# Analysis of the Macroscopic Behavior of Server Systems in the Internet Environment
## 3 Analysis of the Macroscopic Behavior of Server Systems in the Internet Environment  
### Macroscopic Behavior Model  
There are many studies on the analysis of various server systems run in the Internet environment. In these studies, generally, a single queuing system model is adopted as a model of a server system [16; 28]. Hence, in this paper, we define a model to analyze the macroscopic behavior of a server system based on a single queuing system model, as shown in Figure 4a. This model can also be applied to a cloud system that consists of multiple server systems, as shown in Figure 4b. The behavior of data processing for the server system of Figure 1 is modeled as  
\[ \begin{split} n^{\prime}=\lambda+\rho\cdot n\\ n=\rho\cdot n+T\cdot n,\end{split}  \tag{1}\]  
where \(\lambda\) is the arrived data, \(n\) is the processed data at time \(t\), \(n^{\prime}\) is data processed at time \(t+1\), \(\rho\) is the reprocessing ratio, and \(T\) is the throughput at time \(t\). Then, the ratio of data processing at time \(t\), \(W\), is defined as  
\[W(n\to n+1)=\lambda+\rho\cdot n. \tag{2}\]  
\[W(n\to n-1)=\rho\cdot n+T\cdot n. \tag{3}\]  
Let \(P(n,t)\) be the probability density of finding state \(n\) at time \(t\). Then, the change of \(P(n,t)\) at time \(t\) is defined as  
\[ \begin{split}\frac{[P(n,t+1)-P(n,t)]}{t}\cong\frac{\partial}{ \partial t}P(n,t)\\ = -[(\lambda+\rho\cdot n)+(\rho\cdot n+T\cdot n)]\cdot P(n,t)\\ +[(\lambda+\rho\cdot(n-1)]\cdot P(n-1,t)\\ +[(n+1)\cdot(\rho+T)]\cdot P(n+1,t).\end{split}  \tag{4}\]  
The first term of Equation (4) represents the effects that state \(n\) changes to states \(n+1\) and \(n-1\) at time \(t\). By contrast, the second and third terms represent the effect that states \(n+1\) and \(n-1\) change to \(n\) at time \(t\). Next, considering the following relations:  
\[x=\frac{n}{\text{N}^{\prime}},\ \mu=\frac{\lambda}{\text{N}^{\prime}},\ \text{and}\ \varepsilon=\frac{1}{\text{N}}\ll 1,\]  
Figure 4: System model in the internet environment. where N is the maximum amount of data to be processed and \(x\) is the activity factor of the system, Equation (4) is rewritten as  
\[ \begin{split}\varepsilon\frac{\partial}{\partial t}P(x,t)=-[\mu+ \rho\cdot x+(\rho+T)]\cdot P(x,t)\\ +[u+\rho\cdot(x-\varepsilon)]\cdot P(x-\varepsilon,t)\\ +[(x+\varepsilon)\cdot(\rho+T)]\cdot P(x+\varepsilon,t).\end{split}  \tag{5}\]  
Using a well-known procedure [29], stochastic Equation (5) is transformed into the Fokker-Plank equation with respect to fluctuation of activity factor \(\xi\) (see Appendix A):  
\[\frac{\partial}{\partial t}P(\xi,t)=\frac{1}{2}\cdot[(\mu+\rho\cdot x_{0})+( \rho+T)\cdot x_{0}]\cdot\frac{\partial^{2}}{\partial\xi^{2}}P(\xi,t)+\frac{ \partial}{\partial x_{0}}(T\cdot x_{0})\cdot\frac{\partial}{\partial\xi^{2} }[\xi\cdot P(\xi,t)], \tag{6}\]  
where \(x_{0}\) is the steady state of activity factor \(x\) and fluctuation \(\xi\) is defined as  
\[\xi=\frac{1}{\sqrt{\xi}}\cdot(x-x_{0}). \tag{7}\]  
Furthermore, the evolution of \(x_{0}\) is given as (see Appendix A)  
\[\frac{\partial x_{0}}{\partial t}=\mu-T\cdot x_{0}. \tag{8}\]  
Predict Changes of the Behavioral Property Using the Variance of the Fluctuation of the Activity Factor  
The variance of the fluctuation of the activity factor, \(\sigma_{\xi}^{2}\), is defined as  
\[ \begin{split}\sigma_{\xi}^{2}=\langle\xi^{2}\rangle-\langle\xi \rangle^{2}\\ \frac{\partial}{\partial t}\sigma_{\xi}^{2}=\frac{\partial}{ \partial t}\langle\xi^{2}\rangle-2\langle\xi\rangle\cdot\frac{\partial}{ \partial t}\langle\xi\rangle.\end{split}  \tag{9}\]  
Using the Fokker-Plank Equation (6) formalized in the previous section, Equation (9) is rewritten as follows (see Appendix B):  
\[\frac{\partial}{\partial t}\sigma_{\xi}^{2}=[\mu+(2\rho+T)\cdot x_{0}]-2\cdot \frac{\partial}{\partial x_{0}}(T\cdot x_{0})\cdot\sigma_{\xi}^{2}. \tag{10}\]  
Next, the behavioral properties of the steady states of the activity factor are analyzed.