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Simulation and queuing models for resoource allocation in the UBC Libraries Knudsen, Cindy 2005

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SIMULATION A N D QUEUEING MODELS FOR RESOURCE A L L O C A T I O N IN THE U B C LIBRARIES by CINDY K N U D S E N - Bachelor of Science (Mathematics and Statistics), University of Regina, 2000 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF SCIENCE IN BUSINESS ADMINISTRATION in THE F A C U L T Y OF G R A D U A T E STUDIES ( M A N A G E M E N T SCIENCE) THE UNIVERSITY OF BRITISH C O L U M B I A April 2005 © Cindy Knudsen, 2005 A B S T R A C T This thesis describes two projects completed at the Centre for Operations Excellence (COE) for Darrell Bailie Manager, Finance and Facilities, and Dwight Tanner, Administrator, Special Projects of UBC Libraries. The cost of maintaining a sizable collection of reference material facilitates the need for U B C Library branches to be more cost effective. This thesis studies two services the library offers with this goal in mind. The first study considers the photocopier service available at Koerner Library. There are copy rooms on four floors of the library with different levels of demand. Furthermore, the peak demand for the photocopiers occurs for only a short period. Queueing theory and simulation are used to model the system to determine the optimal number of photocopiers needed in each copy room to meet the peak demand, while still maintaining an acceptable service level. This thesis shows a reduction in photocopiers is feasible for Koerner Library and offers two types of charts to use when making decisions on size and allocation of the photocopier fleet. The first chart quantifies changes in the mean time in the system (i.e. the mean waiting time and service time), the mean time in the queue, the mean queue length, and the utilization of the photocopiers when the number of photocopiers on a floor is decreased or increased from the current number of photocopiers. The second chart indicates how many photocopiers are needed to meet a certain level of customer service, for each Copy room. The second study considers the services offered at circulation desks in Koerner Library, Woodward Biomedical Library, and David Lam Library and the self-charge machines at Koerner Library. The demand for their circulation desks is variable not only throughout the day, but also throughout the week and the year. Demand is assessed and grouped into similar periods. Queueing theory is used to determine a staffing schedule to satisfy the demand for each of these periods, while still maintaining an acceptable service level. With the self-charge machine, a patron can check out his/her own material, thereby circumventing the circulation desk. This thesis shows the reliability of the machines is very limited and a large number of patrons will have to continue their transactions at the circulation desk. TABLE OF CONTENTS A B S T R A C T T A B L E O F C O N T E N T S I I L I S T O F T A B L E S V L I S T O F F I G U R E S V I A C K N O W L E G E M E N T S V I I I I N T R O D U C T I O N . I I L I T E R A T U R E R E V I E W I I I P H O T O C O P I E R A N A L Y S I S III. 1 METHODS : III. 1.1 Data Collection III. 1.2 Analysis of Status Quo III. 1.2.1 Graphs .-III. 1.2.2 Utilization III. 1.2.3 Demand Variation III. 1.3 Analysis Of Service Times 1 III. 1.3.1 Outlier Test For Tuesday 1 III. 1.3.2 Outlier Test For Floor 1 Tuesday III. 1.3.3 Determining Service Time Best-Fit Distributions III. 1.3.4 Determining Service Time "Worst Case" Distributions 1 III. 1.3.5 Determining Service Time "Mean Case" Distributions III. 1.4 Analysis Of Inter-Arrival Times 1 III. 1.4.1 Determining Inter-Arrival Time Best-Fit Distributions 1 III. 1.4.2 Determining Inter-Arrival Time "Worst Case" Distributions 1 III. 1.4.3 Determining Inter-Arrival Time "Mean Case" Distributions 1 ///. 7.5 Analysis Of Mean Arrival Rate and Mean Service Rate 1 III. 1.6 Queueing Models and Simulation Models 2 III. 1.6.1 Assumptions for Queueing Model 2 III. 1.6.2 Notation for Queueing Model 2 III. 1.6.3 Formulas for Queueing Model 2 III. 1.6.4 Inputs for Queueing Model 2 III. 1.6.5 Simulation 2 111.2 RESULTS 2 III. 2.1 Analysis of Status Quo system :. 2 III. 2.2 Queueing Model ; 2 111.2.2.1 Queueing Analysis of "Worst Case" 2 111.2.2.2 Queueing Analysis of "Mean Case" 2 III.2.3 Simulation 2 111.3 INTERPRETATION OF RESULTS 2 111.4 S U M M A R Y 3 III. 4.1 Comments 3 III.4.2 Further Work : 3 I V H U M A N R E S O U R C E S A N A L Y S I S 33 IV. 1 M E T H O D S 33 IV. 1.1 Data Collection 34 I V . l . 1.1 Transformation of Data 35 IV. 1.2 Analysis of Status Quo 36 I V.l. 3 Development of Baseline 36 IV.1.4 Period Formulation 37 IV. 1.4.1 Grouping of Like Months 37 IV. 1.4.2 Grouping of Like Days of the Week 38 IV. 1.4.3 Grouping of Like Hours of the Day 39 IV. 1.5 Calculation of Arrival Rates 41 IV. 1.6 Development of Staffing Rules 42 IV. 1.7 Comparison and Sensitivity Analysis 42 IV. 1.8 Self-Charge Assessment 42 IV.1.8.1 Data Collection for Self-Charge Machines......... 43 IV. 1.8.2 Assessment of Reliability By Machine 44 IV. 1.8.3 Assessment of Reliability By Scan Barcode Placement 44 IV. 1.8.4 Determination of Interaction Effect 44 IV.2 R E S U L T S 46 IV.2.1 Status Quo 46 IV. 2.2 Baseline 46 IV.2.3 Periods 46 IV.2.3.1 Grouping of Like Months .46 IV.2.3.2 Grouping of Like Days of the Week 48 IV.2.3.3 Grouping of Like Hours of the Day 50 IV.2.3.4 Summary of Groupings Per Branch 52 IV. 2.4 Arrival Rates 53 IV. 2.5 Results of Queueing Analysis 55 IV.2.5.1 Staffing Rules 55 IV. 2.6 Comparison and Sensitivity Analysis 57 IV.2.6.1 Change in Percentage of Renewal Transactions 57 IV.2.6.2 Change in Percentage of Money-Collected Transactions 58 IV.2.6.3 Change in Average Number of Transactions Per Client 59 IV.2.6.4 Change in Service Time 59 IV. 2.7 Self-Charge Machines 60 IV.3 I N T E R P R E T A T I O N O F R E S U L T S 62 IV.4 S U M M A R Y 64 IV.4.1 Comments 64 IV.4.2 Further Work : 64 R E F E R E N C E S 66 iv APPENDIX.... 68 I Event-Driven Graphs 68 II Time-Driven Graphs 71 III Service Time Graphs 85 IV Probability Density Function for an Exponentional distribution ,. 