Radio Resource Allocation Fairness in Cooperative Cognitive Radio Relay Networks

Abstract

Imbalance and unfairness in resource allocation and utilization can lead to a degraded network environment where wireless network users experience poor quality of service (QoS) within the wireless networks. In multi-user wireless radio networks with constrained wireless network resources, such as power and bandwidth resources, uniform allocation of power or bandwidth may lead to unfairness in network resource management. The cooperative cognitive radio relay networks’ (CCRRN) radio resource allocation problem among multi-users is investigated in this paper through combined resource optimization. To facilitate fairness in the multi-user networks, the maxi–min criterion is employed. The max–min fairness ensures that when the wireless network attains max–min utility fairness, no user’s utility can be improved without degrading the utility performance of any other user. To maximize the worst-case user (max–min) capacity in the CCRRN using the power and bandwidth allocation strategy, a combined optimal resource allocation (COPBA) scheme is proposed. The max–min resource allocation optimization problem developed is observed to be convex. Efficient computer-based numerical optimization solver packages, such as SNOPT, exist that can solve convex optimization problems efficiently. The MATLAB software provides a robust integrated environment where the implementation of the SNOPT solver can be executed. Using the uniform bandwidth optimal power allocation (UBOPA) scheme as a benchmark, the fairness metric of the COPBA is evaluated and compared. Furthermore, the results showed the COPBA outperforming the UBOPA scheme.

Appendix

For notation simplicity, we rewrite the lower bound constraints C1 and C2 of $\textbf{P2}$ as;

$$\begin{aligned} \hspace{-80pt} Q_1(b_1,p_1) = R^i_{SR_k} -z \ge 0, \end{aligned}$$

(30)

and

$$\begin{aligned} \hspace{-80pt} Q_2(b_2,p_2) = R^i_{R_kD_i} -z \ge 0, \end{aligned}$$

(31)

respectively. The concave functions are negative semi-definite functions. In the optimization problem P2, the $Q_1(b_1,p_1)$ and $Q_2(b_2,p_2)$ bound the epigraph function of the optimization objective function. Our goal is to prove that they are concave functions with the facts that the functions are negative semi-definite. The function Q(x, y) is said to be negative semi-definite, if and only if the Hessian of Q(x, y) has eigenvalues ( $\psi _l$) that are non-positive. That is $\psi _l \le 0, l=1,2,...L.$, where the Hessian matrix F(x, y) is of dimension L. Also, $R^i_{SR_k} = b_1 \log \left( 1+ \frac{p_1 g_1}{N_0b_1}\right)$ and $R^i_{R_kD_i} = b_2 \log \left( 1+ \frac{p_2 g_2}{N_0b_2}\right)$.

Firstly, for the negative semi-definite proof of $Q_1(b_1, p_1)$, we define the Hessian matrix $H_{Q_1(b_1,p_1)}$ as;

$$\begin{aligned}{} & {} H_{Q_1(b_1,p_1)} \triangleq \left[ \begin{array}{cc} \frac{ \partial ^2 Q_1(b_1,p_1)}{\partial b^2_1} &{} \frac{ \partial ^2 Q_1(b_1,p_1)}{\partial b_1 p_1} \\ \frac{ \partial ^2 Q_1(b_1,p_1)}{\partial p_1 b_1} &{} \frac{ \partial ^2 Q_1(b_1,p_1)}{\partial p^2_1} \end{array} \right] , \end{aligned}$$

(32)

$$\begin{aligned}{} & {} \quad \frac{ \partial ^2 Q_1(b_1,p_1)}{\partial b^2_1} = - \frac{g^2_1p^2_1}{(b_1(b_1 N_0 +p_1 g_1)^2)}, \end{aligned}$$

(33)

$$\begin{aligned}{} & {} \quad \frac{ \partial ^2 Q_1(b_1,p_1)}{\partial p^2_1} = - \frac{b_1 g^2_1}{(b_1 N_0 +p_1 g_1)^2}, \end{aligned}$$

(34)

$$\begin{aligned}{} & {} \quad \frac{ \partial ^2 Q_1(b_1,p_1)}{\partial b_1 \partial p_1} = \frac{ g^2_1 p_1}{(b_1 N_0 +p_1 g_1)^2}, \end{aligned}$$

(35)

$$\begin{aligned}{} & {} \quad \frac{ \partial ^2 Q_1(b_1,p_1)}{ \partial p_1 \partial b_1} = \frac{ g^2_1 p_1}{(b_1 N_0 +p_1 g_1)^2}, \end{aligned}$$

(36)

Therefore, we have the Hessian matrix $H_{Q_1(b_1,p_1)}$ as;

$$\begin{aligned} H_{Q_1(b_1,p_1)} =\left[ \begin{array}{cc} - \frac{g^2_1p^2_1}{(b_1(b_1 N_0 +p_1 g_1)^2)} &{} \frac{ g^2_1 p_1}{(b_1 N_0 +p_1 g_1)^2} \\ \frac{ g^2_1 p_1}{(b_1 N_0 +p_1 g_1)^2} &{} - \frac{b_1 g^2_1}{(b_1 N_0 +p_1 g_1)^2} \end{array} \right] . \end{aligned}$$

(37)

