# No ex-queue-ses: understanding queue formation in the graduate operations management classroom.

INTRODUCTION

Providing an introduction to queuing theory is an important part of the graduate operations management (OM) course-as it can be viewed as the counterpart of inventory theory in the design of a service operation. While the mathematics behind the basic results of queuing theory are quite elegant in their simplicity, they remain beyond the normal scope, abilities and interests of students in a graduate OM course. As a result, texts rely on spreadsheets (e.g., the Performance.xls spreadsheet provided by Anupindi et. al., 2012) as a means of enabling students to develop an intuitive understanding of some of the field's key principles.

While such methods can be effective, there often remains a gap in the students' Understanding-which keeps them from successfully integrating the text/lecture concepts and the more advanced results utilized in the spreadsheet models. This study discusses an experiential exercise that has proven effective in bridging the gap between lecture concepts and the spreadsheets that are often utilized to solve basic problems.

PREVIOUS STUDIES

The exercise described in this study was inspired, in part, by the boy scouts' matchstick game detailed in Goldratt (1992) and which illustrates the impact and interaction of statistical fluctuations and dependent events on a process flow. Umble and Umble (2001) and Johnson (2002a, 2002b) report on the successful use within the classroom of exercises based on this game. The simulations described by these authors, as well as Ammar and Wright (1999), principally focus on balanced lines-i.e., those in which the average processing capacities of each step are equal.

For example, while Tommelein et. al. (1998) use altered dice to change the variability of the processing capacity, they only assess the impact on balanced lines. Similarly, while Johnson (2002a) does consider some scenarios in which the line is not perfectly balanced, the purpose of this is to assess the impact of the location of the bottleneck on process performance (and the location of queue formation within the process). The figures of merit (and focus of analysis) in these exercises tend to be the average throughput and work in process inventory of the process(es) considered.

The purpose of this simulation exercise-and the contribution of this study-is to help students to understand better the process of queue formation itself and, specifically, the impact of process utilization on queue formation. In the scenarios described below, a single process step is analyzed in isolation (i.e., like the single-queue, single-server problem that typically represents the first unit of analysis in students' introduction to queuing). Moreover, and more importantly from a queuing perspective, the average input or arrival rate to the process step under study is strictly less than its average capacity. This meets the requirement for process stability and isolates the effect of process utilization on queue formation for easier analysis and understanding by the students.

BACKGROUND OF THIS STUDY

The Basic Simulation Exercise

The queuing simulation utilizes sets of "Gamers' dice"-which typically include 4-, 6-, 8-, 10-, 12-, and 20-sided dice, as shown in Figure 1. The rolls of the dice determine the number of units entering and able to be processed by a particular process step during a given time period, as illustrated in Figure 2. Poker chips represent the units to be processed in this simulation, although other markers could be used. Alternatively, students may simply keep track of the units on paper, but this is not always as effective. While the arrivals are not exponentially distributed in time as is normally assumed in queuing (i.e., Poisson arrival rate), the results obtained with the uniformly-distributed dice rolls adequately serve the purposes of this illustration.

Students form two-person teams-with one student rolling the die to determine the number of arrivals during a given time period and the other rolling to determine the number of units that can be processed during the same time period. One "period" of the simulation is conducted as follows:

* Student 1 rolls the "arrival die" to determine the number of units that arrive for processing during the period-and moves this number of poker chips into the queue. (This is recorded as the "Arrivals" on the worksheet provided and shown in Figure 3.)

* Student 2 rolls the "processing die" to determine the number of units that could be processed during the period. (This is recorded as the "Capacity" on the worksheet.)

* The number of units that are processed is determined as the minimum of the number that began in the queue plus the number of arrivals in the period and available capacity in the period. (This is recorded as the "Processed" units on the worksheet.) This number of poker chips is removed from the queue and moves onto the next (untracked) process step.

* The units remaining in the queue ("Final Queue"-which becomes the "Initial Queue" for the next period) are calculated by subtracting the "Processed" units from the sum of the "Initial Queue" and "Arrivals."

* "Utilization" is calculated by dividing "Processed" units by "Capacity" available.

Students can typically capture the results of 50-100 periods within 15-20 minutes at which point they can calculate the average queue length and process utilization. These results may, then, be graphed to illustrate the impact of process utilization on the average queue length.

Extending the Simulation Exercise

Once students have mastered the relationship between utilization and average queue length, the basic simulation scenario can be extended to the following process design question: Given an arrival quantity determined by two 4-sided dice-with two processing servers with capacity each determined by 6-sided dice-which process design would provide a shorter queue (i.e., waiting time)? In Figure 4a, the arrivals are split into separate queues for each server. In Figure 4b, the two servers share a common queue.

To simulate the process in Figure 4, students use two 4-sided dice for the arrivals and two 6-sided dice for the process capacity. In the case of the Figure 4a setup, each server may draw only from its own queue-whereas either server may draw from the shared queue in the Figure 4b setup. Using the same 4- and 6-sided dice rolls for both configurations allows students to compare the two queuing disciplines directly, and the performance superiority of the 4b configuration will become evident. The instructor can, then, help students to link this result with their real-world experiences of when they have (e.g., banks) and have not (e.g., grocery stores) seen this in action and why each design may have been chosen.

If spreadsheet-based simulation has been introduced previously in the course or curriculum (e.g., as in Johnson and Drougas (2002c) or Johnson (2011)) then students may replicate the results of the in-class simulation in a spreadsheet model. A sample spreadsheet model is available for download at http://domin.dom.edu/faculty/ajohnson/queues.xls.