86 V Probability Density Function for a Shifted Exponentional Distribution 86 VI Chi Square Test s 87 VII Kolmogorov-Smirnov Test 88 VIII Outlier Test 89 IX Service Time Data for Tuesday 90 X Box Plot for Tuesday 91 XI Service Time Data for Floor 1 Tuesday 91 XII Box Plot for Floor 1 Tuesday 92 XIII Arrival Rate Graphs 92 X I V Inter-Arrival Time Graphs 94 X V Queueing Analysis "Worst Case" Charts 96 X V I Queueing Analysis "Worst Case" Matrices 97 XVII Queueing Analysis "Mean Case" Charts 98 XVIII Queueing Analysis "Mean Case" Matrices 99 X I X Simulation comparison and screen shots 100 X X Monthly Graphs 103 X X I Daily Graphs 107 XXII Comparison of Proportions of Two Independent Samples 110 XXIII Binary Categorical Logistic Regression 111 X X I V Minimum Number of Servers Charts For Baseline 112 X X V Mean Waiting Time In Queue Charts For Baseline 115 X X V I Minimum Number of Servers Charts For Period Combinations 121 X X V I I Mean Waiting Time in Queue Charts for Period Combinations 122 XXVIII Comparison of Minimum Number of Servers 124 v LIST O F T A B L E S Table 1: Classification of Usage Levels of Each of the Three Copy Rooms 9 Table 2: Service Time Best-Fit Distributions : 12 Table 3: Service Time "Worst Case" Distributions 13 Table 4: Probabilities of Shifted Exponential Distributions 13 Table 5: Service Time Non-Shifted Exponential "Worst Case" Distributions 14 Table 6: Floor/Day Service Time Distributions , 14 Table 7: Service Time "Mean Case" Distributions 15 Table 8: Probabilities of Shifted Exponential Distributions 15 Table 9: Service Time Non-Shifted Exponential "Mean Case" Distributions 15 Table 10: Scenario Sets for Analysis of Inter-Arrival Times 16 Table 11: Inter-arrival Time Best-Fit Distributions 17 Table 12: Inter-arrival Time "Worst Case" Distributions 17 Table 13: Floor/Day Inter-arrival Time Distributions 18 Table 14: Inter-arrival Time "Mean Case" Distributions 18 Table 15: Probabilities of Shifted Exponential Distributions 19 Table 16: Inter-arrival Time Non-Shifted Exponential "Mean Case" Distributions 19 Table 17: Fraction of Use of Each Usage Level for Each of the Three Copy Rooms 24 Table 18: Fraction of Use Between 8:30 and 10:30 of Each Usage Level for Each of the Three Copy Rooms 25 Table 19: Total Arrivals By Month For Koerner Library 37 Table 20: Total Arrivals By Month For Woodward Biomedical Library 37 Table 21: Total Arrivals By Month For David Lam Library 38 Table 22: Total Arrivals By Day Of The Week For Koerner Library 38 Table 23: Total Arrivals By Day Of The Week For Woodward Biomedical Library 39 Table 24: Total Arrivals By Day Of The Week For David Lam Library 39 Table 25: Total Arrivals By Hour Of The Day For Koerner Library 40 Table 26: Total Arrivals By Hour Of The Day For Woodward Biomedical Library 40 Table 27: Total Arrivals By Hour Of The Day For David Lam Library 41 Table 28: The Number Of Successes For The Self-Charge Machines 43 Table 29: The Number Of Successes For The Self-Charge Machines In Batches Of 10 44 Table 30: Breakdown Of Periods For Koerner Library 52 Table 31: Breakdown Of Periods For Woodward Biomedical Library 53 Table 32: Breakdown Of Periods For David Lam Library 53 Table 33: Summary Of Calculation Process For Koerner Library 54 Table 34: Summary Of Arrival Calculation Process For Woodward Biomedical Library... 54 Table 35: Summary Of Arrival Calculation Process For David Lam Library 55 Table 36: Staffing Rules For Koerner Library 56 Table 37: Staffing Rules For Woodward Biomedical Library 56 Table 38: Staffing Rules For David Lam Library 57 Table 39: Range Of Allowable Change In Percentage Of Renewal Transactions 58 Table 40: Range Of Allowable Change In Percentage Of Money-Collected Transactions.. 58 Table 41: Range Of Allowable Change In Average Number Of Transactions For A Client 59 Table 42: Range Of Allowable Change In Service Time 60 v i LIST O F FIGURES Figure 1: Current Set-up of the CopyRoom on Floor 1 at Koerner Library 6 Figure 2: Current Set-up of the Copy Room on Floor 2 at Koerner Library 6 Figure 3: Current Set-up of the Copy Room on Floor 4 at Koerner Library 7 Figure 4: Number of Users on Floor 2 10 Figure 5: Mean Arrival Rate and Mean Service Rate For Floor 2 19 Figure 6: Estimated Marginal Means Of The Probability Of Success 45 Figure 7: Total Arrivals By Month For Koerner Library 47 Figure 8: Total Arrivals By Month For Woodward Biomedical Library 47 Figure 9: Total Arrivals By Month For David Lam Library 48 Figure 10: Total Arrivals By Day Of The Week For Koerner Library 49 Figure 11: Total Arrivals By Day Of The Week For Woodward Biomedical Library .... 49 Figure 12: Total Arrivals By Day Of The Week For David Lam Library 50 Figure 13: Total Arrivals By Hour Of The Day For Koerner Library 50 Figure 14: Total Arrivals By Hour Of The Day For Woodward Biomedical Library 51 Figure 15: Total Arrivals By Hour Of The Day For David Lam Library 52 ACKNOWLEGEMENTS First and foremost, I would like to thank NSERC and the Centre for Operations Excellence for the financial support that allowed me to pursue my MSc degree. This aid during my studies was greatly appreciated. I learned a lot while working on the two projects for UBC Libraries. I would like to thank the COE for giving me the extraordinary opportunity to work on the applied projects for my thesis. I would like to thank Dr. Martin Puterman and Dr. David Glenn for all of the support, encouragement, and advice they have freely given to me throughout both of these projects. Without their contributions to both the projects and the thesis, all might not have been completed successfully. I would also like to thank Darrell Bailie and Dwight Tanner from U B C Libraries for providing me with two challenging projects and their trust and support in me straight through to project completion. Finally, I would like to thank my friends, family, and colleagues for putting up with me while I completed my MSc. Their support and understanding was greatly appreciated. I would also especially like to thank Dr. Judith McDonald who convinced me that I could and should do my Masters. viii / INTRODUCTION This thesis describes two projects completed at the Centre for Operations Excellence (COE), an applied research centre in the Faculty of Commerce at the University of British Columbia, for Darrell Bailie Manager, Finance and Facilities, and Dwight Tanner, Administrator, Special Projects of U B C Libraries. The U B C Library is a conglomerate of 10 branches throughout the University of British Columbia Campus (three of which are Koerner Library, Woodward Biomedical Library, and David Lam Library), as well as 4 off campus. The branches contain books, journals and series subscriptions, maps, videos, microforms, plus other material in their collections. The aggregate size of the collection is substantial, making the U B C Library one of the largest research libraries in Canada. The branches not only offer a diverse collection of information, they also provide web-based services, research advice and instruction, as well as other essential services such as photocopying and printing. The cost of maintaining a sizeable collection, such as the one found at U B C Libraries, and the various services offered facilitates the need for the branches to be cost effective in the services they provide. This thesis considers two types of services for a subset of the 14 branches. The first study considers the photocopier service available at Koerner Library. A large number of reference materials, such as books and journals, are not available for use outside of the branch in which they are stored. The branches do, however, allow photocopying of the material for external use and provide photocopiers for this occasion. Patrons may also use the photocopiers for purposes other than this. Koerner Library has 19 photocopiers available for public use and the