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In a cognitive wireless network with spectrum underlay, secondary links can transmit at the same spectrum as primary links as long as the interference to the primary links is below a pre-negotiated interference threshold. This paper studies the effect of setting different interference thresholds on the transmission rate of the secondary links and how secondary transmissions affect the primary links. An optimization problem is formulated to find the maximum achievable rate for the secondary links given the interference threshold, and an analytical model is developed to find the interference level at the primary link receiver. Our results show some interesting relationship among the interference threshold, transmission rate of secondary links, and transmission power increase of the primary links in such networks.