A production/inventory system consisting of a single-stage facility producing multiple product types and a warehouse is considered. The demand process is assumed to be Poisson and the excess demand is backordered. Products have a priority structure and the processor is shared according to a switching rule. (R, r) continuous-review inventory control policies are used to start and stop production of each product. Production of a product continues until its target level is reached. Then, a switch-over takes place to the highest priority product among the ones that need the processor's attention. A set-up takes place every time a switch-over occurs. Both processing and set-up times are assumed to have arbitrary distributions. The number of switch-overs among the high priority products is restricted in case low priority products need attention. The objective is to obtain performance measures such as the average inventory and backorder levels as well as die probability of having a backorder upon arrival for each product. We develop an approximation procedure which emphasizes the characterization of the delay each product type experiences in receiving the processor's attention, once they place a request for it. The approximating arguments are based on independence assumptions.