demand is extremely variable. Furthermore, Koerner Library experiences peak demand of its photocopiers for only a transitory period. The focus of this project is to analyze the usage of the photocopiers in Koerner Library and to determine the number of photocopiers needed for the branch to meet peak demand. The second study considers the services offered at circulation desks in Koerner Library, Woodward Biomedical Library, and David Lam Library. The primary function of staff manning the circulation desk in a UBC Library branch is to facilitate the checkout of allowable items, renew items already on charge, discharge material on loan, and collect money for fines. Certain branches also have a self-charge machine. With this machine, a patron can check out his/her own material, thereby circumventing the circulation desk. Each of the three branches studied contain a circulation desk that experiences variable demand within the day, the week, and the year. Furthermore, Koerner Library and Woodward Biomedical Library contain self-charge machines that help alleviate the demand of the circulation desks. The focus of this project is to analyze the demand of the circulation desks in Koerner Library, Woodward Biomedical Library, and David Lam Library to determine staffing rules for periods of different levels of demand and to assess the reliability of the self-charge machines located in Koerner Library and Woodward Biomedical Library. 1 // LITERA TURE REVIEW Most literature relating to analysis of photocopying or photocopiers within a library setting falls into three categories: study of inter-library loan photocopy requests, analysis of copyright issues, and evaluation of photocopiers. Studies of the first type address the volume of photocopies made by request and the quality of those photocopies. Studies of the second type discuss photocopying and how it relates to copyright issues. An example of this is seen in Morton (1996) and Rasmussen (1990). The former gives an excerpt of a question and answer period regarding self-serve photocopiers and copyright issues, while the latter discusses how the availability of both photocopiers and video/recorders make copyright infringement profitable and why libraries must be vigilant about copyright infringement. Studies of the third type, as seen in Swart (1995) and McKern (1989), assess different photocopiers that are available to meet the needs of the libraries. Though all three areas are interesting, none are similar to the study done for this thesis. There are a few studies done on queueing disciplines in a university library setting. Most studies separate different areas of the library where queues may occur, such as circulation, reference desk, photocopier, etc., and study each of these queueing problems separately. This approach is similar to that taken in the project for this thesis. A few articles, such as Wilson (1990), study the usage of photocopiers from a qualitative perspective. This is usually accomplished by surveying users or librarians to determine i f the available services are meeting demand adequately. Responses for surveys give the perception of operation standards experienced by users, which may differ from that which was actually experienced. Many studies done through survey also allow users to give suggestions as to how changes in operations could result in higher user satisfaction. Although surveys are mainly used for analysis of photocopier usage within a library, this method of data collection gives more of an anecdotal view of self-serve photocopier operations, and his highly susceptible to biases when considering cost and optimality. Smith (1981) investigates the use of mixed integer programming, along with queueing networks and utility theory, to optimally allocate resources within a library. This paper considers the layout of a library for the optimization of staff, equipment, and space and uses a comprehensive approach to attack the resource allocation problem. The article gives a methodology for allocation of resources within a library layout and it considers a particular example with statistics from Champaign Public Library. This approach to solving a library resource allocation problem considers the demand of all resources found in a library, whereas the thesis considers the demand of only two resources separately. Hence, the paper gives a higher-level approach of solving the resource allocation problem than is needed for the study of this thesis. While the methodology cited in this paper is not applicable to the project for this thesis, it does show the extent to which the resource allocation problem can be considered. Guo (2003) develops a queueing model and a simulation model to determine how many workstations are needed in three of the branches of University of British Columbia Library. Guo considers periods of different levels of usage and determines the minimum number of 2 workstations needed to meet the criteria. The workstation problem Guo presents is similar to the photocopier allocation problem described in this thesis. Both deal with non-movable servers and variable demand. They both also use queueing theory to establish a solution. The arrival rates for the study done by Guo are unknown and are estimated. This is not the case for the photocopier analysis. Both service time and arrival rates are observed during a peak period. In the paper by Guo (2003), regression, queueing, and simulation models are used to develop a set of staffing rules for the reference desks of three branches of U B C Library. Guo determines the correlation between circulation charge data and reference data. This correlation is used to derive the arrival rate for the reference desks for the queueing model and the simulation model. Guo presents staffing rules stating how many staff should be at the desk to meet the service criteria. A similar approach is presented in this thesis to determine the staffing rules for the circulation desk. However, no correlation between reference desk data and circulation data is needed as direct calculations from the electronic data produce a reasonable estimate of arrival rates for the circulation desk. A study and survey by Alston (1996) investigates the problem of queues at Chelmsford Public Library located in Chelmsford Essex in the U K . The purpose of that study was to determine and maintain an operational standard of queue times using existing resources within the library. The author considers a comprehensive approach that contains both theoretical and practical steps in solving the problem. The main theoretical tool used to solve the problem was queueing theory. The practical issues needed to solve the problem considered such things as changes to operations in a way that reduces idle time of a server and allows extra servers to be available i f the situation calls for it. This study is similar to the circulation desk project done for UBC Libraries. Alston had the opportunity to collect data on the discharge counter in the Chelmsford Library, whereas data collection