We study online task allocation for multi-robot, multi-queue systems with asymmetric stochastic arrivals and switching delays. We formulate the problem in discrete time: each location can host at most one robot per slot, servicing a task consumes one slot, switching between locations incurs a one-slot travel delay, and arrivals at locations are independent Bernoulli processes with heterogeneous rates. Building on our previous structural result that optimal policies are of exhaustive type, we formulate a discounted-cost Markov decision process and develop an exhaustive-assignment actor-critic policy architecture that enforces exhaustive service by construction and learns only the next-queue allocation for idle robots. Unlike the exhaustive-serve-longest (ESL) queue rule, whose optimality is known only under symmetry, the proposed policy adapts to asymmetry in arrival rates. Across different server-location ratios, loads, and asymmetric arrival profiles, the proposed policy consistently achieves lower discounted holding cost and smaller mean queue length than the ESL baseline, while remaining near-optimal on instances where an optimal benchmark is available. These results show that structure-aware actor-critic methods provide an effective approach for real-time multi-robot scheduling.