A polynomial rooting approach for analysis of competition among secondary users in cognitive radio networks

The competitive coexistence of a large number of secondary users in heterogeneous cognitive radio networks (CRN) is a challenging issue. In this paper, we address this issue by developing an efficient approach to analyze the competition among the secondary users. Based on the Markov model bank that we developed recently, we can change the evaluation of the secondary user throughput into studying a special root of a polynomial that is much more simplified. Under this framework, we study the competition among the secondary users in three special cases: homogeneous CRN with identical secondary users, heterogeneous CRN with a group of identical secondary users plus an outlier, and heterogeneous CRN with two different groups of secondary users. In each case, we derive the polynomial, prove the uniqueness of the root, and derive the throughput of each secondary user based on the root. Furthermore, we derive closed-form approximate solutions to the root, based on which constrained optimization can be formulated to study the optimal competition strategies for the secondary users. The optimization results are in water-filling principles and can be derived in closed-form expressions. Simulations are conducted to verify the analysis results.