Stability of real-time lot-scheduling and machine replacement policies with quality levels

In this note we show how the main result of Lou et al. [2] can be generalized to an unreliable machine with various quality levels. We consider a single machine which can process parts of di erent types. The machine can process at most one part at a time and it needs a setup period whenever it switches from one part-type to another. Parts enter the system in type-speci c input rates and require type-speci c processing times. The quality of the machine deteriorates according to a continuous-time Markov process and the only way to improve the quality is by replacing the machine by a new one. We postulate a simple form of real-time replacement and scheduling policies, and derive conditions ensuring the stability of the system under these policies.