Analysis of Spectrum Access Strategy with Multiple Cross-Layer Considerations in Cognitive Radio Networks

Abstract

In the multi-channel cognitive radio network, we build an M/M/1/K packet delay queueing model integrating multiple cross-layer considerations for secondary users. To minimize the total average packet delay, the optimal probability vector for spectrum access is determined by genetic algorithm. Furthermore, the total average packet loss rate is derived with another consideration that the limited tolerable retransmission attempt for ARQ. Numerical result proves that our proposed spectrum access strategy outperforms the strategies of random, equal probability and inverse proportion. Finally, under the proposed spectrum access strategy, the influences of related cross-layer parameters on average packet delay and packet loss rate are illustrated.

Appendix: Proof of Non-convexity of \(s({\mathbf{Q}})\)

Since $s({\mathbf{Q}})$ is twice differentiable and the definition domain of $s({\mathbf{Q}})$ (dom s) is convex, $s({\mathbf{Q}})$ is convex if and only if its Hessian $\nabla ^2s({\mathbf{Q}})$ is positive semidefinite: for all $q\in$ dom s. Here we will give the opposite side, that is, its Hessian is indefinite.

$$\begin{aligned}{}[\nabla ^2 s({\mathbf{Q}})]_{ij} = \left\{ {\begin{array}{ll} {\frac{{\partial ^2 s({\mathbf{Q}})}}{{(\partial q_i )^2 }}} &{} \quad i = j \\ 0 &{} \quad i \ne j \\ \end{array} } \right. , \end{aligned}$$

(19)

where $\frac{{\partial ^2 s({\mathbf{Q}})}}{{(\partial q_i )^2 }} = g''(q_i )q_i + 2g'(q_i )$ is the eigenvalue of the Hessian matrix. If there are both positive and negative eigenvalues, the Hessian is indefinite. We can obtain that the first and second-order derivates of g(q) are divisions of two polynomials in q with positive and negative coefficients, respectively. And in form, the denominator has a higher power in q than numerator. Here we don’t display the derivates results, since its verbosity. We just take for example when $K_s=1$ to prove that not for all $q\in$ dom s, the eigenvalues are nonnegative.

For $K_s=1$, it is derived that

$$\begin{aligned} \frac{{\partial ^2 s({\mathbf{Q}})}}{{(\partial q_i )^2 }} = \frac{{4a^6 q_i^5 - 8a^4 q_i^3 }}{{(a^2 q_i^2 - 2aq_i {\text { + }}1)^4 \mu _s }}. \end{aligned}$$

(20)

Then $\frac{{\partial ^2 s({\mathbf{Q}})}}{{(\partial q_i )^2 }} \ge 0$ if and only if $q_i^2\ge \frac{2}{a^2}$, that is, $q_i\ge \frac{\sqrt{2}\mu _s^2}{{\mathbf{r}}^T {\mathbf{1}}[1 - (1 - \mu _s )^{D + 1} ]}$. Therefore, non-convexity of $s({\mathbf{Q}})$ holds.