This paper addresses the beam allocation problem in a switched-beam based massive multiple-input-multiple-output (MIMO) system working at the millimeter wave (mmWave) frequency band, with the target of maximizing the sum data rate. This beam allocation problem can be formulated as a combinatorial optimization problem under two constraints that each user uses at most one beam for its data transmission and each beam serves at most one user. The brute-force search is a straightforward method to solve this optimization problem. However, for a massive MIMO system with a large number of beams N, the brute-force search results in intractable complexity O(NK), where K is the number of users. In this paper, in order to solve the beam allocation problem with affordable complexity, a suboptimal low-complexity beam allocation (LBA) algorithm is developed based on submodular optimization theory, which has been shown to be a powerful tool for solving combinatorial optimization problems. Simulation results show that our proposed LBA algorithm achieves nearly optimal sum data rate with complexity O(K logN). Furthermore, the average service ratio, i.e., the ratio of the number of users being served to the total number of users, is theoretically analyzed and derived as an explicit function of the ratio N=K.
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