The eigenvalues $\psi _1$ and $\psi _2$ of $H_{Q_1(b_1,p_1)}$ are the first principal minors of $H_{Q_1(b_1,p_1)}$. Hence,

$$\begin{aligned}{} & {} \psi _1 = - \frac{g^2_1p^2_1}{(b_1(b_1 N_0 +p_1 g_1)^2)} <0, \end{aligned}$$

(38)

$$\begin{aligned}{} & {} \quad \psi _2 = - \frac{b_1 g^2_1}{(b_1 N_0 +p_1 g_1)^2}<0. \end{aligned}$$

(39)

The Hessian matrix determinant is the last eigenvalue $\psi _3$.

$$\begin{aligned} \psi _3 = \begin{array}{|cc|} - \frac{g^2_1p^2_1}{(b_1(b_1 N_0 +p_1 g_1)^2)} &{} \frac{ g^2_1 p_1}{(b_1 N_0 +p_1 g_1)^2} \\ \frac{ g^2_1 p_1}{(b_1 N_0 +p_1 g_1)^2} &{} - \frac{b_1 g^2_1}{(b_1 N_0 +p_1 g_1)^2} \end{array} = 0. \end{aligned}$$

(40)

Therefore, with $\psi _1 <0$, $\psi _2 <0$ and $\psi _3 = 0$. Consequently, $H_{Q_1(b_1,p_1)}$ is negative semi-definite. Thus in $B^i_S$ and $P^i_S$, $R^i_{SR_k}$ is a concave function.

Secondly, for the negative semi-definite proof of $Q_2 (b_2, p_2)$, we the Hessian matrix $H_{Q_2(b_2,p_2)}$ as;

$$\begin{aligned}{} & {} H_{Q_2(b_2,p_2)} =\left[ \begin{array}{cc} \frac{ \partial ^2 Q_2(b_2,p_2)}{\partial b^2_2} &{} \frac{ \partial ^2 Q_2(b_2,p_2)}{\partial b_2 p_2} \\ \frac{ \partial ^2 Q_2(b_2,p_2)}{\partial p_2 b_2} &{} \frac{ \partial ^2 Q_2(b_2,p_2)}{\partial p^2_2} \end{array} \right] , \end{aligned}$$

(41)

$$\begin{aligned}{} & {} \quad \frac{ \partial ^2 Q_2(b_2,p_2)}{\partial b^2_2} = - \frac{g^2_2p^2_2}{(b_2(b_2 N_0 +p_2 g_2)^2)}, \end{aligned}$$

(42)

$$\begin{aligned}{} & {} \quad \frac{ \partial ^2 Q_2(b_2,p_2)}{\partial p^2_2} = - \frac{b_2 g^2_2}{(b_2 N_0 +p_2 g_2)^2}, \end{aligned}$$

(43)

$$\begin{aligned}{} & {} \quad \frac{ \partial ^2 Q_2(b_2,p_2)}{\partial b_2 \partial p_2} = \frac{ g^2_2 p_2}{(b_2 N_0 +p_2 g_2)^2}, \end{aligned}$$

(44)

$$\begin{aligned}{} & {} \quad \frac{ \partial ^2 Q_2(b_2,p_2)}{ \partial p_2 \partial b_2} = \frac{ g^2_2 p_2}{(b_2 N_0 +p_2 g_2)^2}, \end{aligned}$$

(45)

Thus, the Hessian matrix $H_{Q_2(b_2,p_2)}$ is;

$$\begin{aligned} H_{Q_2(b_2,p_2)} =\left[ \begin{array}{cc} - \frac{g^2_2p^2_2}{(b_2(b_2 N_0 +p_2 g_2)^2)} &{} \frac{ g^2_2 p_2}{(b_2 N_0 +p_2 g_2)^2} \\ \frac{ g^2_2 p_2}{(b_2 N_0 +p_2 g_2)^2} &{} - \frac{b_2 g^2_2}{(b_2 N_0 +p_2 g_2)^2} \end{array} \right] . \end{aligned}$$

(46)

The corresponding eigenvalues $\psi _1$ and $\psi _2$ of $H_{Q_2(b_2,p_2)}$ are the first principal minors of $H_{Q_2(b_2,p_2)}$. Hence,

$$\begin{aligned}{} & {} \psi _1 = - \frac{g^2_2p^2_2}{(b_2(b_2 N_0 +p_2 g_2)^2)} <0, \end{aligned}$$

(47)

$$\begin{aligned}{} & {} \quad \psi _2 = - \frac{b_2 g^2_2}{(b_2 N_0 +p_2 g_2)^2}<0, \end{aligned}$$

(48)

while the Hessian matrix determinant is the eigenvalue $\psi _3$.

$$\begin{aligned} \psi _3 =\begin{array}{|cc|} - \frac{g^2_2p^2_2}{(b_2(b_2 N_0 +p_2 g_2)^2)} &{} \frac{ g^2_2 p_2}{(b_2 N_0 +p_2 g_2)^2} \\ \frac{ g^2_2 p_2}{(b_2 N_0 +p_2 g_2)^2} &{} - \frac{b_2 g^2_2}{(b_2 N_0 +p_2 g_2)^2} \end{array} =0. \end{aligned}$$

(49)

Therefore, with $\psi _1 <0$, $\psi _2 <0$ and $\psi _3 = 0$, thus $Q_2(b_2,p_2)$ is negative semi-definite. Since both $R^i_{S R_k}$ and $R^i_{R_k D_i}$ are concave functions, thus the objective function of optimization problem $\textbf{ P2}$ is a concave function.