## 3 Analytical Solution of the Queueing System M(\(\lambda\))/Ek(\(\mu\))/I/\(\infty\)
### The Probability of State of the Queueing System \(M(\lambda)/E_{k}(\mu)/\infty\)  
The establishment of the elementary case of a queueing system with a finite state number \(X_{0}\), \(X_{1}\),..., \(X_{n}\), with the exponentially distributed inter-arrival-time \(E(\lambda)\) and the service time distributed according to [PERSON]'s distribution \(Er(k,\mu_{k})\), \(k\in\) N, is shown in Figure 5.  
Assume that the queueing system is ergodic and that it is in a stationary operating mode. The probabilities of the state \(X_{0}\), \(X_{1}\),..., \(X_{n}\), in the stationary mode of operation, are denoted by the large letter \(P_{i}(t)=P_{i}=const\), \(i\in[1,\,n]\). The probabilities of the abstract states \(h_{ij}\) are denoted by the small letter \(p_{i,j}(t)=p_{i,j}=const\), \(i\in[1,\,k-1],j\in[1,\,n]\). For the X\({}_{0}\) state, the balance (differential) equation of the study-state (stationary operating mode) is (5)  
\[X_{0}:0=-\lambda P_{0}+\mu_{k}p_{k-1,1}\Leftrightarrow p_{k-1,1}=\frac{\lambda }{\mu_{k}}P_{0} \tag{5}\]  
For the abstract HMS \(h_{k-1,1},h_{k-2,1},\ldots,\,h_{2,1},h_{1,1}\) which are between the states \(X_{0}\  
For the state of \(X_{2}\), the balance equation of the study-state is (9)  
\[X_{2}:0=\lambda P_{1}-\lambda P_{2}-\mu_{k}P_{2}+\mu_{k}p_{k-2,2} \Leftrightarrow p_{k-1,2}=\frac{-\lambda P_{1}+\lambda P_{2}+\mu_{k}P_{2}}{\mu _{k}}\Leftrightarrow p_{k-1,3}=\left(\frac{\lambda}{\mu_{k}}\right)^{3}P_{0} \tag{9}\]  
Prescriptively, in accordance with the standard solution procedure, we can obtain solutions for all remaining system states (10):  
\[ \begin{array}{l}\cdots\cdots\cdots\cdots\cdots\\ X_{n-1}:0=\lambda P_{n-2}-\lambda P_{n-1}-\mu_{k}P_{n-1}+\mu_{k}p_{n,2} \Leftrightarrow p_{k-1,n}=\left(\frac{\lambda}{\mu_{k}}\right)^{n}P_{0}\\ \\ \left\{\begin{array}{l}h_{k-1,n}:0=-\mu_{k}p_{k-1,n}+\mu_{k}p_{k-2,n} \Leftrightarrow p_{k-1,n}=p_{k-2,n}\Leftrightarrow p_{k-2,n}=\left(\frac{ \lambda}{\mu_{k}}\right)^{n}P_{0}\\ \\ \cdots\\ \end{array}\right.\\ \end{array}  \tag{10}\] \[\cdots\]  
The probability value \(P_{0}\) can be obtained from the normative Equation (11):  
\[\underbrace{P_{0}+P_{0}\sum\limits_{j=1}^{k-1}\frac{\lambda}{\mu_{k}}}_{s_{0} =P_{0}\sum\limits_{j=1}^{k-1}\frac{\lambda}{\mu_{k}}}+\underbrace{P_{0}\frac{ \lambda}{\mu_{k}}}_{P_{1}}+P_{0}\sum\limits_{j=1}^{k-1}\left(\frac{\lambda}{ \mu_{k}}\right)^{2}+\ldots+\underbrace{P_{0}\left(\frac{\lambda}{\mu_{k}} \right)^{2}}_{P_{n}}+P_{0}\sum\limits_{j=1}^{k-1}\left(\frac{\lambda}{\mu_{k} }\right)^{n}+\ldots=1 \tag{11}\]  
By solving the sum in (11), we can obtain (12)  
\[\underbrace{P_{0}+(k-1)P_{0}\frac{\lambda}{\mu_{k}}}_{s_{0}}+\underbrace{P_{0} \frac{\lambda}{\mu_{k}}+(k-1)P_{0}\left(\frac{\lambda}{\mu_{k}}\right)^{2}}_{ s_{1}}+\ldots+\underbrace{P_{0}\left(\frac{\lambda}{\mu_{k}}\right)^{n}+(k-1)P_{0} \left(\frac{\lambda}{\mu_{k}}\right)^{n+1}}_{s_{n}}+\ldots=1 \tag{12}\]  
The following geometric order is obtained (13):  
\[P_{0}+kP_{0}\frac{\lambda}{\mu_{k}}+kP_{0}\left(\frac{\lambda}{\mu_{k}}\right) ^{2}+kP_{0}\left(\frac{\lambda}{\mu_{k}}\right)^{3}\ldots=1 \tag{13}\]  
where the condition \(\lambda<\mu_{k}\) gives the equation for the \(P_{0}\) (14) calculation. We introduce the tag \(\frac{\lambda}{\mu_{k}}=\rho_{k}\):  
\[P_{0}+kP_{0}\sum\limits_{j=1}^{\infty}\left(\frac{\lambda}{\mu_{k}}\right)^{j}= P_{0}+kP_{0}\left(\frac{1}{1-\rho_{k}}-1\right)=P_{0}\left(\frac{1-\rho_{k}+k \rho_{k}}{1-\rho_{k}}\right)=1 \tag{14}\]  
The probability value of \(P_{0}\) is equal (15) to  
\[P_{0}-kP_{0}+P_{0}\frac{k}{1-\rho_{k}}=1\Leftrightarrow P_{0}=\frac{1-\rho_{k} }{1-\rho_{k}+k\rho_{k}} \tag{15}\]  
The probabilities \(h_{k-1,j}\), \(h_{k-2,j}\),..., \(h_{2,j}\), \(h_{1,j}\) (\(j\in\) N) are abstract probabilities. The analytical process takes them to the catalytic at each transition from the state \(X_{j}\) to the state \(X_{j-1}\), for \(j\in[1,\infty)\). Therefore, the probabilities of \(p_{i,j}\) states \(h_{i,j}\) are abstractly transmitted only for the realization of the service by Erlang's distribution.