## 2 System Model  
We consider a downlink cellular system with \(N\) BSs covering a geographical area in which \(M\) devices are needing to be served. Let \(D\) and \(C\) denote the set of MTC and HTC devices, respectively. The downlink data rate of each HTC device \(c\in C\) associated to BS \(j\) is denoted as \(R_{cj}\). We also assume a minimum downlink data rate requirement that should be fulfilled (i.e., \(R_{cj}\geq R_{c}^{min}\)). Similarly, \(R_{dj}\) denotes the downlink data rate of each MTC device \(d\in D\), whereas \(R_{dj}\geq R_{d}^{min}\) refers to each MTC device's downlink minimum data requirement. Additionally, in line with long-term evolution (LTE)/4G and new radio (NR)/5G radio interfaces, we assume orthogonal frequency division multiple access (OFDMA) as an access method in the air interface so that the total system bandwidth \(W\) is divided into \(K\) resource blocks according to a frequency reuse pattern. Thus, a subset of \(K_{j}\) resource blocks is assigned to each BS \(j\in N\). Furthermore, each device \(i\in M\) is expected to be served by a single BS, which implies that fractional assignments are not allowed in our modeling. As for radio resources, each BS \(j\) has a maximum transmit power \(P_{j}^{max}\). Without a loss of generality, we assume that the total transmission power of each BS is equally distributed, on average, over all available \(K_{j}\) subchannels at BS \(j\). This assumption is mainly intended to indicate that BSs are supposed to make use of all available subchannels in the same way (i.e., there is no subchannel more favored that another [32]). Hence, the mean interference power that is seen in each subchannel is expected to be, on average, the same. We also assume that each BS is constrained by a backhaul capacity of the wireless/wired link connecting each BS with the core network. Such backhaul link capacity at each BS could be defined, for instance, in terms of the maximum achievable data rate in the air interface. Over such a basis, the suitability of the assignment of the device \(i\) to BS \(j\) is modeled in terms of utility functions and resource cost functions, which have been successfully used in the modeling of resource allocation problems. The above described system modelling is valid for any number of mobile devices and/or different minimum data rate requirements, as well as resource constraints. Hence, for a given system's configuration, in this work the BS assignment problem can be regarded as finding the most appropriate assignment solution that satisfies each device's QoS requirement and also fulfill resource constraints. In what follows, we elaborate on the considered utility and resource cost functions. We first define a utility function \(u_{ij}\) that represents the suitability of the assignment in terms of the signal-to-interference-plus-noise ratio (SINR) achieved by device \(i\) being assigned to BS \(j\), over the resource block \(k\), which is computed as:  
\[SINR_{i,j,k}=\frac{P_{i,j,k}G_{i,j,k}}{I_{i,j,k}+\eta} \tag{1}\]  
where \(P_{i,j,k}\) is the transmit power of BS \(j\) over the given resource block \(k\), \(G_{i,j,k}\) denotes the channel gain between device \(i\) and BS \(j\) over resource block \(k\), which accounts for path loss, shadowing, and noise figure, \(\eta\) is the thermal noise level, and \(I_{i,j,k}\) is the inter-cell interference experienced by the device at BS \(j\) on resource block \(k\). Such inter-cell interference can be computed as:  
\[I_{i,j,k}=\sum_{n=1,n\
eq j}^{n=N}P_{m\
eq i,m,k}G_{i,n,k} \tag{2}\]  
where \(P_{m\
eq i,n,k}\) is the transmission power of the \(n\) interfering BSs, on resource block \(k\) assigned to device \(m\
eq i\). Accordingly, if device \(i\) is served by the BS \(j\), the achievable transmission rate \(r_{i,j,k}\) in terms of bps/Hz can be written as:  
\[r_{i,j,k}=\frac{W}{K_{j}}\log_{2}\left(1+SINR_{i,j,k}\right) \tag{3}\]  
The utility function \(u_{ij}\) quantifies the device's satisfaction with respect to the achievable bit rate defined in Equation (3).