In this paper, we establish a many-sources large deviations principle (LDP) for the stationary workload of a multi-queue single-server system with simplex capacity, operated under a stabilizing and non-idling maximum-weight scheduling policy. Assuming a many-sources sample path LDP for the arrival processes, we establish an LDP for the workload process by employing Garcia’s extended contraction principle that is applicable to quasi-continuous mappings. The LDP result can be used to calculate asymptotic buffer overflow probabilities accounting for the multiplexing gain, when the arrival process is an average of i.i.d. processes. We express the rate function for the stationary workloads in term of the rate functions of the finite-horizon workloads when the arrival processes have i.i.d. increments.