# A Novel Power-Saving Transmission Scheme for Multiple-Component-Carrier Cellular Systems
## 4 Proposed Scheme
### Fair Bandwidth Allocation (FBA)  
The respective fairness control parameters of the type-\(m\), \(m=1\), \(2\), users in round \(s\), denoted as \(v^{(m)}(s)\), must then be obtained to control the fairness index for each type of user around the respective target values. The value of \(v^{(m)}(s)\) can be expressed in the form of a stochastic gradient search:  
\[\
u^{(m)}(s)=\
u^{(m)}(s-1)-\eta^{(m)}\cdot(\Phi^{(m)}(s)-\Phi^{(m)}_{\text{ target}}) \tag{8}\]  
for \(m=1\), \(2\), respectively, where \(\eta^{(m)}\) denotes the step-size parameters for type-\(m\) users.  
Since the inherent differing characteristics of the fairness indexes for the RT users and NRT users are respectively used in the calculations for evaluating those indexes, the radio resources are dynamically divided into two parts based on the ratio of the type-\(m\) users' required data rate to the required data rate of all the users in the scheduling queue. In addition, the design includes the stipulation that each type of user has their own region of greater priority than the other type of users. By dint of this approach, certain relative levels of fairness between the RT users and NRT users can be maintained. Specifically, we use \(\Re_{\text{w}}^{(m)}(s)\) to denote the subset of type-\(m\) users in \(\Re_{\text{w}}(s)\) and use \(\bar{\tau}_{\text{req}}^{(m)}(s)\) to denote the average required data rate of type-\(m\) users in \(\Re_{\text{w}}^{(m)}(s)\). In addition, \(b_{i}(s)\) is defined, for round \(s\), as the number of non-occupied sub-channels of CC-\(i\). Next, \(b_{\text{NRT}_{i}}(s)\) is used to indicate the number of sub-channels of CC-\(i\) for which the NRT users are given higher priority than the RT users to share in during round \(s\); this term can be calculated as follows:  
\[b_{\text{NRT},i}(s)=\left|b_{i}(s)\frac{\bar{\tau}_{\text{req}}^{(1)}(s)\left| \Re_{\text{w}}^{(1)}(s)\right|}{\sum\limits_{m=1}^{2}\bar{\tau}_{\text{req}}^{ (m)}(s)\left|\Re_{\text{w}}^{(1)}(s)\right|}\right. \tag{9}\]  
Meanwhile, \(b_{\text{RT},i}(s)\) denotes the number of sub-channels of CC-\(i\) for which the RT user sessions are given a higher priority than the NRT user sessions to share in during round \(s\); this term can be calculated as follows:  
\[b_{\text{RT},i}(s)=b_{i}(s)-b_{\text{NRT},i}(s) \tag{10}\]  
Next, we schedule waiting user session requests in the scheduling queue by starting from the unassigned sub-channels of the active CCs based on SAD with the index \(i=1\) in the configuration \(z\). Referring to the aforementioned study by [PERSON] and [PERSON] [8], the optimal fair assignment vector, which is indicated by tuple(\(i^{*}j^{*}k^{*}\)), can be searched via:  
\[\text{tuple}(i^{*},j^{*},k^{*})=\arg\max\limits_{i,j,k\in\Re_{\text{w}}(s)}( w_{k}(s)\widetilde{\gamma}_{k}(i,j,s)) \tag{11}\]  
under the assumption that equal power allocation is applied for each sub-channel. Note that in Equation (11), \(w_{k}(s)\) denotes the weighting factor for user-\(k\) in \(\Re_{\text{w}}(s)\) during round \(s\). With appropriate modification of the weighting factor presented in the aforementioned study by [PERSON] and [PERSON] [8] to fit our model, \(w_{k}(s)\) can be calculated as follows:  
\[w_{k}(s)=\left(\frac{d_{k}(s)}{\bar{\tau}_{k}(s)}\right)^{v^{(m)}(s)},\ \forall k\in\Re_{\text{w}}(s) \tag{12}\]  
The set of optimal tuple(\(i^{*}j^{*}k^{*}\))s is the solution that results in a value of \(\Phi^{(m)}(s)\) closest to the \(\Phi^{(m)}_{\text{target}}\) value.