Optimal robust configuration in cloud environment based on heuristic optimization algorithm

Multi-server system

Portions of this text were previously published as part of a preprint (Zhou, Chen & Kuang, 2023). In this article, we consider an M/M/m multi-server queuing system, which is shown in Fig. 1. The notations used are listed in Table 1. For the multi-server system, it has m identical servers with speed s (hundred billion instructions per hour). When a customer arrives the queue with service requests, it will be served immediately as long as any server is available. Due to the randomness of the arrival of the customers, we consider the time it takes customers to arrive at the system as a random variable with an independent and identically distributed (i.i.d.) exponential distribution whose mean is 1/λ. In other words, the service requests follow a Poisson process with arrival rate λ. The task execution demand is measured by the number of instructions, which can be denoted by a the exponential random variable j with mean $\overline{j}$. Thus, the execution time of a task on the multi-server system can also be considered as an exponential random variable t = j/s with mean $\overline{t}=\overline{j}/s$. Moreover, the service rate can be denoted as $\mu =1/\overline{t}=s/\overline{j}$ and the utilization factor is defined as $\rho =\lambda /m\mu =\lambda /m\cdot \overline{j}/s$, which means the percentage of the average time when the server is in busy state. Since the total number of service requests brought by customers is M, which determines the upper limit of the service requests being waited or processed in the M/M/1//M queuing system, Let p_k denotes the probability that there are k service requests (waiting or being processed) in the M/M/m queuing system. Then, we have

where (2)${\displaystyle p_0={\left(\sum _{k=0}^{m-1}\frac{{\left(m\rho \right)}^k}{k!}+\frac{{\left(m\rho \right)}^m}{m!}\frac{1}{1-\rho }\right)}^{-1}.}$

To ensure the ergodicity of queue system, it is obviously that the condition 0 < ρ < 1 should be satisfied.

On this basis, when all servers in a multi-server system are occupied by executing service requests, newly arrived service requests must wait in the queue. We express the probability (i.e., the probability that a newly arrived service request have to wait) as follows. (3)${\displaystyle p_q=\sum _{k=m}^{\infty }{p}_k=\frac{p_m}{1-\rho }=p_0\frac{{\left(m\rho \right)}^m}{m!}\cdot \frac{1}{1-\rho }.}$

Waiting time distribution

When enjoying the services, customers always want to have shorter waiting time. Therefore, it is necessary to model the waiting time of customers, such that it can be better quantified. With this premise in mind, we denote W as the waiting time of the service request arriving at the multi-server system. Notice that, the probability distribution function (pdf) of waiting time in M/M/m queuing system has already been fully considered in previous literatures, which is shown as follow Mei et al. (2018). (4)${\displaystyle f_W\left(t\right)=\left(1-p_q\right)u\left(t\right)+m\mu p_m{e}^{-\left(1-\rho \right)m\mu t}}$ where $p_m=p_0{\left(m\rho \right)}^m/m!$ and $u\left(t\right)$ is unit impulse function, which is defined as (5)${\displaystyle u_z\left(t\right)=\left\{\begin{array}{cc}\hfill z,\hfill & \hfill 0⩽t⩽\frac{1}{z}\hfill \\ \hfill 0,\hfill & \hfill t>\frac{1}{z}\hfill \end{array}.\right.}$

The function $u_z\left(t\right)$ has the following property (6)${\displaystyle {\int }_0^{\infty }{u}_z\left(t\right)dt=1}$

namely, $u_z\left(t\right)$ can be treated as a pdf of a random variable with expectation (7)${\displaystyle {\int }_0^{\infty }t{u}_z\left(t\right)dt=z{\int }_0^{1/z}tdt=\frac{1}{2z}.}$

Let z → ∞ and define $u\left(t\right)=\underset{z\to \infty }{\overset{}{lim}}{u}_z\left(t\right)$. It is clear that any random variable whose pdf is $u\left(t\right)$ has expectation 0.

