Bandwidth Resource Competition in Cooperative Cognitive Wireless Communications

Abstract

In this paper, we analyze the behavior of Cooperative Cognitive Wireless nodes for the bandwidth resource competition using the ecological models of mutualism. We transform our cooperative cognitive wireless communication (CCWC) system as mutually interacting species. The two types of species are primary and secondary users. The Primary users have under-utilized licensed bandwidth to be shared with the secondary users. On the other hand, secondary users are willing to behave as possible relays for cognitive primary users, in exchange of shared licensed bandwidth. So, instead of pricing model, the proposed system is analyzed with the traditional barter system (exchange of commodities). We analyzed our proposed system based on three ecological models. Our first model is based on the coalitional form of Lotka–Volterra equations, where predator-prey role of mobile nodes are replaced with cooperation as in symbiosis. The second model deals with the population dynamism of strategically interacting partner for conflicting bandwidth resource. The third model adds the dynamism for the varying amount of bandwidth to be shared among the individual mobile users. For each type, a mathematical model is derived and then simulated for the equilibrium position analysis. The obtained results show that sharing of bandwidth resource is essential for the existence of CCWC systems.

Appendix Proof for the Convergence of Cooperative Cognitive Wireless Systems

Theorem 1

(Cooperative Cognitive System Convergence): The orbits of cooperative cognitive system go to infinity or converge to some rest point.

proof

According to the theory of dynamical systems, let us denote the quadrants of ℜ² with

$$ \begin{aligned} D_{1}&= \Re_{+}^{2}, \\ D_{2}&=\left \lbrace {\left ({p,s} \right ):p\,\leqslant\,0,s\,\geqslant\,0} \right \rbrace,\\ D_{3}&= -\Re_{+}^{2}, \end{aligned} $$

and,

$$ \begin{aligned} &D_{4}= {-D}_{2}.\\ &\hbox{If},\;p(t_{0}),\;s(t_{0}) \subset D_{1},\;\exists t_{0} \subset R\\ &\hbox{Then}, \;p(t) \subset D_{1} \forall \geqslant t_{0} \end{aligned} $$

If, for example,

$$ \begin{aligned} p(t) &= 0 \hbox{\;but\;} s(t) \geqslant 0, \,\hbox{then}\\ \ddot p &= \frac{\partial f_{p}}{\partial p}\dot{p}+\frac{\partial f_{p}}{\partial s}\dot{s} \end{aligned} $$

and hence

$$ (\dot{p} \geq 0) $$

With a similar type of reasoning, we can show that,

$$ \dot{p}(t_{0}),\;\dot{s}(t_{0}) \Rightarrow \dot{p}(t),\;\dot{s}(t) \subset D_{3}, \;\forall t \geq t_{0}. $$

If,

$$ p(t_{0}),\;s(t_{0})\subset D_{2}, $$

Then either, $\dot{p}(t),\;\dot{s}(t)$ enters D ₁ or D ₃ for further duration, or stays in D ₂, ∀ t ≥ t ₀; Same is the case for $\dot{p}(t),\;s(t)\subset D_{4}.$ So, in all the situations, the population variables p(t) and s(t) are always monotonic variables, with convergence point to some particular limit or toward infinity. Hence, the convergence of given cooperative cognitive communication system.