Equilibrium probabilities in queueing models of automated manufacturing systems

Abstract

A natural class of mathematical models for modern automated manufacturing systems such as semiconductor wafer fabricators and service system is queueing network. There have been many studies to analyze such systems, but exact analysis is hard to be applied as the system becomes complicated. While performance bounds, simulation and approximations are employed to provide useful information for the system, but do not completely characterize the steady state behavior. In this dissertation, two queueing models are analyzed with exact method. We obtain Markov chain models for the systems, thus the equilibrium probabilities can be obtained. First, flow line models, also called tandem queue, are analyzed. Flow line models have been studied for decades, but there have been few exact results on their steady state behavior. For flow lines with at most three server, some exact steady state results have been obtained, however they cannot be extended to allow for more servers. As such, approximations based on aggregation or decomposition methods are generally employed. By focusing on flow lines with deterministic service durations,we conduct exact analysis for the system. We focus on two performance metrics: equilibrium probability for delay in each servers and maximum production rate. Each will be studied with different assumptions. For the equilibrium probability for delay in each servers, we start with the investigation of recursions for customer delay based on exact decomposition methods. We demonstrate that the delay a customer faces in each server possesses a Markovian property. For discrete-time flow lines, we obtain a multidimensional discrete-time time-homogeneous Markov chain for the delays; there are an infinite number of balance equations for the equilibrium probabilities. Exploiting a similarity between our system and the GI/D/1 queue allows us to reduce these to a finite number of balance equations that can be solved numerically. To our k...