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Manufacturing systems have become increasingly complex over recent years. This volume presents a collection of chapters which reflect the recent developments of probabilistic models and methodologies that have either been motivated by manufacturing systems research or been demonstrated to have significant potential in such research.
The editor has invited a number of leading experts to present detailed expositions of specific topics. These include: Jackson networks, fluid models, diffusion and strong approximations, the GSMP framework, stochastic convexity and majorization, perturbation analysis, scheduling via Brownian models, and re-entrant lines and dynamic scheduling. Each chapter has been written with graduate students in mind, and several have been used in graduate courses that teach the modeling and analysis of manufacturing systems.



1. Jackson Network Models of Manufacturing Systems

Focusing on discrete-part batch manufacturing systems, we illustrate the relevance and usefulness of Jackson network models. The emphasis is on demonstrating the effectiveness of the models in capturing the fundamental qualitative and structural features of batch manufacturing, so as to provide insight and support for the design and efficient operation of the systems. In particular, we study the relationship between the production rate and the work-in-process, the implications of resource sharing, and the different schemes in workload allocation and assignment. We also bring out the connections to other familiar subjects such as likelihood ratio ordering, majorization and arrangement orderings, and coupling techniques.
John A. Buzacott, J. George Shanthikumar, David D. Yao

2. Hierarchical Modeling of Stochastic Networks, Part I: Fluid Models

Decision processes in complex operations are frequently hierarchical. It is natural, therefore, for models that support such decisions to be hierarchical as well, and here we explore such a hierarchy. Specifically, we are concerned with stochastic flow networks, typified by queueing networks, which will be analyzed within a framework that distinguishes three aggregation levels of time and state-space: a microscopic level that acknowledges individual particles (which is the level at which queueing network models are typically set up), a macroscopic level, in which the network is approximated by a deterministic fluid model, and an intermediate mesoscopic level, that quantifies the deviations between the micro and the macro in terms of stochastic diffusion approximations. It might be useful to start with an example, of a closed nonparametric Jackson queueing network, that attempts to concretize these levels.
Hong Chen, Avi Mandelbaum

3. Hierarchical Modeling of Stochastic Networks, Part II: Strong Approximations

The goal in this part is to establish strong approximations for a family of open queueing networks. We cover, in particular, nonparametric Jackson networks. These are the classical open Jackson queueing networks, but without the parametric assumptions of exponential interarrivai and service times (Section 2.5.2 of Part I). The basic results are Functional Strong Approximations (FSAT, Theorem 3.4.1) and a Functional Law of Iterated Logarithm (FLIL, Theorem 3.4.2). These readily imply fluid approximations, formalized by a Functional Strong Law of Large Numbers (FSLLN, Corollary 3.4.3), and diffusion approximations, formalized by a Functional Central Limit Theorem (FCLT, Corollary 3.4.4).
Hong Chen, Avi Mandelbaum

4. A GSMP Framework for the Analysis of Production Lines

We present the basics of the generalized semi-Markov process (GSMP), focusing on its scheme and related structural properties. We illustrate the various applications of these properties through a detailed study on a serial production Une under the so-called generalized kanban control: the production of each stage is controlled by three parameters that are upper limits on the work-in-process inventory, finished-job inventory, and the overall buffer capacity. In particular, we show that the system satisfies monotonicity, convexity/concavity, and line reversibility, all of which have useful implications in system design. Furthermore the service-completion epochs at all stages satisfy a subadditive ergodicity, which guarantees the existence of long-run average cycle times (or, equivalently, the existence of a well-defined production rate).
Paul Glasserman, David D. Yao

5. Stochastic Convexity and Stochastic Majorization

Convexity and submodularity are useful second-order properties in optimization. Majorization and the related arrangement ordering are key properties that support pairwise-interchange arguments. Together these properties play an important role in resource allocation, production planning, and scheduling. Here we present the essentials of the recently developed theory of stochastic convexity and stochastic majorization, emphasizing their interplay. We illustrate the many applications of the theory through examples in production systems, including a random yield model, a joint setup problem, a manufacturing process with trial runs, a production network with constant work-in-process (WIP) and WIP-dependent production rate, and scheduling in tandem production lines and parallel assembly systems.
Cheng-Shang Chang, J. George Shanthikumar, David D. Yao

6. Perturbation Analysis of Production Networks

This chapter treats the problem of evaluating the sensitivity of performance measures to changes in system parameters for a class of stochastic models. The technique presented, called perturbation analysis, evaluates sensitivities from sample paths, based either on a simulation or on real data.
The chapter begins with an overview, based on a single-machine model. It proceeds to address underlying theoretical issues and then examines a variety of examples of production networks. It concludes with a discussion of issues arising in the estimation of steady-state sensitivities.
Paul Glasserman

7. Scheduling Networks of Queues: Heavy Traffic Analysis of a Bi-Criteria Problem

We consider bi-criteria scheduling problems for three queueing systems (a single queue, a two-station closed network, and a two-station network with controllable inputs) populated by various customer types. The objective is to minimize the long run expected average value of a linear combination of the customer sojourn time and the sojourn time inequity. The inequity at time t is the sum of squares of the pairwise differences of the total number of customers in the system at time t of each type divided by their respective arrival rates. Brownian approximations to these three scheduling problems are solved, and the solutions are interpreted in order to obtain scheduling policies. Simulation results show that the second objective criteria tends to equalize the mean sojourn times of the various customer types, and may lead to a reduction in sojourn time variance. The simulation results also show that in the network settings, in contrast to the single queue case, there are priority sequencing policies that significantly reduce the variance of sojourn times relative to the first-come first-served policy.
Lawrence M. Wein

8. Scheduling Manufacturing Systems of Re-Entrant Lines

Re-entrant lines axe manufacturing systems where parts may return more than once to the same machine, for repeated stages of processing. Examples of such systems are semiconductor manufacturing plants. We consider the problems of scheduling such systems to reduce manufacturing lead times, variations in the manufacturing lead times, or holding costs.
We assume a deterministic model, which allows for bursty arrivals. We show how one may design scheduling policies to help in meeting these objectives. To reduce the mean or variance of manufacturing lead time, we design a class of scheduling policies called Fluctuation Smoothing policies. To reduce the holding costs in systems with set-up times, we introduce the class of Clear-A-Praction scheduling policies.
We study the stability and performance of these scheduling policies. We illustrate how scheduling policies can be unstable in that the levels of the buffers become unbounded. However, we show that all Least Slack policies, including the well known Earliest Due Date policy and all the Fluctuation Smoothing policies, are stable.
P. R. Kumar
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