# Modeling and Evaluation of Power-Aware Software Rejuvenation in Cloud Systems
## 6 Performance Evaluation  
In this section, the desired output measures that can be obtained using the proposed SAN model are defined, and then the proposed model is evaluated. The output measures are computed applying the Markov reward approach [41]. In this approach, appropriate reward rates are assigned to each feasible marking of a SAN model, and then the expected accumulated reward is computed in the steady state. To this end, the proposed analytical model is solved with the Mobius tool [33]. In the following formulas, \(Mark(P_{i})\) returns the number of tokens inside place \(P_{i}\). The ratio of serving VMs to accepted requests is defined as the ratio of the steady-state number of VMs that are being served to the steady-state number of accepted VM requests in the cloud system. This measure is computed using Equation (1):  
\[\small RSA=\sum_{i=1}^{N}E[Mark(P_{CurrentVM_{S_{i}}})]/(E[Mark(P_{Init})]+E[ Mark(P_{Mig})]+\sum_{i=1}^{N}E[Mark(P_{CurrentVM_{S_{i}}})]) \tag{1}\]  
The other measure that is obtained by applying the proposed SAN model is the probability of VMM failure, which can be computed by the following equation:  
\[\small PR_{VMMF}\ =\ \sum_{i=1}^{N}E[Mark(P_{Failedi})]/N \tag{2}\]  
One of the main goals of introducing rejuvenation in cloud systems is to reduce the mean number of failed VMs, which can be computed by the product of the mean number of VMs that are being served on each VMM and the probability of VMM failure in the steady state. The mean number of failed VMs can be calculated as follows:  
\[NFV=\big{(}\sum_{i=1}^{N}E\Big{[}Mark\Big{(}P_{CurrentVM_{S_{i}}}\Big{)}\Big{]}/N \big{)}*\big{(}\sum_{i=1}^{N}E[Mark(P_{Failedi})]/N\big{)} \tag{3}\]  
The _total power consumption_ of cloud servers is another measure which can be computed by the proposed model and is the mean value of power consumed by PMs, VMMs, and VMs of the data center. It can be computed by Equation (4). \[TPC=\rho_{b}*(N-\ E[Mark(P_{Idle})])+\rho_{v}*\sum_{i=1}^{N}E[Mark(P_{CurrentVM _{S_{i}}})] \tag{4}\]  
where \(\rho_{b}\) is the average power consumed by a switched-on PM running a VMM, \(\rho_{v}\) is the average power consumed by a VM, and \(N\) denotes the number of PMs in the system. In order to evaluate the proposed model, we assumed a system with 3 PMs and the maximum of 3 VMs per each VMM. In this setting, the input parameter \(T_{1}\) is 8, which means a VMM of age 8 could be rejuvenated if other VMMs have enough capacity to accept its VMs. The parameter \(T_{2}\) is set to 10, which means a VMM of age 10 has to be rejuvenated instantaneously. In other words, regardless of the capacity of other VMMs, the active VMs of a VMM aged 10 should be placed on the migration queue, and the VMM should be rejuvenated. The other values set for the input parameters of the proposed model are presented in Table 4. Most of the values used herein as input parameters of the proposed SAN model are in the range of the values considered in other related work [10; 11; 15; 22; 29; 36; 37; 39; 42]. In order to compare different situations, the proposed model was evaluated in three modes: (1) pure power saving, (2) pure rejuvenation, and (3) with simultaneous application of rejuvenation and power saving, called power-aware rejuvenation. Modes 1 and 2 are the baselines for our main proposed scheme, which is mode 3. In mode 3, all five sub-models of Figure 2 are used. In mode 1, the rejuvenation sub-model shown in Figure 2d is omitted. In mode 2, the power saving sub-model represented in Figure 2c is removed, and the model is adjusted so that no token would be placed in place PIdle. The results obtained from solving the proposed model in the three modes are presented in Figure 3.  
Figure 3a shows the average probability of VMM failure in the proposed model in the three evaluation modes. One of the main causes of VMM failure is the increasing number of served VMs and the errors introduced due to VMM aging. With the pure rejuvenation scheme, aged VMMs are