on the circulation desk was not available for the project done for this thesis. As a result, the benchmarking methods and the formulation of the inputs for the queueing models used in Alston's study were not feasible for this project. A novel approach needed to be taken to compensate for these shortcomings. Alston had a similar viewpoint regarding accuracy versus usefulness of a model. It was felt that a simple solution that is operationally easy to implement was much more beneficial than one that was extremely accurate. Both projects incorporated this viewpoint in solving their respective queueing problem. The U B C Library project and Alston's study both determined staffing levels during different time-periods. However, Alston's study only considers variable periods throughout the day and throughout the week. The U B C Library project has variability in demand throughout the day and throughout the week, but also throughout the year. Alston compensates for the limitations of queueing theory, such as the assumption that the system is in steady state, by changing operations via allotting extra servers during peak periods. This was incorporated into the author's solution to allow for a reduction in idle time of a server. This approach was not needed in the project done for this thesis as the staffing levels indicate the minimum number of staff needed at the desk to meet the required service level. Any idle time experienced by these personnel would be utilized in undertaking other duties that can be accomplished at the circulation desk. The final major difference between the study done by Alston and the study done for this thesis is that Alston investigates the problem of queues in a public library 3 setting, whereas the project for this thesis is concerned with the problem of queues in a university library setting. The study done by Alston is reassuring as it demonstrates successful application of queueing theory to solve a queueing problem within a library setting. Mansfield (1981) correlates and contrasts queueing theory and queueing behaviour within an academic library, pointing out limitations of queueing theory in this area of application. The author discusses psychological factors of queueing behaviour and suggests ways to minimize queueing frustration experienced by both clients and staff. Mansfield makes the generalized statement that a library client will not stand waiting in a single line for the next available server, but will approach any server at the circulation desk. At UBC Libraries, this is not true. In some branches, there are queueing guide ropes in place that enforce single line queueing. At other branches, clients inevitably queue up in a single line and the server waits on the next client in the line. As a result, there is single line queueing for the next available server. The author also makes the statement that, since there are not always clients waiting to be served, a single queue is not appropriate for libraries. Again, this observation does not hold for UBC Libraries. During times when there is an idle server, there are other duties that server can do while at the circulation desk. Thus, queue length does not matter. Mansfield reveals an interesting point in that it might be advantageous to prioritize clients based on their service time. That is, if a client's service time is expected to be lengthy, it might be beneficial to delay service to that client and to assist a few clients with shorter service times before returning to complete the transaction of the first client. Although the author discusses queueing problems and queueing theory within a library, there is no solution sought regarding staff scheduling. A study by Ashley (1995) utilizes Lotus 123 and linear programming to develop weekly staffing schedules for a university library's reference and circulation desk. The paper captures the essence of the strategy that is employed by most research when solving staff scheduling problems for a university library; that of actually determining the staff schedule. This is inherently different from the project done for the thesis as the project does not determine the actual schedule, but determines how many personnel are required at the circulation desk during different time-periods. In the paper, the author already knows the staffing requirements that need to be met in a time-period, whereas that was to be determined in the project for UBC Libraries. The model developed by Ashley is currently being used at the University of Kansas at the Government Documents and Map Library for staff scheduling at the circulation and reference desks. Warwick (1998) takes on a different approach in using queueing theory in a library setting. The author gives an extension to models proposed by Morse (1968) that employs queueing theory to model circulation behaviour. Demand occurs when a client arrives to look for a book and the book is the service required. In this setting, the elapsed time that the client borrows the book from the library is the service time. Unsatisfied demand creates a queue in the case that a reservation is made for the book. Warwick notes that there are two types of arrival processes: the arrival rate of clients to borrow the book and the arrival rate of clients to the queue in the case that the book is not available. The author develops a model with the latter in mind that describes book circulation and reservation process. Applications of this 4 model could be utilized to determine optimal levels of copies of books. Furthermore, the model can be used to determine the effects of loan period and number of available copies of the book on client satisfaction and reservation system activities. Although the model is still in development, it illustrates that client behaviour can be captured when studying a queueing system within a library. Similarities between this problem and that of the circulation desk problem indicate that client behaviour can also be captured when studying a queueing system of a circulation desk. The project for the thesis was not concerned with client behaviour at this point, but the paper did show a successful application of queueing theory in a library setting and a possible approach to determine optimality taking into account client behaviour. 