Energy-efficient resource allocation for multi-RAT networks under time average QoS constraint

Abstract

In this paper, we propose an energy-efficient resource allocation algorithm for multiple radio access technologies (multi-RAT) networks, where the user equipments (UEs) transmit data over multiple radio interfaces for exploiting the complementary advantages of different RATs. In this scenario, the resource allocation is formulated as a stochastic energy efficiency (EE) maximization problem. More specifically, the time average quality of service constraint is considered to provide the flexible resource allocation over the time-varying fading channels. The virtual queue is introduced for each UE to deal with the time average transmission requirement. By adopting Lyapunov optimization technique and fractional programming theory, the non-concave EE maximization is converted into a mixed integer nonlinear optimization (MINO) problem. After that, the continuity relaxation and Lagrange dual methods are used to find the solution of the MINO problem. Then, we develop an EE-based dynamic joint subcarrier and power allocation algorithm, which does not require any prior knowledge of the channel state information. In addition, the performance bounds of the EE and virtual queue are provided. Our simulation results show that the performance of the proposed algorithm is better than other general algorithms.

Appendix A

1.1 Proof of Lemma 1

Proof

From Eq. (13), we obtain

$$\begin{aligned} L({\mathbf {Q}}(t+1))&-L({\mathbf {Q}}(t))\\&=\frac{1}{2}{\mathbf {Q}}(t+1)^{H}{\mathbf {Q}}(t+1) -\frac{1}{2}{\mathbf {Q}}(t)^{H}{\mathbf {Q}}(t)\\&=\frac{1}{2}(\max [{\mathbf {Q}}(t)-{\mathbf {R}}(t),0]+{\mathbf {R}}^{req})^{H}\\&\quad\times (\max [{\mathbf {Q}}(t)-{\mathbf {R}}(t),0]+{\mathbf {R}}^{req})\\&\quad-\frac{1}{2}{\mathbf {Q}}(t)^{H}{\mathbf {Q}}(t) \end{aligned}$$

where the dynamic queue state in Eq. (10) has been used.

If $Q\ge 0$, $f\ge 0$ and $r\ge 0$, we have

$$\begin{aligned} (\max \{Q-r,0\}+f)^{2}\le Q^{2}+r^{2}+f^{2}+2Q(f-r), \end{aligned}$$

then

$$\begin{aligned} L({\mathbf {Q}}&(t+1))-L({\mathbf {Q}}(t))\nonumber \\&\le \frac{1}{2}({\mathbf {R}}^{req})^{H}{\mathbf {R}}^{req} +\frac{1}{2} {\mathbf {R}}(t)^{H} {\mathbf {R}}(t)\nonumber \\&\quad+\,({\mathbf {R}}^{req}-{\mathbf {R}}(t))^{H}{\mathbf {Q}}(t) \nonumber \\&\le B+({\mathbf {R}}^{req}(t)-{\mathbf {R}}(t))^{H}{\mathbf {Q}}(t). \end{aligned}$$

(37)

Because the value of $\frac{1}{2}[({\mathbf {R}}^{req})^{H}{\mathbf {R}}^{req}+{\mathbf {R}}(t)^{H} {\mathbf {R}}(t) ]$ is finite, we take B as the upper bound of it.

Add $-V\mathbb {E}[R_{tot}(t)|\mathbf {Q}(t)]$ to both sides of Eq. (32) and get an expectation, we have

$$\begin{aligned} &{\mathbb {E}}[\Delta ({\mathbf {Q}}(t))-V R_{tot}(t)|{\mathbf {Q}}(t)]\nonumber \\&\quad\le B-V {\mathbb {E}}[R_{tot}(t)|{\mathbf {Q}}(t)]\nonumber \\&\qquad+{\mathbb {E}}[({\mathbf {R}}^{req}-{\mathbf {R}}(t))^{H}{\mathbf {Q}}(t)|{\mathbf {Q}}(t)]. \end{aligned}$$

(38)

Dividing both sides by ${\mathbb {E}}[P_{tot}(t)|{\mathbf {Q}}(t)]$, we obtain

$$\begin{aligned}&\frac{{\mathbb {E}}[\Delta ({\mathbf {Q}}(t))-V R_{tot}(t)|{\mathbf {Q}}(t)]}{{\mathbb {E}} [P_{tot}(t)|{\mathbf {Q}}(t)]}\le \frac{B}{{\mathbb {E}}[P_{tot}(t)|{\mathbf {Q}}(t)]}\\&\quad - \frac{V R_{tot}(t)}{{\mathbb {E}}[P_{tot}(t)|{\mathbf {Q}}(t)]}\\&\quad+ \frac{\sum _{m\in {\mathcal {M}}}{\mathbb {E}} [Q_{m}(t)|(R_{m}^{QoS} -R_{m}(t))]}{{\mathbb {E}}[P_{tot}(t)|{\mathbf {Q}}(t)]} \end{aligned}$$

The proof of Lemma 1 has been completed.