## 3 Analytical Solution of the Queueing System M(\(\lambda\))/Ek(\(\mu\))/I/\(\infty\)
### The Probability of State of the Queueing System \(M(\lambda)/E_{k}(\mu)/\infty\)
Now, the mean number of customers in the system M(\(\lambda\))/E\({}_{k}\)(\(\mu_{k}\))/l/\(\infty\) and in the queue equals (21)  
\[\underbrace{P_{0}+P_{0}\sum\limits_{j=1}^{k-1}\frac{\lambda}{\mu_{k}}}_{s_{0} =P_{0}\sum\limits_{j=1}^{k-1}\frac{\lambda}{\mu_{k}}}+\underbrace{P_{0}\frac{ \lambda}{\mu_{k}}}_{P_{1}}+P_{0}\sum\limits_{j=1}^{k-1}\left(\frac{\lambda}{ \mu_{k}}\right)^{2}+\ldots+\underbrace{P_{0}\left(\frac{\lambda}{\mu_{k}} \right)^{2}}_{P_{n}}+P_{0}\sum\limits_{j=1}^{k-1}\left(\frac{\lambda}{\mu_{k} }\right)^{n}+\ldots=1 \tag{11}\]3  
By substituting (19) with (21), we obtain the basis for comparing the mean number of customers in the system and queue, i.e., for \(k>1\) and \(\rho<1\), the mean number of customers in the system M(\(\lambda\))/E\({}_{k}\)(\(\mu_{k}\))/l/\(\infty\) is always lower than the mean number of customers in the system M(\(\lambda\))/M(\(\mu_{e}\))/l/\(\infty\). The same applies to the mean number of customers in the queue (22):  
\[\underbrace{P_{0}+P_{0}\sum\limits_{j=1}^{k-1}\frac{\lambda}{\mu_{k}}}_{s_{0} =P_{0}\sum\limits_{j=1}^{k-1}\frac{\lambda}{\mu_{k}}}+\underbrace{P_{0}\frac{ \lambda}{\mu_{k}}}_{P_{1}}+P_{0}\sum\limits_{j=1}^{k-1}\left(\frac{\lambda}{ \mu_{k}}\right)^{2}+\ldots+\underbrace{P_{0}\left(\frac{\lambda}{\mu_{k}} \right)^{2}}_{P_{n}}+P_{0}\sum\limits_{j=1}^{k-1}\left(\frac{\lambda}{\mu_{k} }\right)^{n}+\ldots=1 \tag{11}\]4  
Unlike the relation in the previous queueing system solution M(\(\lambda\))/E\({}_{k}\)(\(\mu_{k}\))/l/\(\infty\) (3), the solution with the abstract HMS (21) completely meets the logic of the Pollaczek-Khinchine formula. Using the Pollaczek-Khinchine formula for the queueing system M(\(\lambda\))/E\({}_{k}\)(\(\mu_{k}\))/l/\(\infty\), we can verify the validity of the patterns (21) and the relation (22). Mean service time and variance are distributed according to [PERSON]'s distribution with known parameters of mathematical expectation and standard deviation \(k\)/\(\mu_{k}\) and \(k\)/\(\mu_{k}^{2}\), respectively. The ratio \(\rho\) is \"\(k\)\" times greater than \(\rho_{k}\) (18). If in the Pollaczek-Khinchine formula we reduce the intensity and the variance, we obtain (23), which is identical to the form (22) obtained:  
\[\underbrace{P_{0}+P_{0}\sum\limits_{j=1}^{k-1}\frac{\lambda}{\mu_{k}}}_{s_{0} =P_{0}\sum\limits_{j=1}^{k-1}\frac{\lambda}{\mu_{k}}}+\underbrace{P_{0}\frac{ \lambda}{\mu_{k}}}_{P_{1}}+P_{0}\sum\limits_{j=1}^{k-1}\left(\frac{\lambda}{ \mu_{k}}\right)^{2}+\ldots+\underbrace{P_{0}\left(\frac{\lambda}{\mu_{k}} \right)^{2}}_{P_{n}}+P_{0}\sum\limits_{j=1}^{k-1}\left(\frac{\lambda}{\mu_{k} }\right)^{n}+\ldots=1 \tag{11}\]5  
By applying the Little formula, it follows that, for the same mean service time (18) and the relationships obtained (22), on average, customers spend more time on the M(\(\lambda\))/M(\(\mu_{e}\))/l/\(\infty\) system. The amount of time \(\varphi\) is the same (24), as follows:  
\[\underbrace{P_{0}+P_{0}\sum\limits_{j=1}^{k-1}\frac{\lambda}{\mu_{k}}}_{s_{0} =P_{0}\sum\limits_{j=1}^{k-1}\frac{\lambda}{\mu_{k}}}+\underbrace{P_{0}\frac{ \lambda}{\mu_{k}}}_{P_{1}}+P_{0}\sum\limits_{j=1}^{k-1}\left(\frac{\lambda}{ \mu_{k}}\right)^{2}+\ldots+\underbrace{P_{0}\left(\frac{\lambda}{\mu_{k}} \right)^{2}}_{P_{n}}+P_{0}\sum\limits_{j=1}^{k-1}\left(\frac{\lambda}{\mu_{k} }\right)^{n}+\ldots=1 \tag{11}\]6  
The same logic is applied for the mean wait in the queue. The explanation of this relationship is based on the relationship between the entropy of the exponential and Erlang's distribution. Among all continuous probability distributions with support [0, \(\infty\)), the exponential distribution has the largest entropy. Therefore, the Markov queueing systems M/M/\(n\)/\(\infty\) have the greatest disadvantage in terms of access and service, and the greatest burden is thus on the service channel. In this case, in general, entropy is inversely proportional to the degree of system organization.