# An Improved Radio Resource Management with Carrier Aggregation in LTE Advanced
## 3 System Model  
### Problem Formulation  
This paper implements MLWDF for downlink radio resource management. MLWDF is a throughput-optimal scheduling algorithm which considers packet delay and probability of packet loss along with instantaneous and previous throughput of the user. The scheduler maximizes the utility function \(\sum_{i}\frac{n(i)}{r_{i}(t-1)}\) to achieve better QoS, where \(r_{i}(t)\) is the instantaneous transmission rate of user \(i\) at TTI \(t\) and \(r_{i}(t-1)\) is the previous average transmission rate. For convenience, Table 2 shows the symbols used in this paper. Let \(i,j,k,l\) be the user, CC, RB, and MSC index, respectively where \(i\in Q=\{1,\ldots,Q\}\), \(j\in R=\{1,\ldots,R\}\), \(k\in S=\{1,\ldots,S\}\) and \(l\in T=\{1,\ldots,T\}\). In the above function, \(r_{i}(t)/\overline{r}_{i}(t-1)\) can be denoted as the weighted transmission rate \(W_{i,j,k,l}\) for user \(i\), CC \(j\), RB \(k\), and MSC \(l\). For simplification, the TTI \(t\) is removed from the rest of the paper. The resource scheduling is defined by solving the following function according to the MLWDF algorithm. \[\sum_{i=1}^{Q}\sum_{j=1}^{R}\sum_{k=1}^{S}\sum_{l=1}^{T}\psi_{i,j,k,l}\cdot W_ {i,j,k,l}\cdot\alpha_{i}\cdot D_{i,j} \tag{1}\]  
Subjected to the following conditions:  
\[\sum_{i=1}^{Q}\psi_{i,j,k,l}\leq 1\quad\text{for}\ j\ \in\ R,\ k\ \in\ S,\ l\ \in\ T \tag{2}\]  
\[\sum_{l=1}^{T}\psi_{i,j,k,l}\leq 1\quad\text{for}\ i\ \in\ Q,\ j\ \in\ R,\ k\ \in\ S \tag{3}\]  
\[\sum_{j=1}^{R}\psi_{i,j,k,l}\leq m\quad\text{for}\ i\ \in\ Q,\ k\ \in\ S,\ l\ \in\ T \tag{4}\]  
\[D_{i,j}\leq\tau_{i}\quad\text{for}\ i\ \in\ Q,\ j\ \in\ R \tag{5}\]  
\[\psi_{i,j,k,l}\in\{0,1\} \tag{6}\]  
where \(\psi_{i,j,k,l}\) is a variable to indicate whether or not CC \(j\), RB \(k\), and MSC \(l\) is assigned to user \(i\), \(D_{i,j}\) is the HOL delay of user \(i\) and CC \(j\). The condition in (2) states that each RB of a CC can be assigned to \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Definition** \\ \hline \(i\) & User, \(i\in Q=\{1,\ldots,Q\}\) \\ \(j\) & CC, \(j\in R=\{1,\ldots,R\}\) \\ \(k\) & RB, \(k\in S=\{1,\ldots,S\}\) \\ \(l\) & MSC index, \(l\in T=\{1,\ldots,T\}\) \\ \(m\) & Maximum number of CC can be assigned to a user \\ \(t\) & Time (TTI index) \\ \(\psi_{i,j,k,l}\) & A variable to denote whether or not CC \(j\), RB \(k\) and MSC \(l\) is assigned to user \(i\) \\ \(W_{i,j,k,l}\) & Weighted transmission rate of user \(i\), CC \(j\), RB \(k\), and MSC \(l\) \\ \(D_{i,j}\) & Head of line delay of user \(i\), CC \(j\) \\ \(\xi_{j}\) & Queue length of CC \(j\) \\ \(\delta_{i}\) & Probability of packet loss of user \(i\) \\ \(\Omega\ (j,k)\) & Transmission rate of user currently being allocated with CC \(j\) and RB \(k\) \\ \(g\ (i,j,l)\) & The gain after assigning CC \(j\) and MSC \(l\) to user \(i\) \\ \(\sigma\ (i,j,k,l)\) & Weighted transmission rate metric with packet loss and delay \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Symbols used in the paper. maximum one user. The condition in (3) assures that one user can use only one MSC with assigned CC and RB. The condition in (4) restricts the user not to be assigned more than \(m\) number of CC. The condition in (5) mentions that HOL delay must be lower than the delay threshold. By the definition of the MLWDF algorithm, \(\alpha_{i}\) in Equation (1) is a variable which allows the scheduler to control the distribution of packet delay [18]. Increasing the value of \(\alpha_{i}\) for user \(i\), while keeping the values for other users unchanged, reduces the packet delay for user \(i\) at the cost of increasing the waiting time for other users. The variable \(\alpha_{i}\) is defined in the following Equation (7). \[\alpha_{i}=\frac{-\log\delta_{i}}{\tau_{i}} \tag{7}\]  
Here, \(\tau_{i}\) is the delay threshold for user \(i\) based on their applications and \(\delta_{i}\) represents the probability of packet loss. Delays of the user data packets need to be kept below a specific threshold value.