# Handover-Driven Interference Management for Co-Channel Deployment of Femto- and Macro-Cells
## 4 The HO-FIM Architecture
### Problem Formulation  
To simplify the exposition, a single RB is assumed while formulating problems and describing algorithms. Extension to multiple RBs is straightforward.  
Let \(n_{j}\) be the serving cell of user \(j\) and \(p_{j}\) and \(q_{j}\) be the DL transmitting power to, and the UL transmitting power from, user \(j\), respectively. \(\gamma_{j}\) is the minimum target SINR. For the user set \(\mathcal{U}\), the following vectors are defined:  
* Serving cell vector \(\mathbf{n}=[n_{1},n_{2},\ldots,n_{N}]\);
* DL transmit power vector \(\mathbf{p}=[p_{1},p_{2},\ldots,p_{N}]\);
* UL transmit power vector \(\mathbf{q}=[q_{1},q_{2},\ldots,q_{N}]\); and
* Minimum target SINR vector \(\mathbf{\Gamma}=[\gamma_{1},\gamma_{2},\ldots,\gamma_{N}]\).  
Then, HO-FIM aims to find \(\mathbf{n}\) (by the handover controller), \(\mathbf{p}\), and \(\mathbf{q}\) (by the radio resource controller) that minimize the DL and UL loads of the entire network. The handover-based load minimization problem denoted by \(P1\) is stated as (the curled inequality symbol \(\succeq\) (and its strict form \(\succ\)) represents component-wise inequality):  
**Definition 1** (_Handover-based load minimization problem_).: \[\begin{array}{l}P1:\min_{\mathbf{n}}(\rho(\mathbf{p})+\rho(\mathbf{q}))\\ \text{subject to }\mathbf{p},\mathbf{q}\succeq 0.\end{array}\] (1)  
When solving \(P1\), \(\mathbf{p}\) and \(\mathbf{q}\) are given by the radio resource controller. It is said that \(p_{j}\) is _feasible_ when a SINR higher than \(\gamma_{j}\) is achieved, i.e., \(h_{n_{j},j}p_{j}/\,I_{\mathrm{DL},j}\geq\gamma_{j}\) where \(h_{n_{j},j}\) is a DL channel gain and \(I_{\mathrm{DL},j}\) is the DL interference experienced by user \(j\); \(\mathbf{p}\) is feasible when \(p_{j}\) is feasible for all \(j\in\mathcal{U}\). The feasibility of \(q_{j}\) and \(\mathbf{q}\) is defined likewise. The feasibility conditions of \(\mathbf{p}\) and \(\mathbf{q}\) can be written using matrix notation as \(\mathbf{p}\succeq\mathbf{G}_{DL}\mathbf{p}+\eta\) and \(\mathbf{q}\succeq\mathbf{G}_{UL}\mathbf{q}+\eta\), respectively, when the elements of the normalized channel gain matrices \(\mathbf{G}_{DL}\) and \(\mathbf{G}_{UL}\) are defined as \(G_{DL}(j,j^{\prime})=\gamma_{j}h_{n_{j^{\prime}},j}/\,h_{n_{j},j}(j\
eq j^{ \prime})\) or \(0(j=j^{\prime})\) and \(G_{UL}(j,j^{\prime})=\gamma_{j}h_{j^{\prime},n_{j}}/\,h_{j,n_{j}}(j\
eq j^{ \prime})\) or \(0(j=j^{\prime})\). \(\eta\) is the thermal noise.  
The handover overhead is considered in a separate problem. Let \(\mathbf{n}_{0}\) and \(\mathbf{n}_{\infty}\) denote the initial and limit points of the serving cell assignment, respectively. Let \(\mathbf{x}\) be a handover sequence to reach \(\mathbf{n}_{\infty}\) from \(\mathbf{n}_{0}\) and \(\Delta_{m}^{(\mathbf{x})}(t)\) be the load update step of a macrocell at the \(t\)th iteration when \(\mathbf{x}\) is applied. Then, to minimize the number of handovers: (1) \(\sum_{t=1}^{T}N_{ho}(t)\) should be minimized when \(N_{ho}(t)\) is the number of triggered handovers at the \(t\)th iteration and \(T\) is the number of iterations required to reach \(\rho_{m}^{(-)}\); and (2) the gap between \(\rho_{m}(T)\) and \(\rho_{m}^{(-)}\) should be minimized as well. Thus, the handover minimization problem can be defined as follows:  
**Definition 2** (_Handover minimization problem_).: \[\begin{array}{l}P2.1:\min_{\mathbf{x}}T\\ P2.2:\min_{\mathbf{x}}\left(\rho_{m}^{(-)}-\rho_{m}(T)\right)\\ \text{subject to }\rho_{m}^{(-)}\geq\rho_{m}(T)\end{array}\] (2)  
Where  
\[\rho_{m}(k)=\rho_{m}(0)-\sum_{t=1}^{k}\Delta_{m}^{(\mathbf{x})}(t). \tag{3}\]