## 4 Gaussian Approximation of the Probability Distribution of the Total Resource Amounts
For a system with renewal arrival process. Let us make the following substitutions in (11)  
\[\epsilon=\frac{1}{N},\,u_{i}=\epsilon x_{i},\,i=1,\ldots,n,\,h(z,u_{1},\ldots, u_{n},t)=f_{1}(z,x_{1},\ldots,x_{n},t,\epsilon). \tag{31}\]  
We obtain  
\[ \begin{split}&\epsilon\frac{\partial f_{1}(z,x_{1},\ldots,x_{n},t, \epsilon)}{\partial t}=\frac{\partial f_{1}(z,x_{1},\ldots,x_{n},t,\epsilon)}{ \partial z}+\\ +&\frac{\partial f_{1}(0,x_{1},\ldots,x_{n},t, \epsilon)}{\partial z}\bigg{[}A(z)-1+A(z)\sum_{i=1}^{n}S_{i}(t)(G^{*}( \epsilon x_{i})-1)\bigg{]},\\ & f_{1}(z,x_{1},\ldots,x_{n},t_{0},\epsilon)=R(z),\end{split}  \tag{32}\]  
To find the first-order asymptotic solution  
\[f_{1}(z,x_{1},\ldots,x_{n},t)=\lim_{\epsilon\to 0}f_{1}(z,x_{1},\ldots,x_{n},t, \epsilon),\]  
let \(\epsilon\to 0\) in (32):  
\[ \begin{split}&\frac{\partial f_{1}(z,x_{1},\ldots,x_{n},t)}{ \partial z}+\frac{\partial f_{1}(0,x_{1},\ldots,x_{n},t)}{\partial z}[A(z)-1]= 0.\end{split} \]  
We assume that  
\[ \begin{split}& f_{1}(z,x_{1},\ldots,x_{n},t)=R(z)\Phi_{1}(x_{1}, \ldots,x_{n},t),\\ &\Phi_{1}(x_{1},\ldots,x_{n},t_{0})=1,\end{split}  \tag{33}\]  
where \(\Phi_{1}(x_{1},\ldots,x_{n},t)\) is a scalar differentiable function. Let \(z\to\infty\) in (32):  
\[\epsilon\frac{\partial f_{1}(\infty,x_{1},\ldots,x_{n},t,\epsilon)}{\partial t }=\frac{\partial f_{1}(0,x_{1},\ldots,x_{n},t,\epsilon)}{\partial z}\sum_{i=1 }^{n}S_{i}(t)(G^{*}(\epsilon x_{i})-1). \tag{34}\]  
We use the exponent expansion for functions in the form of (20) and substitute (33) into (34). Furthermore, we divide the resulting expression by \(\epsilon\) and let \(\epsilon\to 0\)  
\[\frac{\partial\Phi_{1}(x_{1},\ldots,x_{n},t)}{\partial t}=\Phi_{1}(x_{1}, \ldots,x_{n},t)j\lambda\sum_{i=1}^{n}x_{i}a_{i}S_{i}(t). \tag{35}\]  
It is easy to see that the solution of Equation (35) is  
\[\Phi_{1}(x_{1},\ldots,x_{n},t)=\exp\bigg{\{}j\lambda\sum_{i=1}^{n}x_{i}a_{i} \int\limits_{t_{0}}^{t}S_{i}(\tau)d\tau\bigg{\}}. \tag{36}\]Furthermore, similarly to the earlier reasoning for the case with an incoming MMPP, we obtain the form of the first-order asymptotic characteristic function \(h(z,u_{1},\ldots,u_{n},t)\)  
\[h(z,u_{1},\ldots,u_{n},t)=R(z)\exp\bigg{\{}jN\lambda\sum_{i=1}^{n}u_{i}a_{i} \int\limits_{t_{0}}^{t}S_{i}(\tau)d\tau\bigg{\}}. \tag{37}\]  
Let us proceed to the construction of the second-order asymptotic. We introduce the function  
\[h(z,u_{1},\ldots,u_{n},t)=h_{2}(z,u_{1},\ldots,u_{n},t)\exp\bigg{\{}jN\lambda \sum_{i=1}^{n}u_{i}a_{i}\int\limits_{t_{0}}^{t}S_{i}(\tau)d\tau\bigg{\}}. \tag{38}\]  
In (11), let us make substitutions  
\[\epsilon^{2}=\frac{1}{N},\,u_{i}=\epsilon x_{i},\,i=1,\ldots,n,\,h_{2}(z,u_{1},\ldots,u_{n},t)=f_{2}(z,x_{1},\ldots,x_{n},t,\epsilon) \tag{39}\]  
and, given (38), we obtain  
\[ \begin{split}\epsilon^{2}\frac{\partial f_{2}(z,x_{1},\ldots,x_ {n},t,\epsilon)}{\partial t}+f_{2}(z,x_{1},\ldots,x_{n},t,\epsilon)j\epsilon \lambda\sum_{i=1}^{n}x_{i}a_{i}S_{i}(t)=\frac{\partial f_{2}(z,x_{1},\ldots,x _{n},t,\epsilon)}{\partial z}+\\ +\frac{\partial f_{2}(0,x_{1},\ldots,x_{n},t,\epsilon)}{\partial z }\bigg{[}A(z)-1+A(z)\sum_{i=1}^{n}S_{i}(t)(G^{*}(\epsilon x_{i})-1)\bigg{]}. \end{split}  \tag{40}\]  
Let us find the asymptotic solution of Equation (40)  
\[f_{2}(z,x_{1},\ldots,x_{n},t)=\lim_{\epsilon\to 0}f_{2}(z,x_{1},\ldots,x_{n},t, \epsilon). \tag{41}\]  
So,  
\[\frac{\partial f_{2}(z,x_{1},\ldots,x_{n},t)}{\partial z}+\frac{\partial f_{2 }(0,x_{1},\ldots,x_{n},t,\epsilon)}{\partial z}[A(z)-1]=0. \tag{42}\]  
We find the function \(f_{2}(z,x_{1},\ldots,x_{n},t)\) as follows:  
\[ \begin{split} f_{2}(z,x_{1},\ldots,x_{n},t)=R(z)\Phi_{2}(x_{1}, \ldots,x_{n},t),\\ \Phi_{2}(x_{1},\ldots,x_{n},t_{0})=1.\end{split}  \tag{43}\]  
We write the function \(f_{2}(z,x_{1},\ldots,x_{n},t)\) as the power expansion  
\[f_{2}(z,x_{1},\ldots,x_{n},t,\epsilon)=\Phi_{2}(x_{1},\ldots,x_{n},t)\Bigg{[} R(z)+j\epsilon\lambda f(z)\sum_{i=1}^{n}x_{i}a_{i}S_{i}(t)+O(\epsilon^{2}) \Bigg{]}, \tag{44}\]  
where \(f(z)\) is some differentiable function.