5 /// PHO TOCOPIER ANAL YSIS Darrell Bailie and Dwight Tanner of UBC Library noted many of the photocopiers in Koerner Library remained idle and felt there are possibly too many photocopiers on hand. In Koerner Library, there are ten photocopiers in the copy room on the first floor, seven photocopiers in the copy room on the second floor, one photocopier in the copy room on the fourth floor, and one photocopier in the copy room on the fifth floor. The photocopiers analyzed in this project are located on floor 1, floor 2, and floor 4. The photocopier on the fifth floor is not used in the analysis as the set-up of the photocopy room on the fifth floor is similar to the set-up of the fourth floor and annual counter readings of the two photocopiers are similar. It is therefore assumed the analysis of the fourth floor copy room is transferable to the photocopier in the fifth floor copy room. The copy room on the first floor of Koerner Library contains ten photocopiers placed throughout the copy room. The placement of these photocopiers within the copy room is shown in Figure 1. Photocopier Figure 1: Current Set-up of the Copy Room on Floor 1 at Koerner Library Figure 2 illustrates the placement of the seven photocopiers contained in the copy room on the second floor. Photocopier Figure 2: Current Set-up of the Copy Room on Floor 2 at Koerner Library 6 The copy room on the fourth floor of Koerner Library contains one photocopier, as illustrated in Figure 3. Photocopier Figure 3: Current Set-up of the Copy Room on Floor 4 at Koerner Library Queueing Theory and Simulation are used to determine the number and allocation of photocopiers in the three copy rooms of Koerner Library during a peak demand period. Furthermore, a series of alternatives for the quantity of photocopiers, as well as other qualitative recommendations, is determined to assist U B C Library's management in decisions regarding reducing operational costs of this type of service. The development of the model of the queueing system for the photocopiers at Koerner Library required four steps. First, necessary data on the queueing system was identified and collected. Second, an understanding of the status quo was established based on available data. Third, the arrival rates and service rates were estimated. The final step included the development of an analytical model and a simulation model of the queueing system. III. 1.1 DATA COLLECTION The data necessary for this analysis consists of the arrival time, the departure time, the identification of the photocopier used, and the number of photocopies made by each user. The collection process included one data collector observing the entire copier room. The observer had a structured data sheet to record the required information for each user in the copy room. Data collection occurred on Tuesday March 13, 2001 from 8:30 to 16:30, Wednesday March 14, 2001 from 8:30 to 16:30, Thursday March 15, 2001 from 10:30 to 14:30, Friday March 16, 2001 from 10:30 to 14:30, and Monday March 19, 2001 from 8:30 to 14:30. Based on discussions with Darrell Bailie and Dwight Tanner of UBC Libraries, the time frame was chosen with the assumption that it is representative of the peak period for photocopier usage at Koerner Library. From historical facts and figures maintained by U B C Libraries, this period within the semester was the busiest throughout the year. This assumption is deemed reasonable, as there was no reason to expect 2001 year will depart from the historical trend. III. I METHODS 7 Thus, the chosen data collection period adequately reflects peak demand experienced at Koerner Library. It was also assumed that the method of data collection would not alter the true value of the data. One way to ensure this was to minimize contact between the data collector and a user as the user was going through the system. The method of data collection used in this study was unobtrusive, as the data collector observed inconspicuously and noted the required information without interfering with the users' access or use of the photocopiers. Thus, user behaviour was not altered by the data collection method chosen. Based on the collected data, two variables were derived: the service time per user and the inter-arrival time between users. The service time data refers to the length of time a user makes use of a photocopier and was calculated by subtracting the arrival time from the departure time. The inter-arrival data refers to the length of time between arrivals of successive users and was calculated by subtracting the arrival times of two consecutive users. III. 1.2 ANAL YSIS OF ST A TUS QUO Three elements were employed for the development of a clear understanding of the status quo. A series of graphs that indicate the activity levels within all of the copy rooms during the period of data collection were created. The utilization of the photocopiers within the copy rooms was determined and these utilization rates were compared for each room at different periods throughout the day. A demand pattern throughout the entire week was established for each copy room. III.1.2.1 Graphs To aid in the understanding of the status quo, two types of graphs were used. Both types indicate how many copiers were in use throughout the day. The first type of graph uses an event-driven approach. Every change in the graph represents either a user arriving or a user departing the system, and the graph changes only when one of these two events occurs. Each graph is representative of the photocopying activity of one floor on one particular day. The graphs for each floor throughout the entire data collection period appear in Appendix I. The second type of graph uses a time-driven approach. Every change in the graph is a moment in time, with the number of users in the room depicted at that point in time. Each graph is representative of the photocopying activity of one floor on one particular day. The graphs of each floor throughout the entire data collection period are in Appendix II. 8 III.1.2.2 Utilization Another component in the understanding of the status quo is to consider the variability in the activity levels for the copy rooms on each of the floors throughout the day and determine the utilization of the photocopiers. The results of this study can be used to enhance UBC Library managements' understanding of the queueing system. One of the practical implications would be the transferability of the results to other analyses (such as the feasibility of variable pricing) they may wish to administrate. Three levels of usage were established for each copy room: very busy, moderate, and light. The portion of time in which a copy room is one of these levels was determined for each day of data collection. Utilization for each level was calculated as the portion of time a copy room is in a certain level during that day divided by the length of time the copy room is under observation that day. The thresholds for very busy, moderate, and light vary by floor since the levels of activity are different for each floor. Thus, different standards are needed for each copy room. Table 1 shows the threshold (number of users in each of the copy rooms) chosen for all usage levels. Floor # Copiers Very Busy Moderate Light 1 10 >7 4 - 7 0 - 3 2 7 >5 3-5 0 - 2 4 1 >1 1 0 Table 1: Classification of Usage Levels of Each of the Three Copy Rooms Utilization between 8:30 and 10:30 is considered next. The rationale for extracting a portion of the day and computing its utilization was to show the magnitude of the variability of utilization throughout the day. This was done by comparing the utilization between 8:30 and 10:30, which was a very slow period, with the utilization throughout the day as a whole. Again, utilization was computed as the length of time a copy room