Non-orthogonal multipe access based multiple-input multiple-output system (MIMO-NOMA) has been widely used in improving user's achievable rate of millimeter wave (mmWave) communication. To meet different requirements of each user in multi-user beams, this paper proposes a power allocation algorithm to satisfy the quality of service (QoS) of head user while maximizing the minimum rate of edge users from the perspective of max-min fairness. Suppose that the user who is closest to the base station (BS) is the head user and the other users are the edge users in each beam in this paper. Then, an optimization problem model of max-min fairness criterion is developed under the constraints of users' minimum rate requirements and the total transmitting power of the BS. The bisection method and Karush-Kuhn-Tucher (KKT) conditions are used to solve this complex non-convex problem, and simulation results show that both the minimum achievable rates of edge users and the average rate of all users are greatly improved significantly compared with the traditional MIMO-NOMA, which only consider max-min fairness of users.