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In this paper we use results on the aggregation of singularly perturbed Markov chains to analyze manufacturing systems. The basis for this analysis is the presence in the system of events and processes that occur at markedly different rates - operations on machines, set-ups, failures, and repairs, etc. The result of the analysis is a set of models, each far simpler than the full model, describing system behavior over different time horizons. In addition, we present a new theoretical result on the computation of asymptotic rates of particular events in perturbed Markov processes, where an "event" may correspond to the occurrence of one of several transitions in the process. We may apply this result to compute effective production rates at different time scales, taking into account the occurrence of setups and failures.

During the past three decades, the success of the Toyota Production System has spurred much research in manufacturing systems engineering. Productivity and quality have been extensively studied, but there is little research in their intersection. The goal of this paper is to analyze how production system design, quality, and productivity are inter-related in small production systems. We develop a new Markov process model for machines with both quality and operational failures, and we identify important differences between types of quality failures. We also develop models for two-machine systems, with infinite buffers, buffers of size zero, and finite buffers. We calculate total production rate, effective production rate (ie, the production rate of good parts), and yield. Numerical studies using these models show that when the first machine has quality failures and the inspection occurs only at the second machine, there are cases in which the effective production rate increases as buffer sizes increase, and there are cases in which the effective production rate decreases for larger buffers. We propose extensions to larger systems.