is at a certain usage level divided by the length of that time period. The same thresholds, indicated in Table 1, were used to classify the usage levels as very busy, moderate, and light. III.1.2.3 Demand Variation The final step for understanding the status quo was to determine i f the demand varied throughout the week and, i f so, which day(s) would be the heaviest in terms of usage and which would be the lightest. For this part of the analysis, only the data for floor 2 was considered, as this was the only floor in which data collection takes place throughout the week. The graph in Figure 4 shows the total number of users in the copy room on floor 2, from 10:30 to 14:30, for each day of the week. Recall data collection occurred from Tuesday to Monday of the following week. 9 Figure 4: Number of Users on Floor 2 III.1.3 ANALYSIS OF SERVICE TIMES Four steps were carried out in this portion of the analysis. First, it was visually determined i f the service times could be well modelled by an exponential distribution. Exponential distributions were needed for both service times and inter-arrivals times to carry out an analysis on an M/M/c queue. This is explained in further detail later in the thesis. Second, Best-Fit Distributions were established. These distributions were used in the third step: determining the "Worst Case" Distributions. The final step was the development of the "Mean Case" Distributions. Both the "Worst Case" Distributions and the "Mean Case" Distributions are inputs for the queueing models and simulation models. For each floor on each day, service times were put in bins of one minute. The corresponding frequencies of the bins were calculated and graphed. These graphs can be seen in Appendix III. The service times seem to follow an exponential distribution. This suggests that service times can be well modelled by an exponential distribution. For the second step, two separate scenarios were examined: consideration of each day separately and consideration of each floor each day separately. These scenarios were chosen for two reasons. The copy room chosen by the user might influence the service time. This was not the case, as is shown in section III. 1.5 of this paper. A variety of scenarios give a variety of service time distributions, allowing the choice of an absolute worst-case service time distribution for use in the final results. An exponential distribution was fit to the service times for all of the scenarios. The distribution fitting was done using the input analyzer from A R E N A . A R E N A is a simulation software package used to model systems and allows one to perform extensive analysis, without disrupting the real systems. The probability density function for an exponential distribution is depicted in Appendix IV and the probability density function for a shifted exponential distribution is depicted in Appendix V . The goodness of the fit was determined by using the Chi-square test and the Kolmogorov-Smirnov test. A description of these tests is given in Appendix VI and Appendix VII respectively. 10 A l l scenarios, except Tuesday and Floor 1 Tuesday, were fit well by an exponential distribution. An outlier test was needed on the Tuesday and Floor 1 Tuesday scenarios to determine i f there were any outliers. Any data value greater than 1.5*IQR beyond the 3rd quartile or less than 1.5*IQR below the 1st quartile was considered an outlier. A description of the outlier test and applicable notation is given in Appendix VIII. III.1.3.1 Outlier Test For Tuesday The six steps of the outlier test to determine i f there were any outliers for Tuesday service times are depicted below. 1. Data was arranged from highest to lowest and is shown in Appendix IX. 2. Sample size: n = 386 i) 1 s t Quartile Q,: c - np/100 = (386*25)/100 = 96.5 96.5 rounded up is 97. Thus, Qi corresponds to the value 120. ii) 3 r d Quartile Q 3 : c = np/100 = (386*75)/100 = 289.5 289.5 rounded up is 290. Thus, Q3 corresponds to the value 510. 3. Find the inter-quartile range (IQR). IQR = Q 3 - Qi = 510 - 120 = 390. 4. Multiply the inter-quartile range by 1.5. S4=1.5*IQR= 1.5*390 = 585. 5. Subtract the value obtained in Step 4 from Q_ and add the value to Q3. S5- = Q, - S4•= 120 - 585 = -465. S5+ = Q 3 + S4 = 510 + 585 = 1095. 6. Note that S5- must be greater than 0. Thus, any value outside the range of 0 to 1095 was an outlier. There were a few outliers in the data set, but mainly one extreme outlier. This data point, 4079, is much larger than any of the other values, as can be seen in the box plot in Appendix X . It was considered an outlier and removed from the data set. 11 III.1.3.2 Outlier Test For Floor 1 Tuesday The same six steps of the outlier test for the data for service times on Floor 1 Tuesday to determine i f there are any outliers were used. The data arranged from highest to lowest for this is shown in Appendix X I . There were a few outliers in the data set, but mainly one extreme outlier. This data point, 4079, is much larger than any o f the other values, as can be seen in the box plot in Appendix XI I . It was considered an outlier and removed from the data set. Thus, in both instances the misfit was mainly caused by the inclusion o f a data point that is an extreme outlier. When this data point was eliminated from consideration, both Tuesday and Floor 1 Tuesday were fit well by an exponential distribution. III.1.3.3 Determining Service Time Best-Fit Distributions From the exponential distributions found above, for each floor on each day the distribution with the longest service time was selected from either the fit for that day for the specific floor or the fit for that day across all floors. For each instance, the hypothesis that the selected service time is an exponential distribution was tested. The Chi Square Test and the Kolmogorov-Smimov Test were used to test this hypothesis. If the selection had a small p-value for either of the tests, the next longest service time was chosen, until there was a p-value for both tests that was adequately large. The resulting selections for each floor on each day were considered the best fits and were called the Service Time Best-Fit Distributions. These distributions are shown in Table 2. The units are in seconds. Floor/Day Mean Service Time Distribution p-value of Chi-Square test p-value of Kolmogorov-Smirnov tests 1/Tues 443 10 + exp(433) 0.545 >0.15 2 /Tues 385 exp(385) 0.688 >0.15 2/Wed 331 2 + exp(329) 0.116 >0.15 4/Wed 327 2 + exp(325) 0.207 >0.15 1/Thurs 455 40 + exp(415) 0.624 >0.15 2/Thurs 401 10 + exp(391) >0.75 >0.15 2/Fri 325 5 + exp(320) 0.486 >0.15 4/Fri 322 5 + exp(317) 0.506 >0.15 2/Mon 395 exp(395) 0.208 0.0815 Table 2: Service Time Best-Fit Distributions 12 It is noteworthy that some of the distributions are shitted distributions. For example, the distribution for Floor 1 Tuesday is 10 + exp (433). This means that on Floor 1 Tuesday, the distribution of the service time is shifted by 10 seconds. That is, all service times will be at least 10 seconds. III.1.3.4 Determining Service