Fundamental Behaviors in Hybrid Deterministic Flow Lines With Applications

Abstract

Since the mid-1960s, it has been known that regular or deterministic flow lines (DFLs) with an arbitrary arrival process possess three essential behaviors: 1) a customer exit time recursion; 2) a server reordering principle; and 3) a delay equivalence with prototype queueing systems. Even though these three fundamental behaviors were discovered decades ago, analogous results for hybrid DFLs (HDFLs)—possessing multiple servers for each stage—have not been unearthed. In this article, we prove the existence of a recursion for customer exit times in HDFLs. This result leads naturally to a server reordering principle. Furthermore, we identify specific conditions under which the total delay faced by a customer in an HDFL is equivalent to that in a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G/D/c$ </tex-math></inline-formula> queue. Exploiting these results, we numerically determine the proportion of HDFL systems with a delay equivalent, study the computational burdens required for HDFL simulation, and assess the performance of approximations for HDFL systems. We hope these newly identified behaviors will lead to improved approximation methods and optimization models for use in modern automated manufacturing systems. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Many modern automated manufacturing systems, such as semiconductor wafer fabricators, use hundreds or thousands of highly complex, costly, and precise tools. These automated tools are often configured as a collection of processing modules served by material handling robots and clustered into a single chassis. One important example of such cluster tools is the clustered photolithography tool (CPT). For certain key performance metrics, including cycle time, residency time, and throughput time, such tools can be well-modeled as a tandem queueing system with random arrivals, regular or deterministic service times, multiple servers devoted to each stage, and finite internal buffers—a so-called HDFL. Such HDFLs also serve as a fundamental model of automated assembly systems. In this work, we prove that the behavior of HDFLs can be characterized by an exit recursion that requires significantly less computation than otherwise. Furthermore, HDFLs possess the surprising property that the customer queueing time is invariant to the order of the stages and, in some cases, is equivalent to a prototype queueing system. These results can be exploited to assess customer delay and to obtain simulation models with about an order of magnitude less computational complexity. We apply our results to a CPT model from the literature to highlight the properties of such tools. Finally, the theory introduced in this article enables a deeper understanding and intuition for such systems.