# Comprehensive Comparison of VNE Solutions Based on Different Coordination Approaches
## 4 Evaluation of Examined Solutions
### Adopted Objective Functions  
The targeted objectives of the solutions participating in our simulations are, hereafter, listed and explained, being a key and critical element of the VNE problem. Specifically, themain interest of the authors of [18] is to maximize the long-term average revenue as given by the following notation:  
\[\lim_{T\rightarrow\infty}\frac{\sum_{t=0}^{T}R(G^{V}(t))}{T} \tag{6}\]  
To achieve this goal, it is crucial to embed incoming VNRs efficiently, such that the substrate resources are minimally occupied, hence, the maximization of revenue. For a VNR denoted as \(G^{V}\), they define its revenue \(R(G^{V}(t))\) at any particular chronological instance \(t\), as:  
\[R(G^{V}(t))=\sum_{i^{l}\in L^{i}}BW(l^{i})+a\sum_{n^{i}\in N^{i}}\text{CPU}(n^{ i}) \tag{7}\]  
where \(l^{i}\) is a virtual link that belongs to the set \(L^{i}\), and \(n^{i}\) is a virtual node that belongs to the set \(N^{i}\). They also introduce the tunable weight \(\alpha\), which allows the InP to maintain a balanced outcome regarding the imminent costs of the given resource parameters, \(BW(l^{i})\) and \(\text{CPU}(n^{i})\).  
The authors of [22] follow a similar approach regarding the principle of their objective. They advocate that the lower the residual capacity of a particular substrate node or link, the higher the value of the objective function. This notion is notated as follows:  
\[\begin{split}&\text{\emph{minimize}}\sum_{uv\in L_{S}}\frac{ \alpha_{uv}}{R_{L_{S}}(u,v)}\sum_{l\in L^{i}}BW(l)\\ &\quad+\sum_{n\in N_{S}}\frac{\beta_{n}}{R_{N_{S}}(n)}\sum_{m\in N ^{i}}\text{CPU}(m)\end{split} \tag{8}\]  
This simplified form of their objective function interprets that, for the given substrate node \(n\) and physical path \(uv\), the respective resources \(R_{N_{S}}\) and \(R_{L_{S}}\) with more residual capacities are preferred over the resources with less residual capacities, thus, achieving higher values of revenue. The parameters \(\alpha_{uv}\) and \(\beta_{n}\) are used as weights in order to control the balancing of the load while embedding a VNR.  
However, in [27], the prime goal was to increase the value of the revenue divided by the total mapping cost as a means to express the quality of a VNR mapping.  
\[maximize\ R(G^{V})/cost(M(G^{V})) \tag{9}\]  
In order to achieve such a measurement for every embedded VNR, the developed algorithm calculates the costs of the node and link mappings, defined in (10) and (11), respectively. The authors associate the computational and bandwidth demands of the processed virtual networks with the constraints of CPU and BW correspondingly. The calculated costs are, therefore, combined in the constraint cost function defined in (12).  
\[cost_{1}(M(G^{V}))=\sum_{n^{i}\in N^{i}}\text{CPU}(n^{i}) \tag{10}\] \[cost_{2}(M(G^{V}))=\sum_{i^{l}\in L^{i}}BW(l^{i})\cdot length(g_{ i})\] (11) \[cost(M(G^{V}))=\sum_{i=1}^{2}cost_{i}M(G^{V}) \tag{12}\]  
The overall revenue of the mapping is denoted as \(R(G^{V})\) and expressed in (13)  
\[R(G^{V})=\sum_{n^{i}\in N^{i}}\text{CPU}(n^{i})+\sum_{l^{i}\in L^{i}}BW(l^{i}) \tag{13}\]The ratio in (9) receives values between 1 and 0, where the value of \(1\) represents an optimal VN embedding and 0 an infeasible one.