Time "Worst Case" Distributions The service time "Worst Case" distributions were determined by taking, for each floor separately, the service time distribution that had the largest mean for that floor found in the Service Time Best-Fit Distributions of Table 2. The resulting three distributions were the service time "Worst Case" Distributions. The distributions are displayed in Table 3. The units are all in seconds. Floor Mean Service Time Distribution 4 327 2 + exp(325) 2 401 10 + exp(391) 1 455 40 + exp(415) Table 3: Service Time "Worst Case" Distributions Instead of using the shifted distribution found, the mean of that distribution was used as the parameter of an exponential distribution. This was done for simplicity and so that the analysis could be completed using an M/M/c model. Recall the probability density function of the shifted exponential distribution, as explained in Appendix V , has the form: 1 ~ y / f(y+9) = — e / p , for > 0, where 9 is some constant. The probability density function, as illustrated in Appendix IV, of the corresponding non-shifted exponential distribution has the form: f ( y ) = - e ~ ^ . B For each of the cases in which there was a shifted exponential distribution, p(0 < y < 9) was calculated. The results are illustrated in Table 4. Not all of the probabilities are small. A comparison between the two types of distributions was done using simulation. The methodology and results are discussed in section III. 1.6.5. The results show that the data is robust and the system would not be significantly affected by using the corresponding non-shifted exponential distribution. Shifted Distribution Non-Shifted Distribution 0 D(0 < V < 0) 2 + exp(325) exp(327) 2 0.0061 10 + exp(391) exp(401) 10 0.0246 40 + exp(415) exp(455) 40 0.0842 Table 4: Probabilities of Shifted Exponential Distributions 13 The resulting (non-shifted) distributions, one for each floor, were used as the service time inputs for the queueing model. The three non-shifted exponential distributions are presented in Table 5. Floor Mean Service Time Distribution 4 327 exp(327) 2 401 exp(401) 1 455 exp(455) Table 5: Service Time Non-Shifted Exponential "Worst Case" Distributions • III.1.3.5 Determining Service Time "Mean Case" Distributions The service time "Mean Case" distributions allow for an overall representation of the days events, as opposed to the focus on the peak period, which was chosen for the "Worst Case" distributions. This format is not as confined as the "Worst Case" distributions and allows for the emergence of conclusions that may be less rigorous. For example, the library's management may want to make decisions regarding the size of the photocopier fleet based on an average day as apposed to the peak period. The first part of this step was to consider the actual fitted distributions for service time per floor per day, with the corresponding p-values of the Chi Square Test and the Kolmogorov-Smirnov Test. These distributions are displayed in Table 6. The units are in seconds. Where the sample was too small to conduct the Chi Square Test, N/A appears in the chart. Floor/Day Mean Service Time Distribution p-value of Chi-Square test p-value of Kolmogorov-Smirnov tests 1/Tues 443 10 + exp(433) 0.545 >0.15 2/Tues 342 exp (342) 0.348 >0.15 2/Wed 331 2 +exp (329) 0.116 >0!l5 4/Wed 249 30 +exp (219) N/A >0.15 1/Thurs 455 40 + exp (415) 0.624 >0.15 2/Thurs 367 10 +exp (357) 0.149 >0.15 2/Fri 325 5 + exp (320) 0.486 >0.15 4/Fri 257 49 + exp (208) N/A >0.15 2/Mon 395 exp(395) 0.208 0.0815 Table 6: Floor/Day Service Time Distributions The second part of this step was the selection of the service time distribution found in Table 6 that corresponded to the largest mean that had a large p-value for both the Chi Square Test and the Kolmogorov-Smirnov Test, for each floor. The resulting three distributions were the 14 Service Time "Mean Case" Distributions. These distributions are displayed in Table 7. The units are all in seconds. Floor Mean Service Time Distribution 4 257 49 + exp(208) 2 395 exp(395) 1 455 40 + exp(415) Table 7: Service Time "Mean Case " Distributions Again, instead of using the shifted distribution found, the mean of that distribution was used as the parameter of an exponential distribution. For each of the cases in which there was a shifted exponential distribution, p(0 < y < 9) was calculated. The results are illustrated in Table 8. Not all of the probabilities are small. A comparison between the two types of distributions was done using simulation. The methodology and results are discussed in section III.1.6.5. The results show that the data is robust and the system- would not be significantly affected by using the corresponding non-shifted exponential distribution. Shifted Distribution Non-Shifted Distribution 0 D(0 < V < 0) 49 + exp(208) exp(257) 49 0.1736 40 + exp(415) exp(455) 40 0.0842 Table 8: Probabilities of Shifted Exponential Distributions The resulting (non-shifted) distributions were used as the inputs for the queueing model. The three non-shifted exponential distributions are presented in Table 9. Floor Mean Service Time Distribution 4 257 exp(257) 2 395 exp(395) 1 455 exp(455) Table 9: Service Time Non-Shifted Exponential "Mean Case " Distributions III.1.4 ANALYSIS OF INTER-ARRIVAL TIMES Four steps were carried out in this portion of the analysis. First, it was visually determined i f the inter-arrival times could be well modelled by an exponential distribution. Recall exponential distributions were needed for both service times and inter-arrivals times to use M/M/c queue theory to derive results. Second, Best-Fit Distributions were established. These distributions were used in the third step: determining the "Worst Case" Distributions. The fourth step was the development of the "Mean Case" Distributions. Both the "Worst Case" Distributions and the "Mean Case" Distributions are used as inputs of the models. 15 For each floor on each day, the number of user arrivals during each half hour block was tallied. The arrival rate for each half hour block was then determined. In order to smooth the graphs, the arrival rates for the half hours were grouped into hour blocks. The frequency for each block of arrival rates was the number of arrivals assigned to that particular block. The resulting blocks and respective frequencies are depicted in the graphs of Appendix XIII. Inter-arrival times were calculated by subtracting the arrival time of a user from the arrival time of the next user. Inter-arrival times were put in bins of one minute. The corresponding frequencies of the bins were calculated and then graphed. These graphs can be seen in Appendix X I V . The inter-arrivals seem to follow an exponential distribution. This suggests that inter-arrival times can be well modelled by an exponential distribution. Six scenarios were considered: Day, Floor Day, Day Hour, Day Bi-hour, Floor Day Hour, and Floor Day Bi-hour. A summary and description is in Table 10. , Scenario Set Description of Data Set Day each day Floor Day each floor