By exploiting the superiority of non-orthogonal multiple access (NOMA), NOMA-aided mobile edge computing (MEC) can provide scalable and low-latency computing services for the Internet of Things. However, given the prevalent stochasticity of wireless networks and sophisticated signal processing of NOMA, it is critical but challenging to design an efficient task offloading algorithm for NOMA-aided MEC, especially under a large number of devices. This paper presents an online algorithm that jointly optimizes offloading decisions and resource allocation to maximize the long-term system utility (i.e., a measure of throughput and fairness). Since the optimization variables are temporary coupled, we first apply Lyapunov technique to decouple the long-term stochastic optimization into a series of per-slot deterministic subproblems, which does not require any prior knowledge of network dynamics. Second, we propose to transform the non-convex per-slot subproblem of optimizing NOMA power allocation equivalently to a convex form by introducing a set of auxiliary variables, whereby the time-complexity is reduced from the exponential complexity to $mathcal{O} (M^{3/2})$. The proposed algorithm is proved to be asymptotically optimal, even under partial knowledge of the device states at the base station. Simulation results validate the superiority of the proposed algorithm in terms of system utility, stability improvement, and the overhead reduction.
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