## 4 Design of RT-CCCS
### CC Selection
#### 4.2.1 Component Carriers Configuration and Switching Process  
The traffic model plays an important role in analyzing the performance of resource scheduling schemes in SA systems. We therefore consider a M/M/S/A process in this paper, which is a dynamic traffic model with Poisson arrival and used in [10]. In this process, the arrival of SUs follows the Poisson distribution, and the service time follows a negative exponential distribution (M for Markov). S is the number of active SUs, and A is the corresponding maximum number of SUs in the CR-based SA system. In SA, all CCs are regarded as one channel for users, so S is equal to 1 in this paper. The admission control in CR-based SA system is assumed to limit the SUs' number of CCs to maximum \(A\) SUs. This leads to the following arrival rate:  
\[\lambda_{su}= \begin{cases}\lambda&0\leq k<A\\ 0&k\geq A\end{cases}  \tag{3}\]  
The service rate, which is the average number of SUs who finish their transmissions during one second, can then be expressed as  
\[\mu_{su}= \begin{cases}\dfrac{C_{su}}{F}&0\leq k<A\\ 0&k\geq A\end{cases}  \tag{4}\]  
Because the SUs and PUs coexist in each CC, the SUs' data transmission time is affected by PUs. \(T_{i}\) denotes the SUs' data transmission time of the \(i\)th CC, \(L_{i,su}\) denotes the length of the SUs' data transmitted in the \(i\)th CC, \(T_{i,pu}\) denotes the PUs' data transmission time of the \(i\)th CC, and \(C_{i}\) is the data throughput of the \(i\)th CC, so the each SU's data transmission time of the \(i\)th CC can be expressed as  
\[T_{i}=\dfrac{L_{i,su}}{C_{i}}+T_{i,pu} \tag{5}\]  
Based on the principle of dispatching policy, the data transmission time of each SUs' packet in every CCs is also equal. \(T\) denotes the SUs' data transmission time when SUs aggregate the all CCs for data transmission, thus  
\[T=T_{i},i=1,2,\ldots,N \tag{6}\]  
Based on Equations (5) and (6), we can obtain the following expression:  
\[ \begin{cases}L_{i,su}=TC_{i}-T_{i,pu}C_{i}\\ \sum_{i=1}^{N}L_{i,su}=F\end{cases}  \tag{7}\]  
Thus, the SUs' data transmission time \(T\) can be obtained when SUs use the CR-based spectrum aggregation for data transmission. \[T=\dfrac{F}{C}+\sum_{i=1}^{N}\dfrac{C_{i}T_{i,pu}}{C} \tag{8}\]  
where  
\[C=\sum_{i=1}^{N}C_{i} \tag{9}\]  
We can find that \(T\) is related to the payload size \(F\), data throughput _C_i of each CC, and PUs' data transmission time \(T_{i,pu}\) of each CC. \(T_{i,pu}\) obeys Erlang distribution and its order is related to \(T\). Thus, the expectations of \(T_{i,pu}\) and \(T_{i,pu}^{2}\) can be obtained. \[ \begin{cases}\text{E}(T_{i,pu})=\frac{\lambda_{i,pu}}{\mu_{i,pu}}\,\text{E}\left(T \right)\\ \text{E}(T_{i,pu}^{2})=\frac{2\lambda_{i,pu}}{\mu_{i,pu}^{2}}\,\text{E}\left(T \right)+\frac{\lambda_{i,pu}^{2}}{\mu_{i,pu}^{2}}\,\text{E}\left(T^{2}\right) \end{cases}  \tag{10}\]  
So, based on Equations (8) and (10), we can calculate the expectation and variance of the SUs' data transmission time \(T\), as follows. \[\text{E}(T)=\frac{F}{C-\sum_{i=1}^{N}\frac{C_{i}\lambda_{i,pu}}{\mu_{i,pu}}} \tag{11}\]  
\[\text{D}(T)=\frac{F\sum_{i=1}^{N}\frac{2C_{i}^{2}\lambda_{i,pu}}{C^{2}\mu_{i,pu }^{2}}}{C(1-\sum_{i=1}^{N}\frac{C_{i}\lambda_{i,pu}}{C\mu_{i,pu}})(1-\sum_{i=1} ^{N}\frac{C_{i}^{2}\lambda_{i,pu}^{2}}{C^{2}\mu_{i,pu}^{2}})} \tag{12}\]  
A M/G/1 process is considered for SUs in the CR-based SA, the arrival of SUs follows the Poisson distribution, and the service times are independent random variables have a common probability distribution. The service times are independent of the arrival process. _T\({}_{w}\)_ denotes the average delay of the SUs' data transmission time. Based on the Pollaczek-Khintchine formula, we can obtain the expression of _T\({}_{w}\)_ as follows.