# Clustering-Based Channel Allocation Method for Mitigating Inter-WBAN Interference
## 3 System Model  
We consider a set of WBANs in an area \(\mathcal{S}\). A cloud server coordinates the channels for the WBANs in \(\mathcal{S}\) so that the amount of interference experienced by the WBANs can be mitigated. We assume that within each WBAN, there is always data to be exchanged between a coordinator and the sensor nodes. Similar to other works [23], we also assume that the unsuccessful data transmission within a WBAN is caused only by the interference among the WBANs. We assume that the control time is divided into discrete time slots of equal size \(\tau_{c}\). At every \(\tau_{c}\), the coordinators of the WBANs send their location information to a cloud server through a cellular link such as an LTE and 5G. Therefore, we assume that a server can receive the messages from the coordinators reliably and in time regardless of the density of the coexisting WBANs. A cloud server performs a channel allocation task with the position information. We denote the set of WBANs in \(\mathcal{S}\) at the \(t\)-th time slot as \(N(t)\). We denote the set of channels that can be assigned to a WBAN as \(K=\{1,2,\cdots,|K|\}\), where \(|K|\) is the cardinality of the set \(K\). The channel allocated to a WBAN \(i\) during the \(t\)-th time slot is denoted as \(k_{i}(t)\in K\). We also denote by \(N_{i}(t)\) the set of WBANs within the interference range of a WBAN \(i\). In other words, if a WBAN \(j\in N_{i}(t)\) uses the same channel as that of \(i\) during the \(t\)-th time slot, \(j\) interferes \(i\). To focus on the influence of the channel allocation method, we assume that each WBAN does not control the data transmission power and denote the transmission power of each sensor node in a WBAN \(i\) as \(p_{i}\). Then, the amount of interference experienced by a WBAN \(i\) during the \(t\)-th time slot is given as  
\[I_{i}(t)=\sum_{j\in N_{i}(t)}p_{j}h_{i,j}(t)\delta_{i,j}(t), \tag{1}\]  
where \(h_{i,j}(t)\) is the channel gain between a WBAN \(i\) and a WBAN \(j\). Since we consider a relatively long time slot, we assume that the small time scale fading averages out and only the path loss influences on \(h_{i,j}(t)\). The function \(\delta_{i,j}(t)\) is used to represent whether WBANs \(i\) and \(j\) are using the same channel or not. In other words, \(\delta_{i,j}(t)=1\) if \(k_{i}(t)=k_{j}(t)\). Otherwise, \(\delta_{i,j}(t)=0\). From the [PERSON]'s capacity formula [18], the data rate per Hertz obtained by a WBAN \(i\) during the time slot \(t\) is given as  
\[\gamma_{i}(t)=\log_{2}(1+\frac{p_{i}h_{i,j}(t)}{I_{i}(t)+N_{0}}), \tag{2}\]  
where \(N_{0}\) is the noise power, which is assumed to be a stationary random process. Accordingly, the total data rates of the WBANs in \(N(t)\) is obtained as  
\[\gamma(t)=\sum_{i\in N(t)}\gamma_{i}(t) \tag{3}\]  
Our goal is to maximize \(\gamma(t)\). From Equations (1) and (2), we observe that we can control \(\gamma(t)\) by manipulating the channel allocation profile \((k_{1}(t),\cdots,k_{|N(t)|}(t))\). Thus, our goal becomes selecting an optimal channel allocation profile that can maximize the total \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & **Operation** & **Signaling** & **Overhead** & **Adaptability** & **DT** & **Approach** \\ \hline
[16,17] & distributed & inter-WBAN & high & moderate & moderate & GC \\
[22] & distributed & inter-WBAN & high & moderate & slow & heuristic, FDMA \\
[10] & distributed & not required & low & low & slow & stochastic learning \\
[23] & distributed & not required & low & low & fast & latin square \\
[24] & centralized & WBAN-Server & moderate & low & fast & heuristic \\
[25] & centralized & WBAN-Server & moderate & low & fast & greedy \\
[26] & centralized & WBAN-Server & moderate & low & fast & KNN, FDMA \\
[27] & hybrid & WBAN-Server & high & low & slow & Stackelberg game \\ proposed & centralized & WBAN-Server & moderate & high & fast & clustering, GC \\ \hline \hline \end{tabular}
\end{table}