## 3 Architecture and Problem Formulation
### System Model
#### 3.2.4 Network Battery Lifetime  
Network battery lifetime is defined as the total time from network deployment to the time when the network is considered to be no longer functional. In the case of MTC-centric design, uRLLC devices have higher priority and therefore they contribute to the determination of the boundary condition of network functionality. A shortest individual lifetime (SIL) is deemed suitable to define and propose the notion of network lifetime defined as:  
\[L_{n}(t^{\rho})=\text{min}L_{k}(t^{\rho}). \tag{10}\]  
Massive connectivity can be partially supported by existing LTE-A cellular networks by adopting a clustering design for mMTC nodes. For simplicity, mMTC devices are assumed to have similar communication requirements including packet size and generation patterns. To maximize the lifetime of nodes clustered together in mMTC groups, all the members of the respective group perform gateway functionality. This helps in reducing the possibility of a single node experiencing battery drainage and causing a network failure condition as per SIL criterion. This approach also ensures the maximum possible SIL for each cluster. Let us denote the average cluster size of these mMTC nodes as \(s\). The cluster forming problem in the context of mMTC nodes is considered at a reference point in time where \(E_{i}(t^{\rho})=E_{o}\) and \(E_{o}\) represent the full battery capacity. The member and gateway transmit power levels are denoted by \(p_{t}^{m}\) and \(p_{t}^{g}\), respectively. In each data transmission period of clustered communication design, a node may either be in the gateway mode or the member mode with probabilities 1/s or (1 - 1/s), respectively. Hence, the expected life of battery for each mMTC node in a cluster centered at a distance \(d_{g}\) from the BS is expressed as the product of the ratio between the remaining battery energy to the averaged energy consumption in each transmission interval and cluster reporting period is defined as:  
\[L_{g}\Big{(}d_{g},\ s\Big{)}=\ \frac{E_{o}\ T_{g}}{\frac{1}{s}\ \hat{E}_{g}+ \Big{(}1-\frac{1}{s}\Big{)}\hat{E}_{m}}, \tag{11}\]  
where:  
\[\hat{E}_{g}=\ E_{s}^{g}+\ \frac{(s-1)\ \hat{D}}{R_{m}}p_{l}+[\ 1+\ \mu_{c}(s-1)] \hat{D}\frac{p_{e}+\eta p_{t}^{g}}{R_{g}}, \tag{12}\] \[\hat{E}_{m}=\ E_{s}+\ \hat{D}\frac{p_{e}+\eta p_{t}^{m}}{R_{m}},\]  
where \(T_{g}\) is the cluster transmission interval, \(\hat{D}\) is the average packet size, \(p_{l}\) is the power consumption in listening mode, and \((s-1)\ \hat{D}p_{l}/R_{m}\) models the energy consumption of the gateway during the reception of data from other nodes in the cluster. In addition, \(\mu_{c}\) is the coefficient for the compression of the packet length on gateway nodes, and \(E_{s}\)\({}^{S}\) is the averaged static energy consumed in gateway mode for a node. \(R_{\text{m}}\) and \(R_{\text{g}}\) are the data rates of the respective nodes communicating within and outside the cluster as provided in an example [37]. To meet the mission-critical service requirements a direct link needs to be established to the BS. The air interface for the proposed joint HTC/MTC design is modeled considered as 3 GPP LTE Release 13 [38] with time-frequency distribution of resources. For an LTE uplink power control mechanism, the transmit power for the uplink for each MTCD can be acquired using the downlink path loss estimation as:  
\[P_{UL}(c_{k},\delta_{k})=\ c_{k}P_{\text{$\theta$}}\delta_{k}\Big{[}2^{\frac{1 2 STS(\zeta_{k},\delta_{k})}{\zeta_{k}N_{\text{$\theta$}}N_{\text{$\theta$}}}}-1 \Big{]}, \tag{13}\]  
where \(c_{k}\) denotes the number of physical resource block pairs (PRBPs) allocated to a node \(k\), \(\beta_{k}\) is the compensation factor and the estimated downlink path loss is defined as \(\theta_{k}\). The number of symbols and subcarriers in a PRBP are denoted as \(N_{s}\) and \(N_{sc}\) respectively. The transport block size (TBS) is the function of \(c_{k}\) and the TBS index, defined as shown in Table 7.1.7.2.1-1 [38].