Further, according to Eqs. (4) and (6), we can calculate the expectation of waiting time as follow. (8)${\displaystyle \begin{array}{cc}T\hfill & ={\int }_0^{+\infty }t\cdot f_W\left(t\right)dt\hfill \\ \hfill & ={\int }_0^{+\infty }t\cdot \left(\left(1-p_q\right)u\left(t\right)+m\mu p_m{e}^{-\left(1-\rho \right)m\mu t}\right)dt\hfill \\ \hfill & ={\int }_0^{+\infty }t\cdot m\mu p_m{e}^{-\left(1-\rho \right)m\mu t}dt\hfill \\ \hfill & =\frac{p_m}{m\mu {\left(1-\rho \right)}^2}\hfill \end{array}.}$

Profit modeling

Cloud service providers play an important role in maintaining the cloud service platform as well as serving customer requests. During the operation of the platform, for the cloud service provider, revenue can be obtained from the customers whose requests have to be served, while operational costs have to be paid for maintaining the platform. In this article, we denote ɛ and C as the revenue and the cost of cloud service provider respectively. Subtracting the former to the latter, the profit of cloud service provider can be obtained, which can be demonstrated in Eq. (9). (9)${\displaystyle G=\varepsilon -C}$ where, G is the profit obtained by cloud service provider. In the following subsections, the revenue and cost of cloud service provider will be modeled in detail.

Robustness analysis

Considering the demands of cloud service providers and customers described in the previous section, most researches have focused on achieving optimal results under certain conditions. However, to the best of our knowledge, perturbation factors have never been discussed in the modeling and solving process of this problem, which may result in some undesirable impacts for cloud service providers and customers. In this section, we will describe the impacts of specific perturbation factors on the profit of cloud service providers and the waiting time of customers, and then design a robust strategy to protect the demands of cloud service providers and customers as much as possible.

The derivation of equations Eqs. (8) and (9) gives us an initial insight into the demands of cloud service providers and customers. Based on this, we need to further quantify and analyze the perturbation factors (i.e., server size and speed, which are often discussed in references about profit and waiting time (Li et al., 2021; Wu et al., 2020; Li et al., 2020) in the system). According to Eqs. (8) and (9), because of the sophisticated structure of p_m, the waiting time of customers and the profit of cloud service providers are difficult to analyzed numerically. For the sake of simplicity, the following approximations are adopted, i.e., ${\sum }_{k=0}^{m-1}\frac{{\left(m\rho \right)}^k}{k!}\approx e^{m\rho }$ (closed-form approximation) and $m!\approx \sqrt{2\pi m}{\left(\frac{m}{e}\right)}^m$ (Stirling’s approximation of m!) (Wang et al., 2022a). Therefore, we get the following closed-form approximation of p_m. (17)${\displaystyle p_m\approx \frac{1-\rho }{\sqrt{2\pi m}\left(1-\rho \right){\left(e^{\rho }/e\rho \right)}^m+1}.}$

Then, substituting Eqs. (17) into (13), we have (18)${\displaystyle F_W\left(D\right)\approx 1-\frac{e^{-m\mu \left(1-\rho \right)D}}{\sqrt{2\pi m}\left(1-\rho \right){\left(e^{\rho }/e\rho \right)}^m+1}.}$

Further, according to Eqs. (17) and (18), we can obtain the approximations of the waiting time of customers and the profit of cloud service providers as follow. (19)${\displaystyle T=\frac{{\left(\lambda e\right)}^m}{{\left(sm\right)}^{m-1}{\left(sm-\lambda \right)}^2{e}^{\frac{\lambda }{s}}\sqrt{2\pi m}+\left(sm-\lambda \right){\left(e\lambda \right)}^m}}$

and, (20)${\displaystyle \begin{array}{cc}G\hfill & =\lambda a\overline{r}\left[1-\frac{e^{-m\mu \left(1-\rho \right)D}}{\sqrt{2\pi m}\left(1-\rho \right){\left(e^{\rho }/e\rho \right)}^m+1}\right]\hfill \\ \hfill \\ \hfill & -m\left(\beta +\delta \left(\rho \xi s^{\alpha }+P^{\ast }\right)\right)\hfill \end{array}.}$

