# QoS-Aware Resource Management in 5G and 6G Cloud-Based Architectures with Priorities
## 3 Transmission Waiting Time FIFO Queue
### Packet Transmission Analysis
\begin{array}{c}\Pi(z)=(p_{m}-\Pi_{m})e^{\lambda(z-1)}+e^{-\lambda}\sum \limits_{n=0}^{\infty}\frac{\Pi_{m}(z\lambda z)^{n}}{n!}+e^{-\lambda}\sum \limits_{n=0}^{\infty}\frac{\Pi_{m+1}(\lambda z)^{n}}{\lambda\cdot\left(n-1 \right)!}+\ldots+e^{-\lambda}\sum\limits_{n=0}^{\infty}\frac{\Pi_{m+n}( \lambda z)^{n}}{\lambda^{n}}\\ \qquad\qquad=\frac{\Pi_{m}(z)-p_{m}z^{m}}{1-z^{m}e^{\lambda(1-z)}}.\end{array}  \tag{19}\]  
where using Laurent power series definition we define the generating function \(\Pi(z)=\sum\limits_{n=0}^{\infty}\Pi_{n}z^{n}\) and by definition of finite Laurent series \(\Pi_{m}(z)=\sum\limits_{n=0}^{m}\Pi_{n}z^{n}\). Since \(0\leq\Pi_{n}\leq 1\), complex function \(\Pi(z)\) is a regular function bounded into the unit circle on the complex space \(|z|\leq 1\). Numerator \(\Pi_{m}(z)-p_{m}z^{m}\) consists of two polynomials of m\({}^{\text{th}}\) order. Both \(\Pi_{m}(z)\) and \(p_{m}z^{m}\) are analytical functions inside the simple curve \(|z|\leq 1\) and also bounded into the unit circle on the complex space \(|z|\leq 1\). Since \(|p_{m}z^{m}|\leq|\Pi_{m}(z)|\) on \(|z|\leq 1\), then both have same number of zeroes inside \(|z|\leq 1\), and since they are polynomials of mth order they have m zeroes inside, \(|z|\leq 1\), denoted as \(z_{1},z_{2},\ldots,z_{m}\), respectively, leading into a closed-form function of \(\Pi(z)\) which can be written as  
\[\Pi(z)=\frac{A(z-z_{1})(z-z_{2})\cdots(z-z_{m})}{1-z^{m}e^{\lambda(1-z)}}= \frac{A(z-z_{1})(z-z_{2})\cdots(z-z_{m-1})\cdot(z-1)}{1-z^{m}e^{\lambda(1-z)}}. \tag{20}\]  
Since z = 1 is one of the roots, the numerator is  
\[\underset{z\to 1}{lim}(\Pi_{m}(z)-p_{m}z^{m})=\underset{z\to 1}{lim} \left(\sum\limits_{n=0}^{m}\Pi_{n}z^{n}-p_{m}z^{m}\right)=\sum\limits_{n=0}^ {m}\Pi_{n}-p_{m}=0. \tag{21}\]  
Furthermore, in (20) factor A is a constant calculated from the total probability condition  
\[\underset{z\to 1}{lim}\Pi(z)=\underset{z\to 1}{lim}\sum\limits_{n=0}^{\infty}\Pi_{n}z^{ n}=\sum\limits_{n=0}^{\infty}\Pi_{n}=1 \tag{22}\]  
\[\underset{z\to 1}{lim}\Pi(z)=\underset{z\to 1}{lim}\frac{A(z-z_{1})\cdots(z-z_{m-1}) (z-1)}{1-z^{m}e^{\lambda(1-z)}}=\underset{z\to 1}{lim}\frac{A(z-z_{1})\cdots(z-z_{m-1})(z-1)} {1-z^{m}e^{\lambda(1-z)}}=1 \tag{23}\]  
Finally, using the Laurent series  
\[ \begin{array}{c}\Pi(z)=\frac{(\lambda-m)(z-z_{1})(z-z_{2})\cdots(z-z_{m})}{ (1-z_{1})(1-z_{2})\cdots(1-z_{m-1})\left[1-z^{m}e^{\lambda(1-z)}\right]} \Rightarrow\\ \sum\limits_{n=0}^{\infty}\Pi_{n}z^{n}=\frac{(\lambda-m)(z-z_{1})(z-z_{2}) \cdots(z-z_{m})}{(1-z_{1})(1-z_{2})\cdots(1-z_{m-1})\left[1-z^{m}e^{\lambda(1 -z)}\right]}.\end{array}  \tag{24}\]  
The expansion of the right part of the previous equation into the Laurent series around z = 0 is essential in order to calculate \(\Pi_{n}\). By \(\Pi_{n},n=0,\;1,\;2\;\ldots n\), we assign the coefficients of \(z^{n}\) after the expansion is performed. Considering the case of m = 1 (MAC/Ethernet transmission scheduler considers each packet as a unique service input), the numerator is simplified into a first-order polynomial with one single real root. \[ \begin{array}{c}\left(\Pi_{m}(z)-p_{m}z^{m}\right)|_{m=1}=0\Rightarrow\left. \left(\sum\limits_{n=0}^{1}\Pi_{n}z^{n}-\sum\limits_{n=0}^{1}\Pi_{m}z\right) \right|_{m=1}=0\\ \Rightarrow\left(\Pi_{0}+\Pi_{1}z-\Pi_{0}z-\Pi_{1}z\right)=0\Rightarrow z=1,\Pi_{0} \
eq 0.\end{array}  \tag{25}\]The polynomial expansion coefficients, after expanding the polynomial into Laurent series around z = 0, become  
\[ \begin{split}&\Pi(z)=\sum\limits_{n=0}^{\infty}\Pi_{n}z^{n}=\frac{( \lambda-1)(z-1)}{1-ze^{\lambda(1-z)}}=\\ &=\left(1-\frac{\lambda}{\mu}\right)+\left(\frac{\lambda}{\mu}-1 \right)\left(1-e^{\frac{1}{p}}\right)z+\left(\frac{\lambda}{\mu}-1\right) \left(e^{\frac{\lambda}{p}}\left(\frac{\lambda}{\mu}-1\right)-e^{2\frac{ \lambda}{p}}\right)z^{2}+\ldots\end{split}