Story, Richard M.

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Management Services: A Magazine of Planning, Systems, and Controls Volume 5 Number 1 Article 6 1-1968 Use of Simulation to Solve a Queueing Problem Richard M. Story Follow this and additional works at: https://egrove.olemiss.edu/mgmtservices  Part of the Accounting Commons Recommended Citation Story, Richard M. (1968) "Use of Simulation to Solve a Queueing Problem," Management Services: A Magazine of Planning, Systems, and Controls: Vol. 5: No. 1, Article 6. Available at: https://egrove.olemiss.edu/mgmtservices/vol5/iss1/6 This Article is brought to you for free and open access by eGrove. It has been accepted for inclusion in Management Services: A Magazine of Planning, Systems, and Controls by an authorized editor of eGrove. For more information, please contact egrove@olemiss.edu. Queueing or waiting line theory can involve complex and time-consuming mathematics. However, this is not always essential. Here’s an example of a quick, effective method of solving a queueing problem —THE USE OF SIMULATION TO SOLVE A QUEUEING PROBLEMby Richard M. StoryUniversity of ConnecticutMathematical waiting-line (queueing) theory is being successfully applied to the solution of many business problems involv­ing arrivals (requirements for ser­vice) and service times (accom­plishing the service). Some common examples: machinists standing in line for tools from a tool crib, cars waiting at a toll booth, or produc­tion parts being held up waiting for inspection.Bottlenecks (waiting for service) and idle capacity (waiting to sup­ply service) both incur costs. The staffing of service facilities for the minimum-cost combination requires forecasting the probable combined costs of waiting time and of service availability before service is actu­ally rendered. These costs, how­ever, frequently are difficult to fore­cast.The simplest queueing problem would involve constant arrivals— say, one every five minutes—and uniform service time—say, ten min­utes per “customer.” The answer is readily apparent: Two “servers” per “customer” will result in no waiting for the customers, no idle­ness for the servers, thus minimum cost. Unfortunately, this situation is extremely rare.More commonly arrivals and ser­vice times are both variable. Some­times the patterns of distribution of arrivals and service times fit cer ­tain standard statistical distribu­tions, for example, Poisson arrivals and negative exponential service times. When this is found to be the case, existing equations and tables can be used to determine the pertinent data.There is still a third category of queueing problem, the situation wherein arrivals and service rates58 Management Services1Story: Use of Simulation to Solve a Queueing ProblemPublished by eGrove, 1968ELAPSED TIME IN MINUTESDistribution of Time Between Arrivals of OperatorsINSPECTION TIME IN MINUTESDistribution of Inspection TimesFIGURE 2FIGURE IELAPSED TIME IN MINUTESCumulative Distribution of Time Between Arrivals of OperatorsFIGURE 3Cumulative Distribution  of Inspection TimesFIGURE 4are neither uniform nor in con­formity with standard distributions. A valuable tool in the examination of this type of problem is simula ­tion,1 a method of duplicating a complicated operation by a set of rules and computations so that al­ternative decisions may be observed in action. This article presents an 1 For other examples of the use of simu­lation see “Simulation in Financial Plan ­ning” by E. N. Khoury and H. Wayne Nelson, M/S, March-April ’65, p. 13, and “Using Simulation to Design a Man ­agement Information System” by Adolph F. Moravec, M/S, May-June ’66, p. 50.2 The Monte Carlo method is discussed  in detail in “Setting Inventory Reorder Points” by Felix A. McCameron, M/S, May-June ’65, p. 25.example of the application of simu­lation via the Monte Carlo method,2 a form of  simulation in which ran ­domly chosen numbers determine the course of the computation.The