In this article, we consider the perturbation factors as the unpredictable changes in server size and speed. Notice that, the utilization factor ρ in Eq. (20) can be expressed by $\rho =\frac{\lambda \bar{j}}{ms}$. By substituting such expression of ρ into Eq. (20), and setting $\bar{j}$ to 1 without loss of generality, (21)${\displaystyle G=\lambda a\left[1-\frac{e^{\left(-sm+\lambda \right)D}{\left(e\lambda \right)}^m}{\sqrt{2\pi m}\left(sm-\lambda \right)e^{\lambda /s}{\left(sm\right)}^{m+1}+{\left(e\lambda \right)}^m}\right]-\left(m\beta +\delta \lambda \xi s^{\alpha -1}+m\delta P^{\ast }\right).}$

For the sake of simplicity, Eqs. (19) and (21) can be rewritten as $T=f\left(m,s\right)$ and $G=g\left(m,s\right)$ respectively. Obviously, it can be found the strong nonlinear characteristics in Eqs. (19) and (21), which are adverse to the numerical analysis of such formulas. In this case, we choose to draw the graphics of $T=f\left(m,s\right)$ and $G=g\left(m,s\right)$ to show the functional relationship intuitively.

By setting λ = 10 hundred service requests per hour, a = 10 dollars per one hundred billion instructions (Note: The monetary unit “dollars” in this article may not be identical but should be linearly proportional to the real US dollars.), β = 1.5 dollars per one hundred billion instructions, δ = 0.1 and P^∗ = 2, and then traversing m ∈ [6, 20], s ∈ [0.5, 2], two surfaces with respect to Eqs. (19) and (21), are shown in Figs. 2A and 2B respectively. As can be seen from Fig. 2A, when server size and speed decrease simultaneously, customers have to take a longer waiting time for their requests to be served, and vice versa. While for Fig. 2B, when server size and speed are large, the service ability of the system far exceeds the demands of customers towards the services, which will lead to unnecessary waste to cloud service providers. As server size and speed decrease, the reduction in the cost will certainly increase the profit of cloud service providers. However, when server size and speed further decrease, the profit of cloud service providers will decrease instead, for the reason that the service ability is too weak to serve enough requests before the deadline, such that parts of the customers will enjoy the services for free.

Now, refer to all the points on the surface in Fig. 2A which satisfy f(m, s) = c₁, where c₁ is a constant, they can be connected and shown as the red curve in Fig. 2A. Similarly, by specifying c₁ ∈ [0.001, 0.1, 1], and mapping the corresponding red curves into (m, s) plane, three solid curves can be drawn in Fig. 3A. Specifically, we set the longest waiting time that customers can tolerate to be 1, which corresponds to the blue curve shown in Fig. 3A. At this time, the curve represents the bottom line of the expected performance degradation caused by the perturbation, and the region above the curve should be less than the acceptable maximum waiting time. It is obvious to find that actual server size and speed should be configured at the region above such a curve. For convenience, we refer to such region as the T-feasible region to claim the determination of such region is constrained by the waiting time of customers.

Similarly, refer to all the points on the surface in Fig. 2B that satisfy g(m, s) = c₂, where c₂ is a constant, they can be connected and shown as the red curve in Fig. 2B. By specifying c₂ ∈ [55, 60, 65], and mapping the corresponding red curves into (m, s) plane, three solid curves can be drawn in Fig. 3B. Specifically, we set the lowest profit that cloud service providers can accept to be 55, which corresponds to the blue curve shown in Fig. 3B. At this time, the curve represents the bottom line of the expected performance degradation caused by the perturbation, and the region included in the curve should be greater than the acceptable minimum profit. It is obvious to find that actual server size and speed should be configured at the region inside this curve. For convenience, we refer to such region as the G-feasible region to claim the determination of such region is constrained by the profit of cloud service providers.

Based on the above discussion, the impacts of the perturbation factors on the profit and waiting time can be described. Now, by arbitrarily determining a point in T-feasible region, which is shown as the red point in Fig. 4A, we call such point as the working point. Namely, considering the case that perturbation is not happening, the actual configuration of server size and speed is equal to the horizontal and vertical coordinates of the red point respectively. However, once the perturbation happens, the fluctuations in server size and speed will result in the deviation of the working point. If the fluctuation is small, the configuration of server size and speed can still be located in T-feasible region. Conversely, the waiting time of customers may exceed the deadline, which is unacceptable for customers.

