# A Fairness Index Based on Rate Variance for Downlink Non-Orthogonal Multiple Access System
###### Abstract  
Aiming at the resource allocation problem of a non-orthogonal multiple access (NOMA) system, a fairness index based on sample variance of users' transmission rates is proposed, which has a fixed range and high sensitivity. Based on the proposed fairness index, the fairness-constrained power allocation problem in NOMA system is studied; the problem is decoupled into the intra cluster power allocation problem and the inter cluster power allocation problem. The nonconvex optimization problem is solved by the continuous convex approximation (SCA) method, and an intra and inter cluster power iterative allocation algorithm with fairness constrained is proposed to maximize the total throughput. Simulation results show that the proposed algorithm can take into account intra cluster, inter cluster, and system fairness, and maximize the system throughput on the premise of fairness. non-orthogonal multiple access; fairness; throughput; power allocation; cluster +
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can be seen, the lower bound of Jain index is affected by the number of users in the system. Reference [28] proposed a fairness index (GUI index) based on the difference between user rate and fair rate. In the GUI index, the fair rate is related to the power allocation in the system, and the fair value can reach the maximum only when all users obtain the fair rate. Although the GUI fairness index can reflect the user power distribution well, its lower bound is affected by the number of users and user channel conditions in the system. Based on the Jain index, references [29; 30] proposed power allocation schemes with the weighted sum of throughput and fairness as the optimization objective and power allocation schemes based on the channel gain ratio of paired users, respectively. Based on the proportional fairness constraint, an iterative algorithm for joint subcarrier and user power allocation with the goal of maximizing throughput is proposed in [31]. A price-based power allocation algorithm using game theory is proposed in [32]. A power allocation scheme constrained by the minimum fairness of edge users is proposed in [33]. A power allocation scheme aiming at maximizing the minimum fairness to achieve a compromise between complexity and throughput is proposed in [34]. References [29; 30; 31; 32; 33; 34] focus on improving the overall fairness of the system but cannot reflect the fairness requirements of individual user cluster. Since the lower bound of the Jain fairness index is affected by the number of users in the system, and the GUI fairness index is affected by the power allocation and channel conditions, the Jain index and GUI index are not suitable for systems with changing number of users and channel conditions. In view of the above, this paper studies the fairness index of a NOMA system and proposes a new fairness index based on sample variance of users' transmission rates of a downlink NOMA system. The proposed index and Jain index are based on the rate, that is, the closer the user rate in the system, the greater the fairness, while the GUI index is related to the user power allocation in the system, that is, the greater the user rate with more power allocation, the greater the fairness. The other difference between the proposed index and the other two indexes is that, when the rate of one user in the system is equal to the transmission rate of the system and the rate of other users is 0, the proposed fairness index is 0, and the Jain fairness index is \(1/K\), while the GUI fairness index needs to meet the equal power allocation and the same channel conditions of all users before it can be taken as 0. In general, the fairness index based on rate variance can evaluate the fairness of systems with different user numbers and has better sensitivity.