Interference Management and Power Allocation for Energy-Efficient Cognitive Femtocell Networks

Abstract

Energy efficiency of radio networks is a very important issue to meet the challenges raised by the high demands of traffic and energy consumption. Both cognitive radio and femtocell have the potential ability for high energy efficiency. However, most of previous works are focused on interference avoidance to guarantee the quality of service (QoS) by power control and spectrum sharing in heterogeneous cognitive radio networks with femtocells, and the energy efficiency aspect is largely ignored. In this paper, we study the energy efficiency aspect of power allocation and interference management in heterogeneous cognitive radio networks with femtocells. Particularly, due to sharing the same spectrum resource for cross-tier cognitive radio networks with femtocells, there is a cross-tier interference between fetmocells and macrocell. In this case, we introduce an interference price to measure the effect of interference. Then we formulate the problem of power allocation and interference management for energy-efficient transmissions in heterogeneous cognitive radio networks with femtocells as a Stackelberg game. And we use the backward induction method to solve the optimal power allocation and price determination. An iteration algorithm based on price updating is proposed to obtain the Stackelberg equilibrium solution. Simulation results are illustrated to demonstrate the Stackelberg equilibrium by the proposed iteration algorithm, and energy efficiency can be improved significantly in the proposed scheme.

Appendix

Proof of Theorem 1

First, $p_{n}\ge 0$ is nonempty and convex. Then we only need to prove the utility function $\pi _{n}$is a concave function. We can get the twice derivation of $\pi _{n}$ as

$$ \frac{\partial^{2} \pi_{n}}{\partial^{2} p_{n}}=-\frac{Wh_{n}^{4}}{\left(\sigma_{n}^{2}+I_{n}+p_{n}h_{n}^{2}\right)^{2}\ln2}<0. $$

(27)

Therefore, based on [34], we know that the utility function $\pi _{n}$ is a concave function. Hence, the non-cooperative power competition game $\mathcal {G}$is a concave game. According to [38, 39], a concave game has at least one Nash equilibrium.

Next, from Eq. (27), we know that $\frac{\partial {\pi_{n}}}{\partial {p_{n}}}$ is a strictly monotonic decreasing function about $p_{n}$. Obviously, we have $\mathop {\lim }\limits _{{p_{n}} \to \infty }{\frac {\partial \pi _{n}}{\partial p_{n}} }<0$, and

$$ \mathop {\lim }\limits_{{p_{n}} \to 0 }{\frac{\partial \pi_{n}}{\partial p_{n}} }=\mathop {\lim }\limits_{{p_{n}} \to 0 }{\frac{Wh_{n}^{2}}{(\sigma_{n}^{2}+I_{n})\ln2}-(\mu_{n}+\lambda g_{nm}^{2})}. $$

(28)

We can get two cases for Eq. (28) as follows: (1). $\mathop {\lim }\limits _{{p_{n}} \to 0 }\frac {\partial \pi _{n}}{\partial p_{n}}\le 0$; (2). $\mathop {\lim }\limits _{{p_{n}} \to 0 }\frac {\partial \pi _{n}}{\partial p_{n}}>0$.

For the first case, the utility function $\pi _{n}$ is a strictly monotonic decreasing about $p_{n}$. Therefore the optimal value are obtained at the left endpoint. In this case, the optimal power allocation is $p_{n}^*=0$. For the second case, we have $\mathop {\lim }\limits _{{p_{n}} \to 0 }\frac {\partial \pi _{n}}{\partial p_{n}}>0$. Therefore, the utility function $\pi _{n}$first increases with $p_{n}$, then at the certain point begins to decrease with $p_{n}$. In this case, the optimal power allocation is $p_{n}^*=\frac {W}{(\mu _{n}+\lambda g_{nm}^{2})\ln 2}-\frac {\sigma _{n}^{2}+I_{n}}{h_{n}^{2}}, n\in \{1,...,N\}$. Hence, Eq. (11) is one of the Nash equilibrium points. Thus, Theorem 1 is proved. □