Moreover, by arbitrarily determining a point in G-feasible region, which is shown as the red point in Fig. 4B, we also call such point as the working point. When the perturbation happens, the fluctuations in server size and speed will result in the deviation of the working point. If the fluctuation is small, the configuration of server size and speed can still be located in G-feasible region. Oppositely, the profit earned by cloud service providers may be less than the lowest value that they accept.

In order to overcome the shortcomings brought by those perturbations, we focus on optimizing the cloud computing system working in the worst situation, which can be explained as the idea of robustness. Benefit from such method, those cloud computing systems working in better situation will attain better performance.

Then, we present an intuitive explanation for the idea of robustness introduced in this article. As can be seen from Figs. 4A and 4B, by denoting the working point as H^∗ , and any point on the curve as H. We define the distance between H^∗ and H as follow to represent the shortest distance from the working point to the boundary. (22)${\displaystyle Dist={∥H^{\ast }-H∥}_2.}$

Notice that, the explanations for the idea of robustness in Figs. 4A and 4B are similar. Without loss of generality, we choose the case shown in Fig. 4A as an example. For the working point arbitrarily selected in T-feasible region, we are always able to find the shortest distance between such point and the curve, which represents the worst situation that cloud computing system works in. In this article, we aim to find the optimal working point, such that the shortest distance between such point and the curve can be maximized.

Deadline aware configuration

In this section, the goal is to ensure that customer requests are processed within a specified time-frame. Perturbations in server size and speed can significantly impact the system’s ability to meet these deadlines. For instance, if server size or speed decreases unexpectedly, the waiting time for customers may increase, leading to missed deadlines. Figure 6A illustrates how the T-feasible region, which ensures acceptable waiting time, is affected by changes in server size and speed.

As can be seen from Fig. 6A, the robustness analysis with respect to the working point in T-feasible region rather than G-feasible region is considered. In this case, we merely pursue the optimal configuration of server size and speed that ensure the strongest robustness of the system towards waiting time of customers, which is equivalent to search working point with the longest robustness radius according to Eq. (23) in T-feasible region.

To achieve the optimal configuration of server size and speed in the system, DBO is introduced to perform heuristic searching in T-feasible region. Please note that the fitness of population generated in each iteration is obtained by Algorithm 2 . The pseudo code of DBO algorithm is given in Algorithm 3 .

 
__________________________________________________________________________________________ 
  Algorithm 3: Robust configuration method based on DBO algorithm_________ 
    Input: The maximum iterations tmax, the size of the population N 
    Output: Optimal position xz andits radius r 
 1  Initialize the population i ← 1,2,...,N; xz ← [mz,sz]; b, k,S,g,e1,e2; 
  2  while t < tmax do 
     3   for  i ← 1 to N do 
    4   if i is ball-rolling dung beetle then 
5   δ ← rand(1); 
6   if δ < 0.9 then 
7   α ←through the α selection strategy; 
8   xt+1 
i    ← xt 
i + αkxt−1 
i    + b|xt 
i − xt 
w|; Obtain the ri of xt+1 
i 
by Algorithm 1; 
9   else 
10   θ ←through the θ selection strategy; 
xt+1 
i    ← xt 
i + tanθ∣ ∣ 
 xti − xt−1 
i   ∣ ∣ 
       ; Obtain the ri of xt+1 
i    by 
Algorithm 1; 
11   end 
12   end 
13   if i is brood ball then 
14   Update boundary selection based on x′ 
g; 
15   Obtain the ri corresponding to xt+1 
i    by Algorithm 1; 
16   end 
17   if i is small dung beetle then 
18   xt+1 
i    ← xt 
i + e1 ( 
   xti − lb′ 
          ) 
        + e2 ( 
   xti − ub′ 
          )  
         ; Obtain the ri of 
xt+1 
i    by Algorithm 1; 
19   end 
20   if i is thief  then 
21   xt+1 
i    ← xt 
l+Sg(|xt 
i − xt 
g| + |xt 
i − xt 
l|); Obtain the ri of xt+1 
i    by 
Algorithm 1; 
22   end 
23   end 
24   if the newly generated r is better than before then 
    25   r ← ri; xz ← xt+1 
i   ; 
26   end 
27   t ← t + 1; 
28  end    