XYZ Company is a company engaged in intermittent manufac­ture to stock. Material in process is checked at centralized inspec­tion stations strategically located throughout the plant. Rather widely varied inspection operations are performed by the inspectors man­ning the inspection stations. Be­cause of the relatively long dis­tances traveled by the operators to the inspection stations and the rela­tively short time usually required for the inspections, it is the prac­tice of the operators to wait for the work to be checked. The op­erators arrive at the station and are attended to on a first-come- first-served basis.The quality manager notices that operators frequently wait in line at one of the inspection stations.January-February, 1968 59 2Management Services: A Magazine of Planning, Systems, and Controls, Vol. 5 [1968], No. 1, Art. 6https://egrove.olemiss.edu/mgmtservices/vol5/iss1/6TABLE ISample of Time Values for Arrivals and InspectionsRandom No.Time Since Last Operator Arrived  (Fig. 3) min.Random  No.Inspection Time (Fig. 4) min.40 5 43 516 4 55 654 6 94 846 5 74 7174 23 456 6 12 209 3 43 510 3 81 781 7 47 673 7 95 864 6 23 449 6 49 674 7 59 636 5 18 312 4 71 750 6TABLE 2Simulation of Conditions at Inspection Station—1 InspectorOperators Arrive atInspection  Begins atInspection  Ends atOperator Waiting TimeInspector Idle TimeNumber of  Operators Waiting8:00 a.m. 8:00 a.m. 8:05 a.m. 0 min. 0 min. 08:05 8:05 8:11 0 0 08:098:118:19 2 0 18:15 8:19 8:26 4 0 28:20 8:26 8:30 6 0 28:24 8:30 8:32 6 0 38:30 8:32 8:3720 28:33 8:37 8:44 4 028:36 8:44 8:5080 . 3  8:43 8:50 8:58 7 0 38:50 8:58 9:02 8 0 28:56 9:02 9:08 6 0 39:02 9:08 9:14 6 029:09 9:14 9:17 5 0 29:14 9:17 9:24 3 0 29:18 9:24 9:30 6 02Duration of sample: 90 min. Operators' waiting time: 73 min.TABLE 3Sample of Time Values for Arrivals and Inspections—2 InspectorsRandom No.Time Since Last Oper.  Arrived* min.Random  No.Inspection Time Inspector 1** min.Random  No.Inspection Time Inspector 2** min.78 7 46 633 5 68 758 6 26 427 4 96854 698 8 82 7 87 785 8 72 752 6 65 648 6 96824 4 82 703 1 33 595 9 82795 9 3 755 6 01 103 1 213122*From Fig. 3; **From Fig. 4He thinks an additional inspector might relieve the condition and consequently cut the cost of idle operator time. He asks one of his staff engineers to study the situa­tion.SimulationThe engineer first collects data showing the characteristics of ar­rivals and service times. Both, he observes, are randomly distributed. The distributions are shown in Fig­ure 1 on page 59 and Figure 2 on the same page.Testing for fit, he finds that nei­ther set of data approximates any of the standard distributions (Pois­son, exponential, etc.). Conse­quently, the use of Monte Carlo simulation is indicated.In order to adhere to the fre­quency patterns shown to exist by the histograms (Figures 1 and 2) and at the same time to permit simulation of the randomness of arrivals and service times, he con­structs cumulative distributions from the original histograms and  converts the frequencies to per­centages. The results are shown in Figure 3 and Figure 4 on the pre­ceding page.He now samples randomly from the cumulative distributions to se­lect specific arrival times and ser­vice responses to employ in simu­lating the inspection station opera­tion. The dotted lines in Figures 3 and 4 illustrate how this is done. A sample of resulting time values is  presented in Table 1, this page.With the information derived in Table 1, the engineer can now simulate the operation of the in­spection station. The results are shown in Table 2 on this page. Operator waiting time over an elapsed period of 90 minutes totals 73 minutes. (Waiting time does not include the time required for the inspection after the inspector is "waited on.”)The operators’ average hourly rate is $4. On the basis of this rate and the results of the simulation, he makes the following calcula­tions:60 Management Services 3Story: Use of Simulation to Solve a Queueing