Modeling and analysis of two-stage manufacturing systems with parallel machines and phase-type cycle time

Modern manufacturing systems are facing an increasingly versatile manufacturing environment. Having multiple parallel machines in each stage of the system is a design strategy that improves the system reconfigurability and resilience and enables the system to respond effectively to the uncertainties and disruptions from various sources. In addition, because of the increased complexity in the manufacturing tasks, many operations in the modern manufacturing systems have non-deterministic cycle time. However, traditional discrete-time Markov Chain-based models usually assume there is only one machine per stage and the cycle time is deterministic. In this paper, we develop a new discrete-time Markov Chain model to analyze the system dynamics and evaluate performance of a two-stage manufacturing system that has multiple parallel machines in each stage and multiple operations assigned to every machine. It is assumed that the time to complete each operation follows a geometric distribution, so the cycle time of each machine follows a discreate phase-type distribution. Simplified one-parameter and two-parameter models are also developed to approximately analyze the system performance, and their relations to the traditional Bernoulli and geometric models are discussed. Numerical examples are provided to illustrate the proposed models and verify their effectiveness.