By setting deadline as D = 0.001, and replacing T in Eq. (19) by it, we draw the boundary of T-feasible region, which is shown as the red curve in Fig. 7. As can be seen from Fig. 7, the working point with the longest robustness radius can be found in the upper-right corner with server size and speed equal to 20 and 2, respectively. An intuitive explanation can be given as below. The system with bigger server size and speed owns stronger quality of service, which will certainly lead to a smaller possibility that waiting time of customers exceeds deadline. However, even if the strongest robustness of the system towards waiting time of customers can be obtained, the price that the system has to pay is the minimization in profit of cloud service providers, which is not benefit for the long-term operation of the system. Therefore, the chosen working point [20, 2] in the upper-right corner not only minimizes the risk of exceeding deadlines but also maximizes the system’s resilience to such perturbations. This balance is crucial for maintaining service quality despite the inevitable uncertainties in server performance.

To prove the feasibility of the DBO algorithm, the robustness radius is selected as the fitness function of these heuristic algorithms. It can be seen from Fig. 8 that the DBO algorithm can achieve the global optimum with the least number of iterations. In addition, after 200 iterations, the robust radius obtained by DBO and ESOA algorithms is the same and the longest, while the effect of WOA is inferior to them. We also confirm this conclusion through Tables 2 and 3.

We perform a robustness analysis of the situation shown in Fig. 6A. At this time, let λ be fixed, and the deadlines are set to D = 0.001, D = 0.1 and D = 1, respectively. It can be found from Fig. 9A that the DBO-based robust configuration method can obtain the Optimal working point at the same location in the three cases where the server size and speed are m_z = 20 and s_z = 2, respectively. The longest robustness radii are 1.073185, 1.371784, and 1.45464, respectively. The intuitive explanation is as follows. The greater the deadline, the less likely the customer’s waiting time will exceed the deadline. Table 2 shows the results of the DBO algorithm and other comparison algorithms for this analysis.

Through Fig. 9B, we describe how the robustness radius of the system changes with λ at a fixed deadline, where the dotted box represents the T-feasible region of the equivalent change when λ changes. Specifically, when λ increases, the robustness radius increases accordingly. This means that in the case of higher arrival rates, the system needs to be robust on a wider scale to ensure that service requests can be processed promptly within a given deadline. Table 3 provides specific values, showing the change of the robustness radius of the system under different arrival rates. These data further confirm the trend in Fig. 9B. The data in the table provides a quantitative reference for system designers to optimize the configuration in practical applications.

Profit aware configuration

In this section, the primary objective is to maximize the profit of cloud service providers. Perturbations in server size and speed can affect the system’s profitability. For example, a decrease in server speed may lead to increased operational costs or reduced service quality, impacting the overall profit. As shown in Fig. 6B, the G-feasible region represents the configurations that ensure acceptable profit. Perturbations can shift the working point outside this region, leading to suboptimal profit outcomes. In this case, we merely pursue the optimal configuration of server size and speed that ensures the strongest robustness of the system towards the profit of cloud service provider, which is equivalent to search working point with the longest robustness radius according to Eq. (23) in G-feasible region.

As can be seen from Fig. 6B, the robustness analysis with respect to the working point in G-feasible region rather than T-feasible region is considered. In this case, we merely pursue the optimal configuration of server size and speed that ensures the strongest robustness of the system towards the profit of cloud service providers, which is equivalent to searching the working point with the longest robustness radius in G-feasible region.

Now, the optimal configuration of server size and speed can be obtained by the algorithm similar to the one applied in the previous subsection. By setting the lowest profit that cloud service providers can accept as G = 65, and substituting it in Eq. (21), we draw the boundary of G-feasible region, which is shown as the blue curve in Fig. 10. As can be seen from Fig. 10, the working point with the longest robustness radius can be found in the center with server size and speed equal to 12.7981 and 1.1153, respectively. Please note that since the boundary of G-feasible region is closed, the idea of keeping the working point away from the boundary as far as possible applied in the previous subsection is not suitable for the case in this subsection. Therefore, the chosen working point in the center not only meets the profit requirements but also maximizes the system’s resilience to such perturbations. This balance is essential for ensuring both profitability and stability in the face of unavoidable uncertainties in server performance.

