## 3 Proposed Framework
#### Scheme 1: M/D/1 with PU delay and non-pre-emptive priority model  
The model described in Section 3.1 of Scheme 1, which incorporates PU delay [4], is further improved by incorporating non-preemptive priority [22]. The PU packet experiences a delay equal to a specific number of time slots (PU delay, d = 1). Consequently, the PU packet is unable to disrupt the transmission in the presence of a served SU packet. We assume an unlimited buffer and zero delay time. Despite the absence of a SU in the queue, the PU packet experiences a delay and is subsequently transmitted in the following time slot. TD, the delay time till the start of a time slot, is not accounted for in this approach. In Scheme 1, the average total waiting time of the tagged PU packet encompasses the average service time as well as the service times of the current PU and SU (if there is one) [23]. The system that does not prioritize or interrupt ongoing tasks. The formula for calculating the average total waiting time, given that the PU delay is i (an integer) timeslots, was developed as follows:  
\[W_{P}^{q}(d)=\frac{1}{2}\left(\frac{\lambda_{P}}{\mu_{P}^{2}}+\frac{\lambda_{ S}}{\mu_{S}^{2}}\right)+\frac{N_{P}^{q}}{\mu_{P}}+\frac{i}{\mu_{S}},\ \forall\ d=i \tag{2}\]  
Next, employing [PERSON]'s hypothesis, \(N^{q}\) is equal to \(\lambda W^{q}\), as follows:  
\[W_{P}^{q(non-premp)}(d)=\frac{1}{2}\left(\frac{\frac{\lambda_{P}}{\mu_{P}^{2}} +\frac{\lambda_{S}}{\mu_{S}^{2}}+\frac{i}{\mu_{S}}}{1-\rho_{P}}\right) \tag{3}\]  
The equation representing the average total waiting time of the processing unit in Scheme 1 is displayed below:\[W_{P}^{q(non-premp)}(d)=X_{P}+\frac{1}{2}\left(\frac{\frac{\lambda_{P}}{\mu_{P}^{2} }+\frac{\lambda_{S}}{\mu_{S}^{2}}+\frac{i}{\mu_{S}}}{1-\rho_{P}}\right) \tag{4}\]  
The average cumulative waiting time of the labeled SU packet is thereafter indicated. The calculation includes the average service time, as well as the service time of the identified SU packet and the combined service times of the PU and SU. The future arrivals at the PU line have no influence on it. \[W_{S}^{q}(d)=\frac{1}{2}\left(\frac{\lambda_{P}}{\mu_{P}^{2}}+\frac{\lambda_{S }}{\mu_{S}^{2}}\right)+\frac{N_{P}^{q}}{\mu_{P}}+\frac{N_{S}^{q}}{\mu_{S}},\ \forall\ d=i \tag{5}\]  
By applying [PERSON]'s theorem, this can be deduced as demonstrated below. \[W_{S}^{q(non-premp)}(d)=\frac{1}{2}\left(\frac{\frac{\lambda_{P}}{\mu_{P}^{2} }+\frac{\lambda_{S}}{\mu_{S}^{2}}}{(1-\rho_{P})(1-\rho_{S})}\right) \tag{6}\]  
Ultimately, when the average service time of SU, \(\overline{X_{S}}\), is included, the resulting statement is as follows:  
\[W_{S}^{(non-premp)}(d)=\overline{X_{S}}+W_{S}^{q(non-prem)}(d) \tag{7}\]  
In this work, the following two equations are derived for simulation purposes, assuming that the service time, denoted as \(X_{S}\), is equal to one time slot for both the service and processing times. \[W_{P}^{(non-premp)}(d)=1+\frac{1}{2}\left(\frac{\frac{\lambda_{P}}{\mu_{P}^{2} }+\frac{\lambda_{S}}{\mu_{S}^{2}}+\frac{i}{\mu_{S}}}{1-\rho_{P}}\right) \tag{8}\]  
\[W_{S}^{(non-premp)}(d)=1+\frac{1}{2}\left(\frac{\frac{\lambda_{P}}{\mu_{P}^{2} }+\frac{\lambda_{S}}{\mu_{S}^{2}}}{(1-\rho_{P})(1-\rho_{S})}\right) \tag{9}\]  
_(ii)._ _Scheme 2: M/D/1 with PU delay and pre-emptive priority model_  
Scheme 2 aims to enhance the average total waiting time of the PU by sacrificing the QoS of the SU [24; 25]. This is because the performance of the SU is influenced by the arrival rate of the PU. This model proposes the combination of the pre-emptive priority system with the approach described in reference [4]. The PU packet will have a delay for a designated number of time intervals. However, following the PU delay, if there is an SU packet being processed, the PU packet has the ability to interrupt the transmission in order to receive its own service. However, the timing of the packets' arrival is a determining factor [26]. The PU packet experiences a delay of one unit of time.