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Non-orthogonal multiple access (NOMA) is considered as one of the predominant multiple access technique for the next-generation cellular networks. We consider a 2-user pair downlink NOMA system with imperfect successive interference cancellation (SIC). We consider bounds on the power allocation factors and then formulate the power allocation as an optimization problem to achieve {$α$-Fairness} among the paired users. We show that {$α$-Fairness} based power allocation factor coincides with lower bound on power allocation factor in case of perfect SIC and $α> 2$. Further, as long as the proposed criterion is satisfied, it converges to the upper bound with increasing imperfection in SIC. Similarly, we show that, for $0<α<1$, the optimal power allocation factor coincides with the derived lower bound on power allocation. Based on these observations, we then propose a low complexity sub-optimal algorithm. Through extensive simulations, we analyse the performance of the proposed algorithm and compare the performance against the state-of-the-art algorithms. We show that even though Near-Far based pairing achieves better fairness than the proposed algorithms, it fails to achieve rates equivalent to its orthogonal multiple access counterparts with increasing imperfections in SIC. Further, we show that the proposed optimal and sub-optimal algorithms achieve significant improvements in terms of fairness as compared to the state-of-the-art algorithms.