Stochastic Modeling

Analysis and Simulation

Barry L. Nelson

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486139944
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

A coherent introduction to the techniques for modeling dynamic stochastic systems, this volume also offers a guide to the mathematical, numerical, and simulation tools of systems analysis. Suitable for advanced undergraduates and graduate-level industrial engineers and management science majors, it proposes modeling systems in terms of their simulation, regardless of whether simulation is employed for analysis. 
Beginning with a view of the conditions that permit a mathematical-numerical analysis, the text explores Poisson and renewal processes, Markov chains in discrete and continuous time, semi-Markov processes, and queuing processes. Each chapter opens with an illustrative case study, and comprehensive presentations include formulation of models, determination of parameters, analysis, and interpretation of results. Programming language–independent algorithms appear for all simulation and numerical procedures.

CHAPTER

1

WHY ARE
WE HERE?

This book describes basic tools for modeling, analysis and simulation of systems that evolve dynamically over time and whose behavior is uncertain. One class of examples is queueing systems, which are systems of customers waiting to use service resources. Our goal is to describe these tools in a way that exploits your common sense and intuition but also enables you to use the mathematics, probability and statistics at your disposal to perform a detailed analysis. At the end of the day you should know a bit more about probability and statistics too.

A brief list of the settings in which these tools have been useful includes:

•  Manufacturing applications, such as capacity planning, inventory control and evaluation of process quality

•  Health-care applications, such as hospital staffing and medical decision making

•  Computer applications, such as designing hardware configurations and operating-system protocols

•  Communication applications, such as sizing message buffers and evaluating network reliability

•  Economic applications, such as portfolio management and forecasting industrial relocation

•  Business applications, such as consumer behavior and product-distribution logistics

•  Military applications, such as combat strategy and predicting ordnance effectiveness

•  Biological applications, such as population genetics and the spread of epidemics

There are many more.

Our approach is to model systems in terms of how we would simulate them, and then recognize situations in which a mathematical analysis can take the place of a simulation. There are three reasons for this approach: Simulation is intuitive, because it describes how a system physically responds to uncertainty; the fundamentals of simulation do not change from application to application, while mathematical analysis depends profoundly on characteristics of the model; and simulation provides a common modeling paradigm from which either a mathematical analysis or a simulation study can be conducted. The primary drawback of simulation is that it produces only estimates of system performance, whereas mathematical analysis provides the right answer, up to the fidelity of the model. Other advantages that favor mathematical analysis are described later in the book.

The next chapter illustrates how we propose to think about dynamic systems that are subject to uncertainty. Chapters 3–4 then provide the basic constructs for modeling such systems and simulating them. Standard methods of mathematical analysis, and when they apply, appear in Chapters 5–8. Finally, Chapter 9 describes some critical issues that arise when simulating models that are not amenable to mathematical analysis.

CHAPTER

2

SAMPLE PATHS

This chapter introduces sample-path decomposition, which is one way to think about dynamic systems that evolve over time. A service-system example is used to illustrate the approach, but sample-path decomposition can characterize the behavior of many types of systems, including those described in Chapter 1. Sample- path decomposition is also a convenient way to formulate mathematical models of systems, models that can be used to evaluate how changes will affect existing systems or predict the performance of systems that do not yet exist. Formulation and analysis of these models is the topic of this book, and sample-path decomposition is the perspective that we employ throughout.

2.1 THE CASE OF THE COPY
ENLARGEMENT

Case 2.1. A national chain of small photocopying shops, called The Darker Image, currently configures each store with one photocopying machine and one clerk. Arriving customers stand in a single line to wait for the clerk. The clerk completes the customers’ photocopying jobs one at a time, first-come-first-served, including collecting payment for the job.

Business is good enough that the parent company of The Darker Image plans to enlarge some stores by adding a second photocopying machine and hiring a second clerk. The second copier could be operated by the new clerk, but since some customers with small copying jobs have complained about having to wait a long time, the company is considering installing the second copier for self-service copying only. The company wants to know which option will deliver better service.

Case 2.1 describes a system that is quite simple compared to the complex systems that engineers and management scientists study every day. Nevertheless, the answer to the company’s question is not obvious. Certainly some additional information is needed to perform a thorough analysis. For instance, how many customers will use a self-service copier? Can a self-service copier provide all the services that a customer wants (such as collating, stapling, or attaching covers)? Should a page limit be placed on self-service copying? Are there differences from store to store, or is a single solution good for all stores?

One important characteristic of the system described in Case 2.1 is that neither option will be the best option all of the time. The system’s performance will be subject to some uncertainty: The number of customers that arrive and when they arrive each day are variable. The nature of the customers’ copying jobs—large or small, simple or requiring special handling—is also unknown. These factors, and others, will affect the service delivered on a particular day, so that a system design that seems superior on one day may be inferior on another day. Therefore, the definition of “better service” is not straight-forward.

Modeling and analysis of systems that are subject to uncertainty is the topic of this book. The mathe...