# Handover-Driven Interference Management for Co-Channel Deployment of Femto- and Macro-Cells
## 5 Control Algorithm
A unique system load \(\rho^{*}\) exists such that \(\rho^{*}\leq\rho^{(\mathbf{n})}\) for \(\forall\mathbf{n}\). Due to the monotonicity of \(\rho\) in \(\mathbf{p}\), a sufficient condition is the existence of a unique power vector \((\mathbf{p}^{*},\mathbf{q}^{*})\) such that \((\mathbf{p}^{*},\mathbf{q}^{*})\preceq(\mathbf{p}^{(\mathbf{n})},\mathbf{q}^{ (\mathbf{n})})\)\(\forall\mathbf{n}\), i.e., \(\rho^{*}=\rho(\mathbf{p}^{*},\mathbf{q}^{*})\). When DL was considered only, the existence and uniqueness of \(\mathbf{p}^{*}\) were proved in Theorem 1 of [16] under the same feasibility condition as defined earlier on \(\mathbf{p}\). A new power vector \(\mathbf{y}\triangleq(\mathbf{p},\mathbf{q})\) can be defined and then the actual problem is transformed into one with an analogous structure to the DL-only problem. Thus, the above proposition follows. Starting from any feasible power vectors \(\mathbf{p}\) and \(\mathbf{q}\), the per-user control algorithm makes the system load \(\rho\) converge to \(\rho^{*}\). The proof follows a procedure similar to the one used in [16]. Consider a condition where, if \(\mathbf{p}(t)\) and \(\mathbf{q}(t)\) are feasible power vectors, \((\mathbf{p}(t+1),\mathbf{q}(t+1))\preceq(\mathbf{p}(t),\mathbf{q}(t))\), and \(\mathbf{p}(t+1)\) and \(\mathbf{q}(t+1)\) are feasible. If this condition is met, \((\mathbf{p}(0),\mathbf{q}(0))\succeq\cdots\succeq(\mathbf{p}(t),\mathbf{q}(t))\), and \(\mathbf{p}(t)\) and \(\mathbf{q}(t)\) are feasible. Hence, the sequence of \((\mathbf{p}(t),\mathbf{q}(t))\) is decreasing and always feasible. Since \(\rho\) is monotonic and bounded below by \(\rho^{*}\), the sequence of \(\rho(\mathbf{p}(t),\mathbf{q}(t))\) converges to \(\rho^{*}\). In what follows, it is shown that the proposed algorithm meets the above condition. Equation (6) can be rewritten as  
\[ \begin{array}{l}\max_{n_{j}}\{\Delta_{i,DL}^{(n_{j})}(\mathbf{p})+\Delta_{i, lL}^{(n_{j})}(\mathbf{q})\}\\ =\rho_{i}(\mathbf{p},\mathbf{q})-\min_{n_{j}}\rho_{i}(\mathbf{p}^{(n_{j})}( \mathbf{p}),\mathbf{Q}^{(n_{j})}(\mathbf{q}))\\ =\sum_{j^{\prime}\in\mathcal{U}_{t}}\rho_{i}(p_{j^{\prime}},q_{j^{\prime}})\\ -\min_{n_{j}}\left\{\sum_{j^{\prime}\in\mathcal{U}_{t}-\{j\}}\rho_{i}(p_{j^{ \prime}},q_{j^{\prime}})+\rho_{i}(P_{j}^{(n_{j})}(\mathbf{p}),Q_{j}^{(n_{j})}( \mathbf{q}))\right\}\\ =\rho_{i}(p_{j},q_{j})-\min_{n_{j}}\rho_{i}(P_{j}^{(n_{j})}( \mathbf{p}),Q_{j}^{(n_{j})}(\mathbf{q}))\end{array}  \tag{8}\]  
where the second equality follows from the additivity of the cell load function. That is, the algorithm minimizes  
\[ \begin{array}{l}\rho_{i}(P_{j}^{(n_{j})}(\mathbf{p}),Q_{j}^{(n_{j})}( \mathbf{q}))\ \forall j,\text{ so }\rho_{i}(P_{j}^{(n_{j}^{\prime})}(\mathbf{p}),Q_{j}^{(n_{j}^{\prime})}( \mathbf{q}))\leq\rho_{i}(P_{j}^{(n_{j})}(\mathbf{p}),Q_{j}^{(n_{j})}(\mathbf{q })).\end{array} \]  
Therefore,  
\[ \begin{array}{l}\rho_{i}(\mathbf{p}(t+1),\mathbf{q}(t+1)) =\rho_{i}(\mathbf{p}^{(\mathbf{n}^{*})}(\mathbf{p}(t)),\mathbf{Q}^{( \mathbf{n}^{*})}(\mathbf{q}(t)))\\ =\sum_{j}\rho_{i}(P_{j}^{(n_{j}^{\prime})}(\mathbf{p}(t)),Q_{j}^{(n_{j}^{ \prime})}(\mathbf{q}(t)))\\ \leq\sum_{j}\rho_{i}(P_{j}^{(n_{j})}(\mathbf{p}(t)),Q_{j}^{(n_{j})}(\mathbf{q }(t)))\\ =\rho_{i}(\mathbf{p}^{(\mathbf{n})}(\mathbf{p}(t)),\mathbf{Q}^{(\mathbf{n})}( \mathbf{q}(t)))\\ \leq\rho_{i}(\mathbf{p}(t),\mathbf{q}(t))\end{array}  \tag{9}\]  
where the last inequality is obtained by the defined monotonicity of the cell load function and the assumed feasibility of \((\mathbf{p}(t),\mathbf{q}(t))\). Equation (9) shows that the cell load decreases monotonically as the algorithm runs through iterations. If \(\mathbf{p}(t)\) is feasible, \(p_{j}(t+1)=\mathbf{G}_{j,DL}^{(n)}\mathbf{p}(t)+\eta\geq\mathbf{G}_{j,DL}^{( n)}\mathbf{p}(t+1)\)since \(\mathbf{p}(t)\succeq\mathbf{p}(t+1)\), thus \(\mathbf{p}(t+1)\) is also feasible and the same holds for \(\mathbf{q}(t+1)\) (\(\mathbf{G}_{j,DL}^{(n)}\) is \(j\)th row vector of \(\mathbf{G}_{DL}^{(n)}\)). Thus, the above condition is met.