# Energy-Efficient BWP Configuration for Multi-Slice Users
## 3 System Model
#### The Cost Function  
The cost function for UE \(u\) selecting BWP configuration strategy \(c\) is given by:  
\[C_{u,c}=\sum_{s\in\{c\#\alpha c\}}\alpha_{u,s}\cdot DST_{u,s,c}+\theta_{c} \cdot SwDl_{u}+\beta_{u}\cdot B_{c} \tag{4}\]  
where  
* \(DST_{u,s,c}\) is the sojourn time of UE \(u\) for retrieving the data of a particular service \(s\) when selecting strategy \(c\). The service \(s\) can be either eMBB denoted by \(eb\) or URLLC denoted by \(uc\).
* \(SwDl_{u}\) is the BWP switching delay of UE \(u\) (also a part of the user's overall sojourn time).
* \(B_{c}\) is the total band scanned by the user depending on its devised strategy \(c\).
* \(\alpha_{u,s}\) and \(\beta_{u}\) are normalizing factors. The former reflects the class of service \(s\) for UE \(u\) and the latter represents the battery level of UE \(u\).
* \(\theta_{c}\) is an indicator variable that equates to one in the presence of BWP switching delay and to zero otherwise, and hence it depends on the selected strategy.  
In the cost function (4), the first term represents the UE's sojourn time in the network, which is necessary for retrieving the data of all its services. That first term is used as a QoS indicator as the higher the user sojourn time, the lower its QoS, since the user will endure a higher delay and will stay active in the network longer, which may increase its energy consumption. Hence, a higher sojourn time increases the cost function. The second term is the BWP switching delay (when applicable), which is also a part of the user's sojourn time in the network. The third term of the cost function represents the scanned BWPs by the UE. The higher this band, the higher the energy consumed by the user for scanning a larger band. In fact, if UE \(u\) chooses the multi-numerology strategy \(MN\), it will scan consecutively BWP 1 and BWP 2 to retrieve the data of each service \(s\), and the corresponding sojourn time for service \(s\) is determined by the following equation:  
\[DST_{u,s,MN}=\frac{V_{u,s}}{Thr_{u,s}} \tag{5}\]  
where  
* \(V_{u,s}\) is the volume of data to be retrieved by UE \(u\) for service \(s\).
* \(Thr_{u,s}\) is the throughput of UE \(u\) for service \(s\).  
Additionally, with the multi-numerology scheme, the BWP switching delay is taken into account (\(\theta_{MN}=1\)) and the total scanned band by the user is the sum of BWP 1 and BWP 2 (\(B_{MN}=B_{BWP1}+B_{BWP2}\)). When UE \(u\) selects the single-numerology configuration \(SN\), it will scan BWP 3 solely to retrieve the data of both services. Therefore, it will have the same achieved throughput for both services, since the same number of PRBs is allocated to the user for each service \(s\). Therefore, \(Thr_{u}\) is independent of the type of service in that case and \(DST_{u,s,SN}=\frac{V_{u,s}}{Thr_{u}}\). Also, \(\theta_{SN}=0\) and \(B_{SN}=B_{BWP3}\) as only one BWP is scanned, which prevents BWP switching. Choosing the multi-numerology configuration will improve user's KPIs and QoS and will increase user's throughput, which in turn will reduce the user's sojourn time (first term) but at the cost of adding a BWP switching delay (secondterm) and increasing the BWP scanned by the user (third term) and hence the ensuing energy consumption. In contrast, a single numerology avoids BWP switching (second term) and reduces the energy consumption, as the BWP scanned by the user (third term) is narrower but increases the user's sojourn time (first term). Additionally, the user sojourn time increases when the number of users choosing the same BWP increases as users equally share limited amount of resources, which reduces the achieved throughput. Hence, an astute load balance of users among available configurations (single or multi-numerology) must be attained.  
This cost function is later used in both the centralized and distributed approaches, which are detailed in subsequent sections.