## 2 System Model
### Proactive-Energy Saving Optimisation Based CO\({}_{2}\) Reduction
\[\gamma_{u}(k+\bar{k})=\left[\frac{P_{t}.G_{u}\bar{G_{u}}\delta\alpha(\Delta_{u })^{-\beta}}{N+\sum\limits_{i\in C}\ P_{t}^{i}\.G_{u}\bar{G_{u}}\delta\alpha(\Delta_{u})^{-\beta}}\right]_{k+\bar{k}}, \tag{8}\]  
where cell transmit power is represented by \(P_{t}\), user equipment gain is represented by \(G_{u}\), transmitting antenna gain seen by user \(u\) is represented by \(G_{u}\), signal varient shadowing is represented by \(\delta\), path loss constant is represented by \(\alpha\), estimated user \(u\) location distance \(M_{k+\bar{k}}\) is given by \(\Delta_{u}\) and path loss exponent is represented by \(\beta\). The next time steps \(k+\bar{k}\) are reliant on the time subscript enclosed within \([. ]k+\bar{k}\) throughout the paper. We assume the shadowing estimated information is available whichwould provide user \(u\) estimated locations with minimal errors. Thus, we calculate the SINR expression of fully loaded cells along with the interference from neighbouring cells for data transmission as  
\[\gamma_{u}(k+\bar{k})=\left[\frac{P_{i}.G_{u}\bar{G_{u}}\delta\alpha(\Delta_{u}) ^{-\beta}}{N+\sum\limits_{i\in C}^{n}\rho_{i}P_{i}^{i}\cdot G_{u}\bar{G_{u}} \delta\alpha(\Delta_{u}^{i})^{-\beta}}\right]_{k+\bar{k}}, \tag{9}\]  
where \(\rho_{i}\) denotes the cell load of a cell \(i\). The process of compensating a distorting factor for received interference power from each active cell yields a particular coupling of the total interference when multiple cells are utilised. Heavily loaded cells are more power interfering contributors than less loaded ones [3,7]. For a HetNet arrangement, instantaneous cell load is the ratio of active Physical Resource Blocks (PRBs) obtained during a Transmission Time Interval (TTI) with the available PRBs available in the cell. Hence, for monitoring UL/DL total PRBs usage, the ratio act as a standard measurement indicator. QoS and achievable SINR are influenced by the number of PRBs allocated in an SC(s). PRBs are directly proportional to the required data rate to maintain QoS. Hence, the more PRBs are assigned to a user, the higher is the QoS and the lower is the SINR. We define the total cell \(c\) load for each time stepping intervals \(k+\bar{k}\) to achieve the required rate of all users of an SC as:  
\[\lambda(k+\bar{k})=\left[\frac{1}{RB_{n}}\sum\limits_{i\in C}^{n}\frac{\tau_{u }}{B^{u}(w)\ log_{2}1+\gamma_{u}}\right]_{k+\bar{k}}, \tag{10}\]  
where \(B^{u}(w)\) is the bandwidth of one resource block, \(RB_{n}\) are the number of resource blocks in a cell \(c\), \(\tau_{u}\) is the transmission rate and \(i\subset C\) are the number of active users \(u\) in a cell \(c\). We define this load as a virtual load which is allowed to exceed one to monitor the cell loading state and give us clear indication. The minimum bit rate \(\tau_{u}\) is required to maintain QoS requirements by continuously serving users \(u\) in each cell \(c\). There are several methods used to calculate the required user throughput by calculating required resources to service the user. In the 3 rd Generation Partnership Project (3 GPP) standards, a mechanism known as QoS Class Identifier (QCI) is used to prioritise active users based on the resource type allocation and requirements. We can define the desired throughput on QCI that would model transmission bit rate \(\tau\) in a more robust way while leveraging network analytics. Furthermore, functional user behaviours, their service request patterns, levels of subscription and their in-use applications would be modelled using transmission bit rate \(\tau\)[5].