Energy Efficiency in QoS Constrained 60 GHz Millimeter-Wave Ultra-Dense Networks

Abstract

Millimeter-Wave (mmWave) communication in ultra-dense networks (UDNs) has been considered as a promising technology for future wireless communication systems. Exploiting the benefits of mmWave and UDNs, we introduce a new approach for jointly optimizing small-cell base station (SBS) - user (UE) association and power allocation to maximize the system energy efficiency (EE) while guaranteeing the quality of service (QoS) constraints for each UE. The SBS-UE association problem poses a new challenge since it reflects as a complex mixed-integer non-convex problem. On the other hand, the power allocation problem is in non-convexity structure, which is impossible to handle with the association problem concurrently. An alternating descent method is thus introduced to divide the primal optimization problem into two subproblems and handle one-by-one at each iteration, where the SBS-UE association problem is reformulated using the penalty approach. Then, path-following algorithms are developed to convert non-convex problem into the simple convex quadratic functions at each iteration. Numerical results are provided to demonstrate the convergence and low-complexity of our proposed schemes.

This research was supported by the MSIT (Ministry of Science and ICT), Korea, under the Grand Information Technology Research Center support program (IITP-2018-2016-0-00318) supervised by the IITP (Institute for Information & communications Technology Promotion).

Appendix: Fundamental Inequalities

As the function $f(x,y)\triangleq \ln (1+1/xy)$ is convex in the domain $\{x>0, y>0\}$ [12], it follows that [16] for every $x>0$, $y>0$, $\bar{x}>0$ and $\bar{y}>0$,

$$\begin{aligned} \displaystyle \ln (1+1/xy)&=f(x,y)\nonumber \\&\ge f(\bar{x},\bar{y})+\langle \nabla f(\bar{x},\bar{y}), (x,y)-(\bar{x},\bar{y})\rangle \nonumber \\&= \displaystyle \ln (1+1/\bar{x}\bar{y})+\displaystyle \frac{1/\bar{x}\bar{y}}{1+1/\bar{x}\bar{y}}(2-x/\bar{x}- y/\bar{y}). \end{aligned}$$

(21)

Reutilizing inequalities in [15], we observe that function $x^2/t$ is always convex under condition of $x > 0$ and $t > 0$, which yields inequality

$$\begin{aligned} \frac{x^2}{t} \ge \frac{2\bar{x}}{\bar{t}}x - \frac{\bar{x}^2}{\bar{t}^2} t. \end{aligned}$$

(22)

Then substituting $x \rightarrow \sqrt{x}$ and $\bar{x} \rightarrow \sqrt{\bar{x}}$, we obtain

$$\begin{aligned} \frac{x}{t} \ge \frac{2\sqrt{\bar{x}} }{\bar{t}} \sqrt{x} - \frac{\bar{x}}{\bar{t}^2} t. \end{aligned}$$

(23)

Lastly, the inequality

$$\begin{aligned} x^2 - x \ge \bar{x}^2 - \bar{x} + (2 \bar{x} - 1)(x - \bar{x}) \end{aligned}$$

(24)

always hold true since $x^2 - x$ is in convex quadratic form [15].