# QoS-Aware Resource Management in 5G and 6G Cloud-Based Architectures with Priorities
## 3 Transmission Waiting Time FIFO Queue
### Packet Transmission Analysis  
We do consider a generalized queue system with one single server, \(\mathrm{m}\) channels (resources) in parallel [26], finite queue transmission length, Poisson \(\lambda\) process arrivals, and independent and uniform service time \(\mu_{0}\) distributed on [0, s], \(s>0\). Transit time effects are ignored in this analysis, and the reasoning behind uniform distributed service time is the really small scheduling delay for every IP MAC packet in the queue, following the strict requirements on 5G transmission, Pbit priority scheduling delays, and performing average uniform service time distribution. For equilibrium in the queueing solution, we do suppose that \(\mathrm{m}>\lambda\). Define \(\Pi_{n}\), the probability of existing specifically \(\mathrm{n}\) IP MAC packets in both queue and service at a given time, \(\tau\), and \(p_{n}\) the probability that no more than \(\mathrm{n}\) packets exists in the model at a given time \(\tau\). Further in our analysis the probability in the unit of time specifically zero packets exist in the queue and \(\mathrm{m}\) packets in service \(\Pi_{0}\) could be calculated as the intersection of (the probability \(p_{m}\) that no more than zero packets exist in the queue as long as \(\mathrm{m}\) packets exist in the server at the beginning of unit of time) and (the probability (Poisson distribution) of zero arrivals during the considered time interval), that is,  
\[\Pi_{0}=p_{m}\cap e^{-\lambda}=p_{m}\cdot e^{-\lambda}. \tag{14}\]  
Along the same lines, the probability that specifically one packets exists in the queue \(\Pi_{1}\) at the unit of time can be calculated as the union of (the intersection of (the probability \(p_{m}\) that no more than zero packets exist in the model as long as \(\mathrm{m}\) packets exist in the server at the beginning of unit of time) and (the probability (Poisson distribution) of one arrival during that interval \(\tau=1/\mu_{0}\))) and (intersection of (the probability \(\Pi_{m+1}\) that exactly one packet exists in the queue and \(\mathrm{m}\) packets are in service at the beginning of unit of time) and (the probability (Poisson distribution) of zero arrivals during that interval)):  
\[\Pi_{1}=\left(p_{m}\cap\lambda e^{-\lambda}\right)\cup\left(\Pi_{m+1}\cap e^ {-\lambda}\right)=p_{m}\cdot\lambda e^{-\lambda}+\Pi_{m+1}\cdot e^{-\lambda}. \tag{15}\]  
Considering the general case, the overall probability that specifically \(\mathrm{n}\) packets exist in the system \(\Pi_{n}\) at the unit of time equals  
\[ \begin{split}\Pi_{n}=p_{m}\cdot\frac{\lambda^{n}}{n!}e^{-\lambda }+\sum\limits_{k=1}^{n}\Pi_{m+k}\cdot\frac{\lambda^{n-k}}{\left(n-k\right)! }e^{-\lambda}\\ =p_{m}\cdot\frac{\lambda^{n}}{n!}e^{-\lambda}+\sum\limits_{k=0}^{ n}\Pi_{m+k}\cdot\frac{\lambda^{n-k}}{\left(n-k\right)!}e^{-\lambda}-\Pi_{m}\cdot \frac{\lambda^{n}}{n!}e^{-\lambda}.\end{split}  \tag{16}\]  
To calculate, then, the probability \(\Pi_{n}\) in the above general formula, we proceed with Laurent series expansion as follows:  
\[\sum\limits_{n=0}^{\infty}\Pi_{n}z^{n}=\sum\limits_{n=0}^{\infty}\left(p_{m} \cdot\frac{\lambda^{n}}{n!}e^{-\lambda}\right)z^{n}+\sum\limits_{n=0}^{ \infty}\left(\sum\limits_{k=0}^{n}\Pi_{m+k}\cdot\frac{\lambda^{n-k}}{\left(n-k \right)!}e^{-\lambda}\right)z^{n}-\sum\limits_{n=0}^{\infty}\Pi_{m}\cdot\frac{ \lambda^{n}}{n!}e^{-\lambda}z^{n} \tag{17}\]\[\sum\limits_{n=0}^{\infty}\Pi_{n}z^{n}=p_{m}e^{-\lambda}\sum\limits_{n=0}^{ \infty}\frac{\left(\lambda z\right)^{n}}{n!}+e^{-\lambda}\sum\limits_{n=0}^{ \infty}\left(\sum\limits_{k=0}^{n}\Pi_{m+k}\cdot\frac{\left(\lambda z\right)^{n }\lambda^{-k}}{\left(n-k\right)!}\right)-\Pi_{m}e^{-\lambda}\sum\limits_{n=0}^ {\infty}\frac{\left(\lambda z\right)^{n}}{n!}. \tag{18}\]  
Following the summations and after appropriate mathematical calculations we simplify (18) into  
\[