# Handover-Driven Interference Management for Co-Channel Deployment of Femto- and Macro-Cells
## 5 Control Algorithm
The load gap monotonically decreases as the algorithm runs through iterations. Proof. : First, a power gap is defined as  
\[ \begin{array}{rl}\delta_{DL}(\mathbf{p}(t))&\triangleq\mathbf{p}(t)- \mathbf{p}(t+1)\\ &=\mathbf{p}(t)-\mathbf{P}^{(\mathbf{n}^{*})}(\mathbf{p}(t))\\ &=\mathbf{p}(t)-(\mathbf{G}_{DL}^{(\mathbf{n}^{*})}\mathbf{p}(t)+\eta^{( \mathbf{n}^{*})})\\ &=(I-\mathbf{G}_{DL}^{(\mathbf{n}^{*})})\mathbf{p}(t)-\eta^{(\mathbf{n}^{*})} \\ &\triangleq\widehat{\mathbf{G}}_{DL}^{(\mathbf{n}^{*})}\mathbf{p}(t)-\eta^{( \mathbf{n}^{*})}.\end{array}  \tag{10}\]  
It has been shown for fixed \(\mathbf{G}_{DL}^{(\mathbf{n}^{*})}\) that, if the largest eigenvalue of \(\mathbf{G}_{DL}^{(\mathbf{n}^{*})}\) is less than unity, the power update procedure converges at a geometric rate [25]. Therefore, \(\delta_{DL}\) is a monotonically increasing function in \(\mathbf{p}\). Similarly, so is \(\delta_{UL}\). Let \(\mathbf{n}(t)\) be the BS assignment at \(t\), then  
\[ \begin{array}{rl}\delta(t)-\delta(t+1)=[\widehat{\mathbf{G}}_{DL}^{(\mathbf{ n}(t))}\mathbf{p}(t-1)-\widehat{\mathbf{G}}_{DL}^{(\mathbf{n}(t+1))}\mathbf{p}(t)] \\ \quad+\widehat{\mathbf{G}}_{UL}^{(\mathbf{n}(t))}\mathbf{q}(t-1)-\widehat{ \mathbf{G}}_{UL}^{(\mathbf{n}(t+1))}\mathbf{q}(t)\\ \geq\widehat{\mathbf{G}}_{DL}^{(\mathbf{n}(t))}\mathbf{p}(t-1)-\widehat{ \mathbf{G}}_{DL}^{(\mathbf{n}(t+1))}\mathbf{p}(t-1)\\ \quad+\widehat{\mathbf{G}}_{UL}^{(\mathbf{n}(t))}\mathbf{q}(t-1)-\widehat{ \mathbf{G}}_{UL}^{(\mathbf{n}(t+1))}\mathbf{q}(t-1)\\ =\mathbf{G}_{DL}^{(\mathbf{n}(t+1))}\mathbf{p}(t-1)-\mathbf{G}_{DL}^{(\mathbf{ n}(t))}\mathbf{p}(t-1)\\ \quad+\mathbf{G}_{UL}^{(\mathbf{n}(t+1))}\mathbf{q}(t-1)-\mathbf{G}_{UL}^{( \mathbf{n}(t))}\mathbf{q}(t-1)\\ =\mathbf{P}^{(\mathbf{n}(t+1))}(\mathbf{p}(t-1))-\mathbf{P}^{(\mathbf{n}(t))} \big{(}\mathbf{p}(t-1)\big{)}\\ \quad+\mathbf{Q}^{(\mathbf{n}(t+1))}(\mathbf{q}(t-1))-\mathbf{Q}^{(\mathbf{n}( t))}(\mathbf{q}(t-1))\end{array}  \tag{11}\]  
where the first inequality follows from the monotonicity of \(\delta\) and \(\mathbf{p}(t)\preceq\mathbf{p}(t-1)\). From Equation (6), \(\mathbf{P}^{(\mathbf{n}(t))}(\mathbf{p}(t-1))+\mathbf{Q}^{(\mathbf{n}(t))}( \mathbf{q}(t-1))\preceq\mathbf{P}^{(\mathbf{n}^{\prime})}(\mathbf{p}(t-1))+ \mathbf{Q}^{(\mathbf{n}^{\prime})}(\mathbf{q}(t-1))\) for all \(\mathbf{n}^{\prime}\), and, thus, Equation (11) is non-negative. Therefore, \(\delta(t)\) monotonically decreases as \(t\) increases. If handed-over users are excluded from consideration for further handovers at subsequent iterations, the system load converges to a sub-optimal solution \(\rho^{\prime}\) such that \(\rho(0)\geq\rho^{\prime}\geq\rho^{*}\). Proof. : The power update procedure under a fixed serving cell assignment is known to converge. That is, even after user \(j\) is handed over, its transmit power at subsequent iterations satisfies \(p_{j}(t)\geq p_{j}(t+1)\), but \(p_{j}(t+1)\geq P_{j}^{(n^{\prime})}(\mathbf{p}(t))\) due to a lack of further handovers. The remaining part of the proof is the same as shown above and is hence omitted. Allowing handover of an unlimited number of users at each iteration may speed up convergence but trigger excessive handover overhead at once. To limit handover overhead per iteration, allowing only a configured number of users to be handed over at each iteration is considered. The modified algorithm selects a user to be handed over as  
\[j^{*}=\arg\max_{j\in\mathcal{U}^{\prime}}\left(\max_{n_{j}}\{\Delta_{m,DL}^{(n _{j})}(\mathbf{p})+\Delta_{m,UL}^{(n_{j})}(\mathbf{q})\}\right). \tag{12}\]  
where \(\mathcal{U}^{\prime}\) is the user set excluding previously-selected ones. User selection is repeated (a user producing the next largest gain is selected next) until the configured number of users to hand over per iteration is reached.