Motivated by semiconductor wafer fabrication, we consider a scheduling problem for a single-server multiclass queue. A single workstation fabricates semiconductor wafers according to a variety of different processes, where each process consists of multiple stages of service with a different general service time distribution at each stage. A batch (or lot) of wafers produced according to a particular process randomly yields chips of many different product types, and completed chips of each type enter a finished goods inventory that services exogenous customer demand for that type. The scheduling problem is to dynamically decide whether the server should be idle or working, and in the latter case, to decide which stage of which process type to serve next. The objective is to minimize the long run expected average cost, which includes costs for holding work-in-process inventory(which may differ by process type and service stage) and backordering and holding finished goods inventory (which may differ by product type). We assume the workstation must be busy the great majority of the time in order to satisfy customer demand, and approximate the scheduling problem by a control problem involving Brownian motion. A scheduling policy is derived by interpreting the exact solution to the Brownian control problem in terms of the production/inventory system. The proposed dynamic scheduling policy takes a relatively simple form and appears to be effective in numerical studies.