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In machine-type communications (MTC), contention-based random access is employed to support a number of MTC devices with a limited number of resource blocks (RBs). Since multiple active devices may transmit signals in the same RB or channel, the collision caused by the presence of multiple signals is inevitable and the detection of collision becomes important. Furthermore, if the arrival time and the number of multiple signals can be estimated, successive interference cancellation (SIC) can be employed in time domain to improve the throughput. In this paper, we focus on the estimation of round-trip delays (RTD) of multiple signals in non-orthogonal random access (NORA) based on the maximum likelihood (ML) criterion. Since the computational complexity of the ML approach is high, we propose a low-complexity approach based on variational inference, which is widely used in machine learning. We also …