McPAO: A Distributed Multi-channel Power Allocation and Optimization Algorithm for Femtocells

Abstract

Efficient radio resource management is a key issue in a multi-channel femtocell system, where femtocell base stations are deployed randomly and will generate interference to each other. In this research, we formulate multi-channel power allocation as a convex optimization problem, in order to maximize the overall system throughput under complex transmit power constraint. We apply the Lagrangian duality techniques to make the problem decomposable and propose a distributed iterative subgradient algorithm, namely Multi-channel Power Allocation and Optimization (McPAO). Specifically, McPAO consists of two phases: (I) a gradient projection algorithm to solve the optimal power allocation for each channel under a fixed Lagrangian dual cost; and (II) a subgradient algorithm to update the Lagrangian dual cost by using the power allocation results from Phase I. This two-phase iteration process continues until the Lagrangian dual cost converges to the optimal value. Numerical results show that our McPAO algorithm can improve the overall system throughput by 18 %, comparing to with fixed power allocation schemes. In addition, we study the impact of errors in gradient direction estimation (Phase I), which are caused by limited or delayed information exchange among femtocells in realistic situations. These errors will be propagated into the subgradient algorithm (Phase II) and, subsequently, affect the overall performance of McPAO. A rigorous analytical approach is developed to prove that McPAO can always achieve a bounded overall throughput performance very close to the global optimum.

Appendix

Proof A

As it is proposed in Section 4 that for a given $\boldsymbol{\lambda}$,

$$\mathop {\lim {\rm{inf}}}\limits_{k \to \infty } {\rm{ E}}\left[ {F\left( {\textbf{y}_{m,n}^{\left( k \right)} } \right)} \right] \le F_n^{^\ast } + r_n , $$

(40)

where $F_n \left( {{ \textbf{x}}_n }|\boldsymbol{\lambda}\right) = \sum\limits_{m = 1}^M {f_{m,n} \left( {{ \textbf{x}}_n }|\boldsymbol{\lambda} \right)} = - \sum\limits_{m = 1}^M \left( R_{m,n} \left( {{\textbf{x}}_n } \right)\right. \left.- \lambda _m e^{x_{m,n} } \right)$.

From Eqs. 34 and 35, we have

$$\begin{array}{rll} t\left( \boldsymbol{\lambda} \right) &=& - d\left( \boldsymbol{\lambda} \right) \\ & =& \min \left\{ - \sum\limits_n^{} {(R_{m,n} \left( {\textbf{x}_n } \right) - \lambda _m e^{x_{m,n} } } + \lambda _m P_m {\rm{/}}N)\right\} \\ & =& \min\left\{ { - \sum\limits_n^{} {T_n } + \boldsymbol{\lambda} \cdot t\left( \boldsymbol{\lambda} \right)} \right\} \\ & =&\min\left\{ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over F} \left( {\bf{x}_n |\boldsymbol{\lambda} } \right) + \boldsymbol{\lambda} ^T G\left( {\textbf{x}_n } \right)} \right\} \\ \end{array} $$

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where $G(\textbf{x}_n ) = \nabla t\left( \boldsymbol{\lambda} \right)$. And then

$$\begin{array}{rll} t\left( \boldsymbol{\lambda} \right) &=& \mathop {\min }\limits_{\textbf{x}_n \in X} \left\{ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over F} \left( {\textbf{x}_n }|\boldsymbol{\lambda} \right) + \boldsymbol{\lambda}^T G(\textbf{x}_n )} \right\} \\ &\le& \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over F} \left( {\bar{\bf{x}}_n }|{\boldsymbol{\lambda}} \right) + \boldsymbol{\lambda}^T G(\bar{ \bf{x}}_n ) \\ &=&\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over F} \left( {\bar{ \textbf{x}}_n }|{\boldsymbol{\lambda}} \right) + \bar{\boldsymbol{\lambda} }^T G(\bar {\textbf{x}}_n ) + \left( {\boldsymbol{\lambda} - \bar{\boldsymbol{\lambda} } } \right)^T G(\bar {\textbf{x}}_n ){\rm{ }}. \end{array} $$

