Convergence of Price-Based Resource Allocation Algorithms in Multicellular Multicarrier Systems

Abstract

Inter-cell interference mitigation is a key challenge in the heterogeneous wireless networks which are expected to use an aggressive frequency reuse factor and a high-density access point deployment to improve coverage and spectral efficiency. In this paper, the problem of resources allocation in multicell Orthogonal Frequency Division Multiple Access wireless system is considered with universal frequency reuse and target of Weighted Sum-Rate Maximization. We address multi cell modified iterative water filling as an iterative power allocation algorithm. Also, a new extension of fixed point implementation of Successive Convex Approximation for Low complExity (SCALE) algorithm to multicellular system [referred to as Multi Cell Fixed point SCALE (MCF-SCALE)] is presented and it has been shown both of them resulted to the same convergence point. It is also demonstrated that using Lagrangian multiplier instead of noise variance in Standard Yates framework (as has been used in some previous papers) is not a suitable method for proving convergence and all the previous results based on this pattern need to be revised. Finally, a new framework is presented for proving the convergence of MCF-SCALE algorithm based on Jacobi iterative algorithm. Moreover, some previous convergence criteria are shown to be interpreted as a special case of this condition.

6 Appendix

Proof of Theorem 4

The row sum norm is used to calculate the norm of the iteration matrix C:

$$\begin{aligned} \left\Vert \mathbf{C}\right\Vert_\infty =\mathop {\max } \limits _{1\le i\le M} \sum \limits _{k=1}^M {\left| {c_{ik}}\right|} =\mathop {\max } \limits _{1\le i\le M} \sum \limits _{ \begin{array}{l} k=1 \\ k\ne i \\ \end{array}}^M {\frac{\left| {a_{ik}}\right|}{\left| {a_{ii}}\right|}} <1 \end{aligned}$$

(58)

Hence, we have convergence. $\square$

Proof of Proposition 4

For equation $\mathbf{D}^{n}\mathbf{q}^{n}=\mathbf{\lambda } \ln 2$, if the iterative Jacobi algorithm (51) is written, the following can be obtained

$$\begin{aligned} q_m^n =\frac{1}{D_{m,m}^n} \left({\lambda _m \ln 2-\sum \limits _{ \begin{array}{l} j=1 \\ j\ne m \\ \end{array}}^M {D_{m,j}^n q_j^n}}\right) \end{aligned}$$

(59)

However, since $q_m^n \,{\mathop {=}\limits ^{\Delta }}\, \left({p_m^n}\right)^{-1}$, this relation has the same form as MCF-SCALE algorithm, as obtained in (29). $\square$