Energy-efficiency maximization in D2D-enabled vehicular communications with consideration of dynamic channel information and fairness

Abstract

Device-to-device (D2D) communications in a cellular network can be implemented on vehicular communications. However, high mobilities of vehicles generate fast channel variations in the communications. Channel description has to be specified in the D2D-enable vehicular communications network. In this paper, the vehicle speed and the sampling time are involved to the channel description such that the dynamic information of vehicle is integrated with the channel gain. By doing so, the channel description is more practical. The power control and spectrum sharing problem is formulated in order to maximize the energy-efficiency (EE) of total vehicle-to-infrastructure (V2I) links. It also attempts to ensure reliable communications in a multi-cellular users multi-D2D users frequency division multiple access (FDMA) cellular environment. The reformulated objective function in the fractional form is proved to be quasi-concave, and it can be solved by the Dinkelbach method which requires low computational complexity. In addition, the Hungarian algorithm is used to determine the optimal match of cellular user equipments (CUEs) and V2V pairs. User fairness is further considered under the goal of maximizing overall energy efficiency. Then, the approach of minimum EE maximization is implemented on all V2I links in order to improve system fairness. Finally, the system performance of the proposed scheme is validated by numerical simulations. The results show that the algorithm is effective to improve the energy efficiency and robustness in the dynamic communications environment.

Appendices

Appendix A

In this section, we prove that the fraction objective function is quasi-concave. When the nature of the quasi-concave problems considered, the numerator and the denominator of objective function should be concave. Obviously, the denominator is concave, since it is a linear combination of the positive power variable and the positive power of circuit. Therefore, we focus on proving the numerator is concave. The proof is given as follows:

$$ \frac{\mathrm{d}\left[\log_{2}\left( 1+\frac{{p_{m}^{c}}g_{m,0}^{c2}}{\sigma^{2}+{p_{k}^{v}}g_{k,0}^{v2}}\right)\right]}{\mathrm{d}{p_{m}^{c}}}=\frac{g_{m,0}^{c2}}{\ln2(\sigma^{2}+{p_{m}^{c}}g_{m,0}+{p_{k}^{v}}g_{k,0}^{v2})}, $$

$$ \frac{\mathrm{d^{2}}\left[\log_{2}\left( 1+\frac{{p_{m}^{c}}g_{m,0}^{c2}}{\sigma^{2}+{p_{k}^{v}}g_{k,0}^{v2}}\right)\right]}{\mathrm{d}{{p_{m}^{c}}}^{2}}=-\frac{{g_{m,0}^{c1}}^{2}}{\ln2(\sigma^{2}+{p_{m}^{c}}g_{m,0}^{c2}+{p_{k}^{v}}g_{k,0}^{v2})^{2}}\leq0, $$

The second derivative of the molecular part of the objective function is less than or equals to zero. This issue is concave. An iterative algorithm, Dinkelbach’s method, can be used to solve a qusi-concave problem.

Appendix B

Assuming that h_k,k is an independent random variable which is in the complex Gauss distribution $\mathcal {C}\mathcal {N}{(0,1)}$, and h_k,k can be expressed as a complex variable with a real part and a imaginary part as,

$$ h_{k,k}=x+y\text{i}, $$

where x and y are identically distributed in $\mathcal {C}\mathcal {N}{(0,\frac {1}{2})}$; x is independent to y. To be specified, the probability density functions of x and y are given as

$$ \left\{ \begin{array}{lll} f_{X}(x)=\sqrt{\pi}e^{-2x} \quad {-\infty}<x<{+\infty},\\ f_{Y}(y)=\sqrt{\pi}e^{-2y} \quad {-\infty}<y<{+\infty}. \end{array} \right. $$

Hence, the combined probability density distribution function of x and y can be formulated as

$$ f(x,y)=f_{X}(x)f_{Y}(y)=\frac{1}{\pi}e^{(x^{2}+y^{2})}. $$

The modulus length of h_k,k is given as

$$ |h_{k,k}|=\sqrt{x^{2}+y^{2}}. $$

If |h_k,k| = 0, f(|h_k,k|) = 0. If |h_k,k| > 0, the probability distribution function $F_{H_{k,k}}(|h_{k,k}|)$ can be formulated as

$$ F_{H_{k,k}}(|h_{k,k}|)=\iint\limits_{\sqrt{x^{2}+y^{2}}\leq |h_{k,k}|}f(x,y) dx dy. $$

Supposing that $x = r{\cos \limits } t$, $y = r{\sin \limits } t$ (0 ≤ r ≤|h_k,k|), the probability distribution function F(|h_k,k|) can be rewritten as

$$ \begin{array}{@{}rcl@{}} F(|h_{k,k}|)&=&{\int}_{0}^{2\pi}{\int}_{0}^{|h_{k,k}|}\frac{1}{\pi}e^{-r^{2}} d\theta dr\\ f(|h_{k,k}|)&=&F^{\prime}(|h_{k,k}|)=2|h_{k,k}|e^{-|h_{k,k}|^{2}}, \end{array} $$

$$ \begin{array}{@{}rcl@{}} F_{H^{2}}(|h_{k,k}|^{2})&=&F(H^{2}\leq|h_{k,k}|^{2})\\ &=&F(0\leq H\leq|h_{k,k}|)\\ &=&2{\int}_{0}^{|h_{k,k}|}|h_{k,k}|e^{-|h_{k,k}|^{2}} d|h_{k,k}|. \end{array} $$

Therefore, the density function of |h_k,k|² is

$$ \begin{array}{@{}rcl@{}} f(|h_{k,k}|^{2})&&=F^{\prime}(H^{2}\leq |h_{k,k}|^{2})=2[|h_{k,k}|e^{-|h_{k,k}|^{2}}]'\\&&=e^{-|h_{k,k}|^{2}}. \end{array} $$