To verify the superiority of the DBO algorithm, ESOA and WOA are adopted to obtain a robustness radius as well. Note that the robustness radius is selected as the fitness function of these heuristic algorithms. It can be seen from Fig. 11 that the longest robust radius obtained by all these algorithms after 200 iterations is basically the same, and the DBO algorithm can achieve the global optimum with the least number of iterations. In addition, we can find from Table 4 that the results obtained by DBO are superior to other comparison algorithms when solving this problem.

Then, we perform the robustness analysis for the case presented in Fig. 6B with the lowest profit that cloud service providers can accept being set to G = 55, G = 60 and G = 65, respectively. As can be seen from Fig. 12A, by simulating the proposed algorithm, the optimal working point can be obtained for three cases, where server size and speed equal to [13.6891, 1.2479], [13.2120, 1.1846], [12.7987, 1.1153], respectively, and the corresponding longest robustness radii are 0.440086, 0.330031 and 0.203571. Then, for the purpose of practical application, we further find the approximate solution in the neighborhood of the optimal working point. Firstly, by setting server size to the floor of the abscissa of the optimal working point, and adopting Algorithm 2 to obtain the corresponding server speed, we have one group of actual optimal working point with server size and speed equal to [13, 1.2891], [13, 1.1982], [12, 1.1733] for the three cases, and the corresponding robustness radii are 0.438730, 0.329844 and 0.201064. Secondly, by setting server size to the ceiling of the abscissa of the optimal working point, and to obtain the corresponding server speed with the same method, we have another group of actual optimal working point with server size and speed equal to [14, 1.2300], [14, 1.1356], [13, 1.1012] for the three cases, and the corresponding robustness radii are 0.439834, 0.328240 and 0.203329. Finally, to compare the robustness radii for each case, the actual optimal working point with a larger robustness radius is chosen to configure the cloud computing platform.

Furthermore, we perform the robustness analysis for the case presented in Fig. 6B when the arrival rate of customers changes. In this case, λ is selected in $\left\{10,12,14,16,18,20\right\}$ and G remains the same. As can be seen from Fig. 12B, by simulating the proposed algorithm, the optimal working point can be obtained for six cases, where server size and speed equal to [13.2120, 1.1846], [16.8255, 1.3126], [20.3930, 1.3997], [23.1644, 1.4676], [27.3285, 1.5135], [30.5062, 1.5577] respectively, and the corresponding longest robustness radii are 0.330031, 0.545380, 0.676727, 0.767042, 0.833406 and 0.884386. Then, for practical application, we further find the approximate solution in the neighborhood of the optimal working point. Similarly, by setting server size to the floor of the abscissa of the optimal working point, and adopting Algorithm 2 to obtain the corresponding server speed, we have one group of actual optimal working point with server size and speed equal to [13, 1.1982], [16, 1.3507], [20, 1.4137], [23, 1.4902], [27, 1.521646], [30, 1.5688] for the six cases, and the corresponding robustness radii are 0.329844, 0.544136, 0.676582, 0.766546, 0.833346, 0.884203. Secondly, by setting server size to the ceiling of the abscissa of the optimal working point, and to obtain the corresponding server speed with the same method, we have another group of actual optimal working point with server size and speed equal to [14, 1.1356], [17, 1.3048], [21, 1.3786], [24, 1.4607], [28, 1.4970], [31, 1.5470] for the three cases, and the corresponding robustness radii are 0.328240, 0.545326, 0.676336, 0.767023, 0.833110 and 0.884336. Finally, according to the selection principle to maximize the robustness radius, the actual optimal working point that makes the robustness radius larger is determined as a better solution to configure the cloud computing system.

Figure 12B depicts how the robustness radius of the system changes with the arrival rate under fixed profit. Specifically, when the arrival rate increases, the robustness radius increases accordingly. This means that in the case of higher arrival rates, the system needs to be robust in a wider range to ensure that service requests can be effectively processed under given profit requirements.

