall economic balance of the company. In many cases, the compensation is set as a percentage (well below 100%) of the service fee (the compensation is therefore a partial refund); this limit guarantees that the outflow related to refunds cannot exceed that same percentage of the revenues. In Table 6, the percentage values set by some providers are reported for the case where the QoS metric is the availability. All providers allow for a larger refund percentage as the QoS degrades more and more. However, for three out of four providers, the refund can never be full. In order to limit the liability of cloud providers within safe bounds, we consider here a similar approach by limiting the unit refund. We set therefore  
\[k_{*}=\gamma F\qquad\gamma\in(0,1), \tag{59}\]  
where the asterisk is a jolly character so that the expression may be applied to any of the three performance parameters, and \(\gamma\) defines the fraction of the fee that the cloud provider may accept to lose due to compensation for each QoS violation. On the other hand, the cloud provider cannot be led to pay too high a premium, since the premium is a fixed periodic payment that erodes the provider's profit. In order to guarantee that the revenues represented by the service fees are not overly reduced, we can set the premium so as not to be larger than a predefined fraction of the service fee itself:  
\[P_{*}<\rho F\qquad\rho\in(0,1). \tag{60}\]  
Since the economic loss \(X_{*}\) is proportional to the QoS metric (which is either \(N_{\text{f}}\), \(N_{\text{lf}}\) or \(U\)), as embodied by Equations (9), (11) and (16), we introduce a generic variable \(L\) to derive a general expression that may then be tailored for the case at hand by replacing \(L\) with \(N_{\text{f}}\), \(N_{\text{lf}}\) or \(U\), respectively. The general constraint on the premium can then be expressed as  
\[ \begin{split} P_{*}&=\mathbb{E}[X_{*}]+\delta\frac {\mathbb{V}[X_{*}]}{2}=k_{*}\mathbb{E}[L]+k_{*}^{2}\delta\frac{\mathbb{V}[L]}{ 2}\\ &=\gamma F\mathbb{E}[L]+\gamma^{2}F^{2}\delta\frac{\mathbb{V}[L]} {2}<\rho F.\end{split}  \tag{61}\]  
This constraint leads to a quadratic inequality in the unit refund \(\gamma\):  
\[F\delta\frac{\mathbb{V}[L]}{2}\gamma^{2}+\mathbb{E}[L]\gamma-\rho<0. \tag{62}\]  
Since the discriminant associated with the quadratic form is  
\[\Delta=\left(\frac{\mathbb{E}[L]}{2}\right)^{2}+\frac{\delta}{2}F\rho\mathbb{ V}[L] \tag{63}\] \begin{table}
\begin{tabular}{l c c} \hline \hline
**Cloud Provider** & **Monthly Uptime [\%]** & **Service Credit [\%]** \\ \hline Amazon 1 & 99–99.99 & 10 \\  & \(<99\) & 30 \\ \hline  & 99–99.99 & 10 \\ Google 2 & 95–99 & 25 \\  & \(<\)95 & 50 \\ \hline  & 99–99.9 & 10 \\ Microsoft Azure 3 & \(<99\) & 25 \\ \hline  & 99.5–99.9 & 10 \\  & 99–99.49 & 25 \\  & 98–98.99 & 40 \\ Rackspace 4 & 97.5–97.99 & 55 \\  & 97–97.49 & 70 \\  & 96.5–96.99 & 85 \\  & \(<\)96.5 & 100 \\ \hline \hline \end{tabular} 1  
\end{table} 
Table 6: Refunds as a percentage of the service fee. and \(\gamma>0\), the solution of the quadratic inequality is  
\[\gamma<\frac{2\sqrt{\Delta}-\mathbb{E}[L]}{\delta F\mathbb{V}[L]}=\frac{\sqrt{1+2 \delta F\rho\frac{\mathbb{V}[L]}{\mathbb{E}^{2}[L]}}-1}{\delta F\frac{\mathbb{V }[L]}{\mathbb{E}[L]}}, \tag{64}\]  
which gives us an upper bound for the unit refund \(\gamma\) to be sustainable. The premium-setting procedure therefore goes through the following steps:  
1. Set the tolerable fraction \(\rho\) of the service fee as in Equation (60);
2. Compute the upperbound \(\gamma\) on the refund (Sections 6.2 and 6.3);
3. Compute the limit unit refund as per Equation (59);
4. Compute the premium (Sections 5.1 and 5.2). In the following subsections, we derive the expression for \(\gamma\) when either outage model applies and for the three QoS metrics.