Preface
1. Fitzsimmons and Fitzsimmons (2011).
Chapter 1
1. Refer to Appendices A and B for more detailed descriptions of the terms and symbols used in the text.
2. Laguna and Marklund (2005), chapter 6.
3. Hillier and Lieberman (2010), chapter 17.
4. Nelson (2010), chapter 8.
5. Some of you are probably asking, “Shouldn’t this value be 50%?” Many of my students ask this question because their familiarity in using normal distributions implies that the average should be at the 50% point. Although this is true for a normal distribution, it is not true for the majority of distributions. For an exponential distribution, the average value corresponds to the 36.78% point and to the 1 − 0.3678 = 63.21% point for the inverse exponential distribution.
6. Abner Krarup Erlang (1878–1929) is a Danish mathematician who developed a number of queuing theory concepts for telephone company applications in the early 1900s. Refer to appendix A for further details.
7. Attributed to the British statistician D. G. Kendall (1918–2007) during the period from 1951 to 1953.
Chapter 2
1. Named after J. D. C. Little, who published a proof of this formula in 1961. The formula had been observed and used by others shortly before Little published his proof.
Chapter 3
1. The single line arrangement is sometimes referred to as a “snake” configuration by some authors because it often requires a winding layout to accommodate its length. A common example is the arrangement at airports before the passenger security checkpoints.
2. Obviously, a new server is likely to not be as productive as an experienced server. However, keep in mind that the waiting line equations are based on a steady-state condition, which implies that all servers have attained the average service rate capability. How long this may take is discussed in more detail in chapter 5.
3. Note the parentheses added around the summation term in the denominator to clarify that the second term is not to be included in the summation. This addresses a commonly observed mistake by many of my students when we first added waiting line topics to our process and operations management courses.
Chapter 4
1. Pollaczek (1930) and Khintchine (1932). See glossary (appendix A).
2. This use of α should not be confused with its use in Equation (1.2) for the exponential distribution.
3. Peck and Hazelwood (1958).
Chapter 5
1. This is done to simplify the equations for calculating P0 and Lq because only ρ and the number of servers s is required in these equations. This also agrees with the P0 and Lq data tables provided for a range of ρ and s values in appendix C. The reader is reminded that ρ here is not the same as ρs = λ/µs, the multiple-channel ρ used by some authors.
2. Named after a sixteenth-century French mathematician, L’Hospital or L’Hôpital. When confronted with an expression where it is not possible to define a limit as a variable approaches a certain value (1 in this case), successive derivatives of the numerator and the denominator with respect to that variable are taken until they converge to a definable limit.
3. The term “finite queuing table” is not exactly correct for this situation since we are discussing finite sources here, not finite queues. The term is more appropriate for our earlier discussion regarding limited line capacity. However, this term is used in many textbooks for both capacity and sourcing issues because it is consistent with its usage by Peck and Hazelwood (1958), a commonly used resource for solving finite sourcing problems.
4. Krajewski et al. (2010).
Chapter 6
1. Bekker (2005) is a good overall reference; Hillier and Lieberman (2010) discuss this in chapter 17, section 17.5.
2. If you are unfamiliar with service blueprints, the article by Shostack (1984) is a good introduction.
3. One utility cost often forgotten is waste management. A good waste management policy can help keep this cost low and reduce the number of staff required to support it.
4. This material was formerly chapter 18 in the previous edition of this reference.
5. McCarran Airport in Las Vegas does this in an effective and entertaining way by showing video clips on conveniently placed video monitors in the security check-in process area. The clips are of local Las Vegas Strip entertainers doing or being corrected for not doing the right thing at different check-in steps.
6. Some Disney theme parks let customers buy tickets for popular rides, where the ticketing system calculates the average waiting time until the ticket holder can board the next available ride and prints that time on the ticket. This allows customers to visit other parts of the park, such as food and retail shops, instead of waiting in line. Longer waits are then not as significant, and park visitors can even go and buy other tickets for rides later in the day. The ticketing system’s job in forecasting the average waiting time is simplified because the service time (the duration of the ride) is essentially constant. The somewhat hidden advantage here is that park revenue from food and retail shops is less affected by lost customer visits while they wait in line at the rides.
7. Abilla (2007).
8. This constrains us to a fixed appointment time during the day because we cannot predict when these urgent-care situations will occur during the day. For doctors who are not expected to handle such drop-in patients, we have more flexibility regarding the length of the appointment times, allowing a mixture of short appointments and long appointments according to patient needs. Then we can even use short-term scheduling rules to schedule the order in which patients are offered appointment times.
9. These values used here should not be construed to represent all medical practices. When a full range of medical services is being provided and urgent care patients must be accommodated, these values are typically larger and have greater variance.
10. As a note of interest, the need to accommodate some urgent-care patients is often greater at the beginning and ending of the week. This is likely caused by the more limited availability of health care professionals on weekends.
11. For those interested in learning more about reservation systems and overbooking, chapter 11 in Fitzsimmons and Fitzsimmons (2011) is a good introduction.
12. Hillier and Lieberman (2010), chapter 17.
13. Haussmann (1970).
14. It should be noted that the benefits of dequeuing can also be achieved by educating the customers waiting in line as to what will be required when they reach the server window, allowing them to fill out such forms in advance while they are waiting. This gives them something to do while also improving line performance.
15. Stevenson (2011), chapter 16.
16. Jacobs and Chase (2011), chapter 19.
17. While we do not show a specific example of how makespan can be used for waiting line management, one example would be estimating how long it would usually take to process customers remaining to be served after normal closing time.
Chapter 7
1. Little (1961).
2. If you are unfamiliar with how to create histograms in Excel using the function provided in Excel’s Analysis ToolPak, refer to the Excel tips section in appendix D.
3. Where this value comes from was commented on earlier for the discussion regarding exponential distributions in chapter 1.
Appendix A
1. Named for D. R. Cox (1995).
2. Named for the Danish mathematician Agner Krarup Erlang (1878–1929). His work with the Copenhagen Telephone Company in the early 1900s provided the foundation for several aspects of modern waiting line analysis.
3. Kendall (1951). This method is commonly indicated by the generic A/B/C/d/e/f notation. The last three characteristics are not shown in many textbooks; those that do use them often do not follow the order given in the definition above.
4. Named for the Russian mathematician Andrey Andreyevich Markov (1856–1922). Sometimes the term Markovian process is restricted to describing sequences of independent random variables with continuous values, such as those defined by exponential distributions. When this is done, similar sequences of discrete variables are referred to as Markov chains.
5. Named for Felix Pollaczek and Aleksandr Khintchine. Each independently developed this formula, in 1930 and 1932, respectively. Many citations of their formula have a wide range of spelling variations for Khintchine.
Appendix D
1. See Fitzsimmons & Fitzsimmons (2011), p. 438, for the reasoning behind the derivation of this expression. For a more convenient method than using 12 random numbers for each value required, use Excel’s NORM.INV function.