Table 5 provides specific values, showing the change in the robustness radius of the system under different arrival rates. These data further confirm the trend in Fig. 12B: as the arrival rate increases, the robustness radius increases. The data in the table provides a quantitative reference for system designers to optimize the configuration in practical applications to ensure the robustness and profitability of the system under various load conditions.

Profit and deadline aware configuration

By summarizing the previous discussions about robustness analysis, the maximization problems of robustness radius with respect to the working point in T-feasible region and G-feasible region are discussed separately. However, when referring to the requirements of cloud service providers and customers, one to be satisfied always comes with the cost of impairing another. Therefore, it is necessary to find the optimal working point by combining with T-feasible region and G-feasible region.

Now, the cases presented in Figs. 6C and 6D are taken into consideration. Please note that the case presented in Fig. 6D is similar to the one in Fig. 6B, the only difference is that the optimal working point selected in the former case satisfies the requirement of customers as well, while the one chosen in the latter case is not. In this subsection, we mainly focus on the case presented in Fig. 6C.

By setting deadline as D = 0.1, and replacing T in Eq. (19) by it, setting the lowest profit that cloud service providers can accept as G = 60, and substituting it in Eq. (21), we draw the boundaries of T-feasible region and G-feasible region, which are shown as the red and blue curve in Fig. 13 respectively.

where, G denotes the lowest profit that cloud service providers can accept, and D denotes the deadline. To solve the multi-objective programming model presented in Eq. (26), Non-dominated Sorting Genetic Algorithms II (NSGA-II) is introduced in this article. The pseudo code for such algorithm is shown in Algorithm 4 .

 
__________________________________________________________________________________________ 
  Algorithm 4: Robust configuration based on NSGA-II algorithm     _________ 
    Input:  interval of server size[mmin,mmax], server speed[smin,smax] 
    Output: optimal h1,h2 
  1  begin 
     2   t ← 1,i ← 1; 
3   Initialize the population in the given interval; 
4   The fitness value Y is calculated through Algorithm 1; 
5   T1 ← perform non-dominated sorting strategy; 
6   Sort f by crowding distance for each rank ; 
7   while t < Max number of iterations do 
    8   Pt ← create parent by T1 using tournament selected; 
9   Qt ← create offspring by Pt using selection, crossover and 
mutation; 
10   Rt ← Pt ∪ Qt ; 
11   The fitness value Yt is calculated through Algorithm 1; 
12   Ft ← calculate all non-dominated fronts of Rt; 
13   Pt+1 ←∅ ; 
14   while |Pt+1|∪∣ ∣ 
 Fit∣ ∣ 
   ≤ N do 
15   Fit ← select ith non-dominated front in Ft using crowding 
distance sorting strategy; 
16   Pt ← Pt+1 ∪ Fit; 
17   i ← i + 1; 
18   end 
19   Tt+1 ← Pt+1 ∪ Ft [1 : (N −|Pt+1|)]; 
20   t ← t + 1; 
21   end 
22  end    

By applying Algorithm 4 in Eq. (26), the result can be shown in Fig. 13. As can be seen from Fig. 14, each point on the Pareto frontier represents an optimal solution for the multi-objective programming model. Without loss of generality, we choose the red point marked in Fig. 14 as the best point on the left side as the optimal working point. On this basis, by setting server size and speed to m_z = 14.5967 and s_z = 1.1518, we can obtain the first fitness function as $\left|r_1-r_2\right|=3.19\times 10^{-5}$ and the second fitness function as r₁ + r₂ = 0.5419. In this case, the longest robustness radius from the working point to the boundaries of T-feasible region and G-feasible region are r₁ = 0.2709 and r₂ = 0.2709 respectively. Then, for practical application, we further find the approximate solution in the neighborhood of the optimal working point. By setting server size to the floor and ceiling of the abscissa of the optimal working point, and adopting Algorithm 4 to obtain the corresponding server speed, we have the actual optimal working point with server size and speed equal to [14, 1.1922] and [15, 1.1260], respectively. Moreover, since the robustness radius corresponding to the two cases are 0.270 and 0.268, respectively, the actual optimal working point with a larger robustness radius is chosen to configure the cloud computing platform.