ProblemPublished by eGrove, 1968TABLE 4Simulation of Conditions at Inspection Station—2 InspectorsOperatorsArrive at:InspectionBegins at:Inspection Ends at:Performed byInspector No.:Operator Waiting Time min.Inspector 1 Idle Time min.Inspector 2 Idle Time min.8:00 a.m. 8:00 a.m. 8:06 a.m. 1 0 0 68:07 8:07 8:14 2 0 6 18:12 3:12 8:16 1 0 0 28:18 8:18 8:26 2 0 6 28:22 8:22 8:30 1 0 0 28:28 8:28 8:35 2 0 5 08:35 8:35 8:42 1 0 0 78:43 8:43 8:49 2 0 7 18:49 8:49 8:57 1 0 0 68:55 8:55 9:02 2 0 2 08:59 8:59 9:04 1 0 0 09:00 9:02 9:09 2 2 5 09:09 9:09 9:16 1 0 0 79:18 9:18 9:19 2 0 3 29:24 9:24 9:27 1 0 5 69:25 9:25 9:27 2 0 0 0Duration of sample: 87 min.Operators' waiting time: 2 min.Two-inspector simulationThe cost of waiting time (nearly $130 a week) appears excessive. Perhaps an additional inspector is needed. To determine whether this expenditure is justified, the engi­neer performs a second simulation, using the original data, with two inspectors manning the inspectionRICHARD M. STORY is associate professor of in­dustrial administration at the University of Con­necticut. A graduate of the University of Ver­mont, he received his M.S. degree from Le­high University, Bethle­hem, Pa., in 1948. Mr.Story is co-author of the textbook Managing Manpower in the Industrial Environment and the author of many articles on various sub­jects pertaining to industry. He is a member of the American Society for Quality Control, the American Production and Inventory Con­trol Society, and the American Association of University Professors.station. Now the total operator waiting time over an elapsed period of 87 minutes is cut to two minutes. The results of this second simula­tion are shown in Table 3 on page 60 and Table 4 on this page.The inspectors’ average hourly rate is $3.85. On the basis of this rate and the results of the simula­tion, the engineer makes the follow­ing calculations:Operator waiting time per 8-hourare far from adequate to produce results yielding a reasonable de­gree of confidence. In actual prac­tice, longer runs (say, a full eight hours) would be simulated, and at least fifty iterations would be conducted for each of the situa­tions—one inspector and two in­spectors.This would be a formidable task to perform by hand. Fortunately, appropriate computer programs are available. Once the original data have been assembled, a com­puter can make short work of the many iterations required.Where waiting lines with irregu­lar arrivals are concerned, casual observation can often be mislead­ing. In the example described in this article it seemed to show the desirability of adding another in­spector; testing this action by simu­lation kept the quality manager from making a mistake that could have cost the company about $1,500 a year at one inspection station alone. The technique is equally ap­plicable to many other problems, ranging all the way from machine maintenance and truck terminal design to timing of traffic signals and scheduling of patients in hos­pital clinics.January-February, 1968 61Operator waiting time per 8-hourday = 48090X 73 = 389 minutes.Cost of waiting time perday = 38960 x $4 = $25.93.day = 480 87 X 2 = 11 minutes.Cost of waiting time perday = 1160 X $4 = $0.73.Cost of added inspector per day =8 X $3.85 = $30.80.Therefore, the total daily cost with two inspectors is $0.73 + $30.80, or $31.53. Comparison of this cost with the previous one ($25.93) indicates that the existing arrangement, with one inspector, is more economical and that the ad­dition of another inspector is not justified.ApplicationNaturally, two simulations based on runs totaling less than two hours4Management Services: A Magazine of Planning, Systems, and Controls, Vol. 5 [1968], No. 1, Art. 6https://egrove.olemiss.edu/mgmtservices/vol5/iss1/6

Use of Simulation to Solve a Queueing Problem

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Use of Simulation to Solve a Queueing Problem

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