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$$\begin{array}{rll} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over F} \left( {\bar{ \textbf{x}}_n }|{\boldsymbol{\lambda}} \right) + \bar \lambda ^T G(\bar {\textbf{x}}_n ) &= &\sum\limits_n^{} {\left( {F_n \left( {\bar { \textbf{x}}_n }|{\boldsymbol{\lambda}} \right) + \sum\limits_{m = 1}^M {\bar \lambda _m { P_m /N}} } \right)} \\ & =& \sum\limits_n^{} {F_n \left( {\bar { \textbf{x}}_n |\bar{\boldsymbol{\lambda} }} \right)}, \end{array} $$

and

$$ F_n \left( {\bar{ \textbf{x}}_n|\bar{\boldsymbol{\lambda} } } \right) \le F_n^\ast \left( {{ \textbf{x}}_n^\ast |\bar{\boldsymbol{\lambda} } } \right) + r_n. $$

So

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over F} \left( {\bar { \textbf{x}}_n }|\bar{\boldsymbol{\lambda}} \right) + \bar{\boldsymbol{\lambda} } ^T G(\bar { \textbf{x}}_n ) \le \sum\limits_n^{} {\left( {F_n^\ast \left( {{ \textbf{x}}_n^\ast |\bar{\boldsymbol{\lambda} } } \right) + r_n } \right)}. $$

(43)

Then substitute Eq. 43 into Eqs. 42,

$$\begin{array}{rll} t\left( \boldsymbol{\lambda} \right)& \le &\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over F} \left( {\bar { \textbf{x}}_n }|{\boldsymbol{\lambda}} \right) + \bar{\boldsymbol{\lambda} } ^T G(\bar { \textbf{x}}_n ) + \left( {\boldsymbol{\lambda} - \bar{\boldsymbol{\lambda} } } \right)^T G(\bar { \textbf{x}}_n ) \\ &\le& \sum\limits_n^{} {\left( {F_n^\ast \left( {\textbf{x}_n^\ast|\bar{\boldsymbol{\lambda} }} \right) + r_n } \right)} + \left( {\boldsymbol{\lambda} - \bar{\boldsymbol{\lambda} } } \right)^T G(\bar { \textbf{x}}_n ) \\ &=&t\left( {\bar{\boldsymbol{\lambda} }} \right) + \left( {\boldsymbol{\lambda} - \bar{\boldsymbol{\lambda} }} \right)^T \nabla t\left( {\bar{\boldsymbol{\lambda} } } \right) + \sum\limits_n^{} {r_n } . \end{array} $$

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This equation yields to our conclusion in Eq. 36. □

Proof B

$\nabla t = - D$ is an ζ -subgradient at $\boldsymbol\lambda ^{\left( l \right)}$. A solution for Exercise 6.3.13 in [2] is presented in [3]. According to the solution,we have

$$ \sup\limits_{\boldsymbol{\lambda} \in \Re ^M } t\left( \boldsymbol{\lambda} \right) - \zeta \le \mathop {\lim \sup }\limits_{l \to \infty } t\left( {\boldsymbol{\lambda}^{\left( l \right)} } \right) \le \sup\limits_{\boldsymbol{\lambda} \in \Re ^M } t\left( \boldsymbol{\lambda} \right). $$

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Together with Eqs. 34 and 35, we have $d(\boldsymbol{\lambda} ) = - t(\boldsymbol{\lambda} )$. Then we have

$$ \inf\limits_{\boldsymbol{\lambda} \in \Re ^M } d\left( \boldsymbol{\lambda} \right) + \zeta \ge \mathop {\lim \inf }\limits_{l \to \infty } d\left( {\boldsymbol{\lambda} ^{\left( l \right)} } \right) \ge \inf\limits_{\boldsymbol{\lambda} \in \Re ^M } d\left(\boldsymbol{\lambda} \right). $$

Thus we accomplish the proof of Theorem 1. The optimal solution to $\max \left( {\sum\limits_n^{} {T_n } } \right)$ could be achieved using our scheme. □