We consider a general model framework for acyclic stochastic processing networks with shared resources that has many applications in telecommunication, computer, and manufacturing systems. A dynamic control policy that utilizes the maximal matching (for scheduling) and the join-the-shortest-queue (for routing) discipline, is shown to maximize the throughput and stabilize the system in a sense called "uniform mean recurrence time property" under fairly mild stochastic assumptions. Owing to the non-Markovian nature of the states, system stability is established using a perturbed Lyapunov function method.