15.7: Summary and Conclusions

15.7 Summary and Conclusions

This chapter has presented several modeling approaches for describing the interactive decision processes that occur in a network of cognitive radios, specifically covering dynamical systems, contraction mappings, standard interference functions, Markov models, games, potential games, and supermodular games. These models and the techniques for establishing whether a cognitive radio network satisfies the conditions of the model are summarized in Table 15.4.

Table 15.4: Presented models
Model (section number)	Basic model	Identification
Dynamical systems (15.3.1)	, evolution equation a(t ^k+1) = f ^t(a(t ^k))	always exists. Solve g for f t. f ^t exists if g satisfies Picard Lindel f theorem
Contraction mappings (in 15.3.2)		Blackwell's conditions
Standard interference function power control (in 15.3.2)		I(p) satisfies positivity, montonicity, and scalability
Finite ergodic Markov chain (in 15.3.3)		? k such that P ^k has all positive entries
Absorbing Markov chain (in 15.3.3)		Apply model definition
Game (15.4.3)		Map from cognition cycle
Potential game (15.5.1)	? u _i( a, b _i) everywhere related to ? V( a, b _i)	(others in Section 15.5.1)
Supermodular game (in 15.5.2)	(1) A _i is a complete lattice (2) u _i is supermodular in ai (3) u _i has increasing differences in ( ai, a _-i)	A _i is a closed interval in R ^ki u _i is twice continuously differentiable on A _i