RESULTS OF THIS STUDY

As discussed earlier, students can typically capture the results of 50-100 periods within 15-20 minutes at which point they can calculate the average queue length and process utilization. In a class of even 20 students, it is possible to test all 10 configurations and to obtain results similar to those shown in Table 1. (Note that the "Arrival" die capacity must be less than the "Process" die capacity for process stability.)

These results may, then, be graphed-as shown in Figure 5-to illustrate the impact of process utilization on the average queue length. While results such as those in Figure 5 are not substantively different from explaining the impact of changes in p (the process utilization) in an equation, Anupindi, et.al. (2012).

[I.sub.i] = [rho][square root of (2(c-1))/[1-[rho]]] x [[[C.sup.2.sub.i]+[C.sup.2.sub.p]]/2], (1)

Evidence of Effectiveness

The primary motivation for the development of the simulation exercise described in this paper was the exam performance of graduate operations management students on queuing-related questions. Specifically, the author used two, multi-part questions (5 questions total) in the vein of those in the instructor's resource guide for Anupindi, et.al. (2012) to assess students' understanding of basic queuing principles. These questions ask students to assess qualitatively and explain the impact on basic queuing measures (e.g., blocking rate, wait time, server utilization, etc.) of various changes in a process' set up (e.g., more servers, fewer available queuing/waiting positions, etc.).

Students' responses were graded, and the number of questions answered completely correctly (i.e., both the direction of and reason for the change) was recorded. As Table 2 shows, the average number of questions answered correctly increased from 2.39 to 3.22 (difference significant with a p-value of 0.0443) after the simulation exercise was introduced within a US-based MBA classroom, with all other subject coverage being equivalent. (The Before/After results for the "US MBA" include two classes taught each way over two academic years.) Based upon this improvement, the simulation exercise was, then, introduced into an executive-format MBA program offered in the European Union, which is shown as "EU MBA" in Table 2. The average number of questions answered correctly increased from 2.72 to 3.85 questions (difference significant with a p-value of 0.0004). In addition, there was a reduction in the variation observed in the number of correctly-answered questions.

CONCLUSIONS

With an experiential learning exercise such as that described in this study, students can develop at least an intuitive understanding of how and why queues form-thereby closing the gap that may exist between the lecture/text treatment of queuing theory and the more advanced results that are often illustrated only in spreadsheet models. It is also worth noting that, even if the results of student learning were not so significantly improved, other arguments could be made to support the use of an experiential learning exercise such as the one presented. These include increasing student engagement on the topic and student satisfaction.

RECOMMENDATIONS FOR FUTURE RESEARCH

While this study focused on student learning in the graduate operations management classroom, the results could be extended-and verified-at the undergraduate level, as well. In addition, the impact/effect of using a spreadsheet-based model could be separately studied.

REFERENCES

Ammar, S., & Wright, R. (1999). Experiential Learning Activities in Operations Management. International Transactions in Operations Research,6(2), 183-197.

Anupindi, R., Chopra, S., Deshmukh, S., Van Mieghem, J., & Zemel, E. (2012). Managing Business Process Flows: Principles of Operations Management, 3rd ed., Prentice Hall, New Jersey.

Goldratt, E., & Cox, J. (1992). The Goal: A process of Ongoing Improvement, 2nd Revisied Edition, North River press, Great Barrington, Massachusetts.

Johnson, A. (2002a). OM and Vegas Night. OR/MS Today, 29(1), 14-15.

Johnson, A. (2002b). OM and Vegas Night. OR/MS Today, 29(2), 14-15.

Johnson, A., & Drougas, A. (2002c). Using 'Goldratt's Game' to Introduce Simulation in the Introductory Operations Management Course. INFORMS Transactions on Education, 3(1), 20-33.

Johnson, A. (2011). Introducing Simulation via the Theory of Records. Decision Sciences Journal of Innovative Education, 9(3), 89-95.

Tommelein, I., Riley, D., & Howell, G. (August 1998). Parade Game: Impact of Work Flow Variability on Succeeding Trade Performance. Proceedings of the 6th Annual Conference of the International Group for Lean Construction,

Arvid C. Johnson

University of St. Francis

Arvid C. Johnson is the President and a Professor at the University of St. Francis in Joliet, Illinois. Prior to this, he was the Dean and a Professor of Management at Dominican University's Brennan School of Business for 5 and 12 years, respectively. Before entering academia on a full-time basis in 2001, he had over 15 years of engineering, manufacturing, and management experience in a variety of business environments-primarily in the defense/aerospace industry. Dr. Johnson has published and spoken extensively in the areas of quantitative analysis, microwave materials processing and electron devices, strategic management, and advanced manufacturing practices.
```
Table 1
Configurations Tested and Sample Results in the Simulation

Maximum Capacity
(# Die Faces)

Arrivals   Processing

4             12
4             10
4             8
4             6
6             12
6             10
6             8
8             12
8             10
10            12

Results

Utilization   Average
Queue

49.7%          0.2
56.8%          0.3
65.0%          0.4
77.7%          0.8
63.8%          0.6
71.8%          0.9
84.0%          2.1
77.0%          1.9
86.7%          3.7
88.1%          6.7

Table 2
Number of Questions Correctly Answered before/after
Introduction of Simulation.

Before Introducing
Simulation

Average   Std. Dev.   n

US MBA    2.39      1.85       18
EU MBA    2.72      1.30       22

After Introducing Simation

Average   Std. Dev.   n

US MBA    3.22      1.13       37
EU MBA    3.85      0.64       25

p-value of
difference

US MBA     0.0443
EU MBA     0.0004
```