# A Novel Power-Saving Transmission Scheme for Multiple-Component-Carrier Cellular Systems
## 3 Problem Description
### Problem Formulation  
In this study, our ideal goal, for each scheduling round, was to minimize the overall power consumed of the BS transceivers, \(P_{\text{total}}(s)\), subject to the stipulation that the fairness indexes for the different type-\(m\) users (_i.e._, \(m=1\), \(2\)), \(\Phi^{(m)}(s)\), needed to be maintained at their respective desired target values, \(\Phi^{(m)}_{\text{target}}\), and, furthermore, that the data transmission rates of the active users, \(r_{k}(s)\), \(\forall k\in\Re_{\text{w}}(s)\cup\Re_{\text{s}}(s)\), had to be achieved at their respective minimum required levels, \(r_{\text{req},k}\). Furthermore, we let \(\beta(s)\) denote the set of available radio resources for round \(s\) and let \(\varpi_{k}(s)\) denote the set of the  
Figure 1: The system model for the present study. The dotted line surrounding the SCCs indicates that those SCCs can be turned off based on the traffic load.  
radio resources assigned to active user-\(k\) in \(\Re_{\text{w}}(s)\cup\Re_{\text{s}}(s)\) during round \(s\). Under those conditions, the optimization problem for each round could then be formulated as follows:  
\[\underset{z,\;\tau_{k}(s),\;\omega_{k}(s)}{\text{minimize}}\;P_{\text{total}}(s)= p^{(z)}(s)+\sum\limits_{k\in\Re_{\text{w}}(s)\cup\Re_{\text{s}}(s)}\sum \limits_{i=1}^{z}\sum\limits_{j=1}^{\ell_{i}}p_{\text{trans},k}(i,j,s)\]  
subject to:  
\(z\in\Omega\);  
\(\Phi^{(m)}(s)=\Phi^{(m)}_{\text{target}}\), \(m=1,\,2\);  
\(\tau_{k}(s)\geq r_{\text{req},k}\), \(\forall k\in\Re_{\text{w}}(s)\cup\Re_{\text{s}}(s)\);  
\(\sum\limits_{k\in\Re_{\text{w}}(s)\cup\Re_{\text{s}}(s)}\sum\limits_{j=1}^{ \ell_{i}}p_{\text{trans},k}(i,j,s)\leq p_{\text{trans},\text{max},i}\), \(i=1,\,2,\,\ldots,\,z\);  
\(\sum\limits_{k\in\Re_{\text{w}}(s)\cup\Re_{\text{s}}(s)}\varpi_{k}(s)\subseteq \beta(s)\), \(\varpi_{x}(s)\cap\varpi_{y}(s)=\varnothing\), \(x\
eq y\).  
Using [PERSON], [PERSON], and [PERSON]'s fairness index formula [29], \(\Phi^{(m)}(s)\) can then be expressed as:  
\[\Phi^{(m)}(s)=\frac{\left(\sum\limits_{k\in\Re^{(m)}(s)}\phi_{k}(s)\right)^{2 }}{\left|\Re^{(m)}(s)\right|\sum\limits_{k\in\Re^{(m)}(s)}\left(\phi_{k}(s) \right)^{2}},\;m\;=\;1,\;2 \tag{2}\]  
where \(\phi_{k}(s)\) denotes the fairness index for the given user-\(k\). Furthermore, according to the aforementioned work by [PERSON] and [PERSON] [8], \(\phi_{k}(s)\) is designed to be calculated by:  
\[\phi_{k}(s)=\frac{\overline{r}_{k}(s)}{d_{k}(s)},\;\forall k\in\Re^{(1)}(s) \cup\Re^{(2)}(s) \tag{3}\]  
where \(\overline{r}_{k}(s)\) denotes the average observed data rate for the given user-\(k\) until round \(s-1\) (which can be determined using various smoothing methods (for an example of a convergence analysis of smoothing methods, please refer to the study by [PERSON] and [PERSON] [30]), such as exponential filtering methods) and \(d_{k}(s)\) denotes the delay in addressing any session request for user-\(k\) till round \(s\).  
At this point, one can ascertain that the considered optimization formulation constitutes an integer-nonlinear programming problem. However, an integer-nonlinear optimization problem is highly complex, meaning that in order to find the optimal solution for such a problem, an exhaustive search is generally required; in other words, a problem of this type can be viewed as numerically tractable. Moreover, no optimal solution will be possible when the traffic load is so high that it exceeds what the system can handle. Consequently, in light of the structure of the considered problem, we instead propose, in the following section, a heuristic power-saving transmission scheme that achieves efficient control of fairness for each type of user while also maintaining the data rates for those users at levels above their respective minimum requirements.