Proportional-fair energy-efficient radio resource allocation for OFDMA smallcell networks

Abstract

This paper investigates the energy-efficient radio resource allocation problem of the uplink smallcell networks. Different from the existing literatures which focus on improving the energy efficiency (EE) or providing fairness measured by data rates, this paper aims to provide fairness guarantee in terms of EE and achieve EE-based proportional fairness among all users in smallcell networks. Specifically, EE-based global proportional fairness utility optimization problem is formulated, taking into account each user’s quality of service, and the cross-tier interference limitation to ensure the macrocell transmission. Instead of dealing with the problem in forms of sum of logarithms directly, the problem is transformed into a form of sum of ratios firstly. Then, a two-step scheme which solves the subchannel and power allocation separately is adopted, and the corresponding subchannel allocation algorithm and power allocation algorithm are devised, respectively. The subchannel allocation algorithm is heuristic, but can achieve close-to-optimal performance with much lower complexity. The power allocation scheme is optimal, and is derived based on a novel method which can solve the sum of ratios problems efficiently. Numerical results verify the effectiveness of the proposed algorithms, especially the capability of EE fairness provisioning. Specifically, it is suggested that the proposed algorithms can improve the fairness level among smallcell users by 150–400 % compared to the existing algorithms.

Appendix: Proof of Theorem 2

Note that the Lagrangian dual function (17) can be decomposed into a $M \times K\times N$ subfunctions, i.e.,

$$\begin{aligned} L(p_{m,k}^n,{\varvec{\lambda ,\gamma ,\zeta }})= & {} \sum \limits _{m \in \mathcal{M}} \sum \limits _{k \in {\mathcal{K}_m}} \sum \limits _{n \in \mathcal{N}} {L_{m,k}^n\left( {p_{m,k}^n,{\varvec{\lambda ,\gamma ,\zeta }}} \right) }\\&\quad +\, \sum \limits _{m \in \mathcal{M}} {\sum \limits _{k \in {\mathcal{K}_m}} {{\mu _{m,k}}} {\left( { - {\mu _{m,k}}{\beta _{m,k}}p_{m,k}^C + {\gamma _{m,k}}P_{m,k}^{\max } - {\lambda _{m,k}}R_{m,k}^{\min }} \right) }} \\&\quad +\, \sum \limits _{n \in \mathcal{N}} {{\zeta _n}I_{Th}^n}, \end{aligned}$$

(30)

where $L_{m,k}^n\left( {p_{m,k}^n,{\varvec{\lambda ,\gamma ,\zeta }}} \right) = \left( {{\mu _{m,k}}{w_{m,k}} + {\lambda _{m,k}}} \right) x_{m,k}^nr_{m,k}^n - \left( {{\mu _{m,k}}{\beta _{m,k}}\xi + {\gamma _{m,k}} + {\zeta _n}g_{M,m,k}^n} \right) x_{m,k}^np_{m,k}^n.$

Due to that problem (13) is a convex optimization problem, the Karush-Kuhn-Tucker (KKT) conditions, i.e.,

$$\begin{aligned} \nabla L_{m,k}^n\left( {p_{m,k}^n} \right)= & \, 0,\quad {\forall n \in \mathcal {N}},\quad {\forall k \in {\mathcal{K}_m}},\quad {\forall m \in \mathcal {M}}, \end{aligned}$$

(31)

$$\begin{aligned} {\lambda _{m,k}}\left( {R_{m,k}^{} - R_{m,k}^{\min }} \right)= & \, 0,\quad {\forall k \in \mathcal{K}_m},\quad \forall m \in \mathcal {M}, \end{aligned}$$

(32)

$$\begin{aligned} {\gamma _{m,k}}\left( {P_{m,k}^{\max } - P_{m,k}^{}} \right)= & \, 0,\quad {\forall k \in \mathcal{K}_m},\quad \forall m \in \mathcal {M}, \end{aligned}$$

(33)

$$\begin{aligned} {\zeta _n}\left( {I_{Th}^n - \sum \limits _{m \in \mathcal{M}} {\sum \limits _{k \in \mathcal{K}_m} {x_{m,k}^np_{m,k}^ng_{M,m,k}^n} } } \right)= & \, 0,\quad \forall n \in \mathcal {N}, \end{aligned}$$

(34)

are both the sufficient and necessary conditions which ensure certain power allocation solution $p_{m,k}^n$ to be optimal [1]. Based on Eq. (33) of KKT conditions, i.e.,

$$\begin{aligned} \frac{{\partial L_{m,k}^n\left( {p_{m,k}^n,\mathbf{{\lambda ,\gamma ,\zeta }}} \right) }}{{\partial p_{m,k}^n}} = 0, \end{aligned}$$

(35)

we can derive the optimal power allocation as

$$\begin{aligned} {p_{m,k}^n}= {\left\{ \begin{array}{ll}\begin{array}{ll} {\left( {WF_{m,k}^n - \frac{{{\sigma ^2} + I_m^n}}{{g_{m,k}^n}}} \right) ^ + }, &{}\quad if\; x_{m,k}^n = 1;\\ 0,&{}\quad if\; x_{m,k}^n = 0; \end{array} \end{array}\right. } \end{aligned}$$

(36)

where ${\left( y \right) ^ + } = \max \left( {0,y} \right)$ and $WF_{m,k}^n = \frac{{\left( {{\mu _{m,k}}{w_{m,k}} + {\lambda _{m,k}}} \right) B}}{{\ln 2({\mu _{m,k}}{\beta _{m,k}}\xi + {\gamma _{m,k}} + {\zeta _n}g_{M,m,k}^n)}}.$