## 4 Gaussian Approximation of the Probability Distribution of the Total Resource Amounts  
It was previously proven that under the asymptotic condition of an infinitely growing service time  
* _For resource queue \(MMPP/GI(2)/\infty\)_: the joint probability distribution of the number of customers and total capacities is a multidimensional Gaussian distribution [34];
* _For resource queue \(GI/GI^{(n)}/\infty\)_: the n-dimensional probability distribution of the total resource amounts is asymptotically Gaussian [29]. To construct a Gaussian approximation of the distribution function of the random process \(\{V_{1}(t),V_{2}(t),\ldots,V_{n}(t)\}\), we use the method of asymptotic analysis with the asymptotic conditions of the growing intensity of the arrival process. In the models under consideration, the flow intensity is represented as \(N\lambda\), where \(\lambda\) is a fixed value and the parameter \(N\) has large values (in theoretical studies \(N\rightarrow\infty\)). The value of \(N\) will be called the parameter of high flow intensity [37]. * For MMPP: to represent matrix of intensity \(\boldsymbol{\Lambda}\) as \(\boldsymbol{\Lambda}_{1}=N\boldsymbol{\Lambda}= \begin{pmatrix}N\lambda_{1}&0& \ldots&0\\ 0&N\lambda_{2}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&N\lambda_{K}\end{pmatrix} \) and the infinitesimal generator matrix \(\mathbf{Q}\) as \(\mathbf{Q}_{1}=N\mathbf{Q}\);
* For renewal arrivals: let us represent \(t=\frac{\xi}{N}\), where \(\xi\) is some non-negative random variable with the distribution function \(A(z)\). Value \(N>0\) is a parameter of high flow intensity, the meaning of which is described above. Then for the distribution function of interval lengths \(t\), we have [38; 39] \[P\{t<x\}=P\{\frac{\xi}{N}<x\}=P\{\xi<Nx\}=A(Nx).\] (9) Then Equation (6) is transformed into \[ \begin{split}&\frac{1}{N}\frac{\partial P(z,w_{1},\ldots,w_{n},t )}{\partial t}=\frac{\partial P(z,w_{1},\ldots,w_{n},t)}{\partial z}+\frac{ \partial P(0,w_{1},\ldots,w_{n},t)}{\partial z}(A(z)-1)+\\ +A(z)\sum_{i=1}^{n}S_{i}(t)\left[\int\limits_{0}^{w_{i}}\frac{ \partial P(0,w_{1},\ldots,w_{i}-y_{i},\ldots,w_{n},t)}{\partial z}dG_{i}(y)- \frac{\partial P(0,w_{1},\ldots,w_{n},t)}{\partial z}\right].\end{split} \] (10) Equation (8), in turn, takes the form \[ \begin{split}&\frac{1}{N}\frac{\partial h(z,u_{1},\ldots,u_{n},t )}{\partial t}=\frac{\partial h(z,u_{1},\ldots,u_{n},t)}{\partial z}+\\ +\frac{\partial h(0,u_{1},\ldots,u_{n},t)}{\partial z}\left[A(z) -1+A(z)\sum_{i=1}^{n}S_{i}(t)(G^{*}(u_{i})-1)\right],\\ h(z,u_{1},\ldots,u_{n},t_{0})=R(z).\end{split} \] (11) We state and prove the following theorems. **Theorem 1**. : _Under the condition of the growing intensity of the arrival process, the joint asymptotic probability distribution of the \(n\)-dimensional process of the total amounts of the occupied resources in the heterogeneous QS is the \(n\)-dimensional Gaussian distribution with parameters that are dependent on the type of arrival process. The expectation vector has the form_  
\[\mathbf{a}=N\lambda\big{[}a_{1}b_{1}\quad\ldots\quad a_{n}b_{n}\big{]}, \tag{12}\]  
_and the covariance matrix has the form_  
\[\mathbf{K}=N(\lambda\mathbf{K}^{(1)}+\kappa\mathbf{K}^{(2)}). \tag{13}\]  
_Here_  
\[\mathbf{K}^{(1)}= \begin{bmatrix}d_{1}b_{1}&\cdots&0\\ \cdots&\cdots&\cdots\\ 0&\cdots&d_{n}b_{n}\end{bmatrix} ,\quad\mathbf{K}^{(2)}= \begin{bmatrix}\beta_{11}a _{1}a_{1}&\cdots&\beta_{1n}a_{1}a_{n}\\ \cdots&\cdots&\cdots\\ \beta_{n1}a_{n}a_{1}&\cdots&\beta_{nn}a_{n}a_{n}\end{bmatrix} ,\] \[a_{i}=\int\limits_{0}^{\infty}ydG_{i}(y),\,d_{i}=\int\limits_{0}^ {\infty}y^{2}dG_{i}(y),\,b_{i}=p_{i}\int\limits_{0}^{\infty}(1-B_{i}(x))dx,\] \[\beta_{ij}=p_{i}p_{j}\int\limits_{0}^{\infty}(1-B_{i}(x))(1-B_{j}(x))dx, \,i=1,\ldots,n,j=1,\ldots,n,\]  
\(\lambda\) _and \(\kappa\) are defined by the type of arrival process:_  
* _For MMPP:_ \[\lambda=\mathbf{r}\mathbf{\Lambda}\mathbf{e},\quad\kappa=2\mathbf{g}(\mathbf{\Lambda} -\lambda\mathbf{I})\mathbf{e},\] (14) \[\mathbf{g}\] _is the solution of system_ \[\left\{