When waiting line (queuing) theory is introduced in business or engineering classes, students are presented fundamental rules and equations that give consistent results for the same set of conditions. This simplifies the initial discussion of often complex issues to enable students to become comfortable with the basic concepts and obtain the average results for various waiting line performance measures. Later, usually at the graduate level, the pesky details regarding the real-life variability in these average values are revealed to explain why the predictions provided by the fundamental rules rarely are exactly true. As many of you no doubt recognize, this last statement also applies to many other disciplines and thus should be considered a basic fact of life.
The tone and scope of this monograph assumes that you, the reader, are a person who has already taken the initiative to increase your understanding of the waiting line and associated service aspects of your business, with the goal of either improving service, reducing operating cost, or both. You have probably reviewed your old college texts, talked to some more experienced colleagues, reviewed the literature available in the local library, and used Internet search engines to find sites that might help. A common result is that you probably noticed that many of the equations provided by different authors do not appear to be the same for a particular performance measure, such as average waiting time or line length. Another outcome is that you often encounter different terms used for what appears to be the same concept. My goal is help you navigate through these apparent discrepancies, introduce you to some new concepts that are likely to be unfamiliar, and help you achieve a better understanding for your business needs. So let us begin.
We begin by reviewing the fundamental rules for different waiting line models and then expand the discussion to cover some of those all-too-frequent occasions where what we observe in actual practice does not seem to agree with the predicted values. These variations should not be viewed with dismay, however, but rather as opportunities for innovating, developing a competitive advantage, and assessing possible business risks.
Some mention of the level of mathematic understanding expected of you is appropriate at this point. Several of the performance expressions will intuitively make sense. Others will require you to accept their validity on faith because the mathematics behind their derivation can be quite daunting to someone who is not a mathematician. For the most part, we will not discuss the derivation of most formulas because this monograph is focused on application. However, we will occasionally need to delve more deeply into how a few formulas are derived so we can understand their limitations in predicting actual business outcomes. In addition, we will develop some expressions that, while not exactly mathematically rigorous, will provide good enough approximations for pragmatic business decisions.
The following is likely to be the most useful and important piece of advice you can gain from reading this monograph: As people gain experience in business, they develop a gut feeling as to what the ballpark estimate of the results of a business analysis will be. Hence when the calculated result does not agree with their estimate, they double-check the calculations. Because of the probabilistic content in waiting line equations, it is more difficult to acquire an intuitive feeling as to what the result should be. Many of the waiting line equations can be quite complex, and the likelihood of typographical errors in equations presented in references and other material is increased. In fact, I discovered formula errors and other formula differences in many of the references consulted during my research for this monograph. Some of these are obvious to an astute reader, but others can take considerable effort to detect.
Therefore, I cannot stress enough how important it is to make sure that the units of measure for each parameter in a waiting line equation are accounted for and that their use balances out into the proper units for the answer. Consistent checking of the units in your answers will help indicate the presence of errors, whether they are caused by an error in the equation, an incorrect entry in the equation, or a calculation error on your part.
Chapter 1 reviews the basic concepts used for waiting line process analysis; discusses the most common probability distributions; and introduces state diagrams that are used to derive many queuing formulas, descriptions and notation for different waiting line models, and basic cost considerations.
Chapter 2 covers the characteristics and analysis of basic single-channel, single-phase models common to small businesses and sections of manufacturing lines. Chapter 3 covers the characteristics and analysis of basic multiple-channel, single-phase models like most of us have encountered in banks and post offices. You can skip these chapters if you feel that you are sufficiently familiar with the basics; but there are some useful clarifications not usually discussed in college textbooks that will be helpful when we address more complex situations.
Chapters 4 and 5 cover more complex concepts related to less common arrival and service distributions, line capacity limitations, limited population applications commonly used for maintenance activities, multiple-server situations, and manufacturing applications. Chapter 5 also includes equations for direct computations of limited population models to allow you to use spreadsheet methods instead of finite queuing tables. Some new nomenclature is introduced to help avoid some confusion that can occur when using such equations.
Chapter 6 focuses on managerial considerations regarding waiting line decisions. The limitation that commonly used waiting line equations predict only average performance is discussed in some detail. Knowing more about performance variability is necessary to enable managerial consideration to reduce its effects on customer service and operating costs. Some possible strategies and methods for reducing variability in both the arrival and the service rates are reviewed, and some cost decisions, such as cost trade-offs between adding service capacity versus process improvements, are discussed. Chapter 6 also discusses some of the softer factors, such as waiting line psychology, priority management, and preferred customer treatment.
In Chapter 7, several tools are discussed. The first is Little’s Law, a useful mathematical expression relating waiting line length to waiting time using the arrival rate. The use of Little’s Law as a handy method to obtain some useful starting data about a waiting line situation without requiring extensive data gathering over extended periods of time is also described. To gain knowledge about performance variability for the evaluation of best- and worst-case scenarios, simulation is required. Some examples show how simulation models can be constructed for several waiting line situations using functions commonly available in Excel 2007 or 2010.
A list of references is provided at the end. Appendix A provides a glossary of terms; appendix B lists symbols used with their definitions; appendix C presents some useful tables and spreadsheet examples for multiple-server applications; and appendix D provides some useful simulation design information for Excel users.