Process drift is a common occurrence in many manufacturing processes where machines become dirty (leading to more contamination) or processing parameters degrade, negatively affecting system performance. Statistical process control tracks process quality to determine when the process has gone out of control (has drifted beyond its specifications). This paper considers the case where parts examined at a downstream inspection station are used to determine when the upstream process is out of control. The manufacturing cycle time from the out of control process to the downstream inspection process influences the detection time that elapses until the out of control process is noticed and repaired. Because an out of control process produces more bad parts, the detection time affects the number of good parts produced and the throughput of the manufacturing system. This situation is common in many industries but no models of the phenomena exist. This paper presents a novel manufacturing system model based on queueing network approximations for estimating the manufacturing cycle time and throughput of such systems. These are important performance measures since they influence economic measures such as inventory costs and revenue. The model can be used for a variety of system design and analysis tasks. In particular, the model can be used to evaluate the placement of inspection stations in a process flow.