each day Day Hour each day each hour Day Bi-hour each day each two hours Floor Day Hour each floor each day each hour Floor Day Bi-hour each floor each day each two hours Table 10: Scenario Sets for Analysis of Inter-Arrival Times The use of the scenarios was done as the library's management wishes to satisfy peak demand, and because the peak time varied per day per floor. Also, some of the scenarios did not have an exponential distribution that is a good fit An exponential distribution was fit to the inter-arrival times for all of the scenarios. Again, the distribution fitting was done using the input analyzer from A R E N A . III. 1.4.1 Determining Inter-Arrival Time Best-Fit Distributions The distribution with the shortest inter-arrival time was selected from one of the six scenarios described above for each floor on each day. For each instance, the hypothesis that the selected inter-arrival time is an exponential distribution was tested. The Chi Square Test and the Kolmogorov-Smirnov Test were used to test this hypothesis. If the selection had a small p-value for either of the tests, the next shortest inter-arrival time was chosen, until there was a p-value for both tests that is adequately large. The resulting selections for each floor on each day were considered the best fits and were called the Inter-arrival Time Best-Fit Distributions. These distributions are shown in Table 11. The units are in seconds. N / A in the chart indicates that the sample was too small to 16 conduct the Chi Square Test. There was no distribution under consideration that had a reasonable p-value for both tests for Floor 2 Mon. However, the data set chosen consisted of only 29 data points. This would result in the Kolmogorov-Smirnov test being a more powerful test than the Chi-Square test. The Chi-Square test p-value of 0.0327 for the distribution chosen is not overly small, so the distribution was assumed to be a reasonable approximation. Floor/Day Mean Inter-arrival Time Distribution p-value of Chi-Square test p-value Kolmogorov-Smirnov tests 1/Tues 83.1 exp(83.1) 0.69 >0.15 2/Tues 77.5 exp(77.5) 0.393 >0.15 2/Wed 69.3 exp(69.3) 0.21 >0.15 4/Wed 501 260+exp(241) N/A >0.15 1/Thurs 129 exp(129) 0.728 >0.15 -2/Thurs 120 exp(120) 0.169 >0.15 2/Fri 96.7 exp(96.7) 0.156 >0.15 4/Fri 96.7 exp(96.7) 0.156 >0.15 2/Mon 132 4+exp(128) 0.0327 >0.15 Table 11: Inter-arrival Time Best-Fit Distributions III.1.4.2 Determining Inter-Arrival Time "Worst Case" Distributions The inter-arrival time "Worst Case" distributions were determined by taking, for each floor separately, the inter-arrival time distribution that had the smallest mean for that floor found in the Inter-arrival Time Best-Fit Distributions in Table 11. The resulting three distributions were the inter-arrival time "Worst Case" Distributions. The distributions are displayed in Table 12. The units are all in seconds. Floor Mean Inter-arrival Time Distribution 4 96.7 ' exp(96.7) 2 69.3 exp(69.3) 1 83.1 exp(83.1) Table 12: Inter-arrival Time "Worst Case" Distributions III.1.4.3 Determining Inter-Arrival Time "Mean Case" Distributions The inter-arrival time "Mean Case" distributions allow for an overall representation of the days events, as opposed to the focus on the peak period, which was chosen for the "Worst Case" distributions. This format is not as conservative as the "Worst Case" distributions and allows for a more liberal analysis of the day's events, as well as for the emergence of conclusions that may be less rigorous. 17 The first part of this step was to consider the actual fitted inter-arrival distributions per floor per day, with the corresponding p-value of the Chi Square Test and the Kolmogorov-Smirnov Test. These distributions are displayed in Table 13. The units are in seconds. Where the sample was too small to conduct the Chi Square Test, N /A appears in the chart. Floor/Day Mean Inter-arrival Time Distribution p-value Chi-Square test p-value Kolmogorov-Smirnov tests 1/Tues 167 exp(167) 0.138 0.0454 2/Tues 124 exp(124) 0.163 0.0858 2/Wed 93.1 exp(93.1) 0.159 0.146 4/Wed 3900 208 + exp(3700) N/A <0.01 1/Thurs 202 exp(202) 0.284 >0.15 2/Thurs 122 exp(122) 0.617 >0.15 2/Fri 103 exp(103) 0.546 >0.15 4/Fri 1760 196+ exp( 1570) N/A >0.15 2/Mon 175 exp(175) 0.127 0.0815 Table 13: Floor/Day Inter-arrival Time Distributions For Floor 1 Tuesday, Floor 2 Tuesday, and Floor 4 Wednesday, the p-values were not large for both the Chi Square Test and the Kolmogorov-Smirnov Test. This was not a hindrance in the analysis since, for each floor, the service time distribution that had a large p-value for both the Chi Square Test and the Kolmogorov-Smirnov Test was used in the end result, and one for each floor did exist. For example, for Floor 1 the distribution found for Floor 1 Thursday was used and Floor 1 Tuesday was not considered at all. The second part of this step was the selection of an inter-arrival time distribution found in Table 13 that corresponds to the smallest mean that had a large p-value for both the Chi Square Test and the Kolmogorov-Smirnov Test, for each floor. The resulting three distributions were the Inter-arrival Time "Mean Case" Distributions. These distributions are displayed in Table 14. The units are all in seconds. Floor Mean Inter-arrival Time Distribution 4 1760 196 + exp(1570) 2 93.1 exp(93.1) 1 202 exp(202) Table 14: Inter-arrival Time "Mean Case" Distributions Again, instead of using the shifted distribution found, the mean of that distribution was used as the parameter of an exponential distribution. For each of the cases in which there was a shifted exponential distribution, p(0 < y < 9) was calculated. The results are illustrated in Table 15. A comparison between the two types of distributions was done using simulation. 18 The methodology and results are discussed in section III. 1.6.5. The results show that the data is robust and the system would not be significantly affected by, using the corresponding non-shifted exponential distribution. Shifted Distribution Non-Shifted Distribution e D(0 < v < 6) 196+ exp( 1570) exp(1760) 196 0.1054 Table 15: Probabilities of Shifted Exponential Distributions The resulting (non-shifted) distributions were used as the inputs for the queueing model. The three non-shifted exponential distributions are presented in Table 16. Floor Mean Inter-arrival Time Distribution 4 1760 exp(1760) 2 93.1 exp(93.1) 1 202 exp(202) Table 16: Inter-arrival Time Non-Shifted Exponential "Mean Case" Distributions III.1.5 ANALYSIS OF MEAN ARRIVAL RATE AND MEAN SERVICE RATE Using the Service Time Best-Fit Distribution from Table 2 and the Inter-arrival Time Best-Fit Distributions from Table 11, the mean arrival rates (the inverse of the mean inter-arrival time) and the mean service rates (the inverse of the mean service time) for floor 2 were graphed with day on the x-axis and value on the y-axis. The units for the mean arrival rate are arrivals per second and the units for the mean service rate are users served per second. This was done to determine i f and how the rates vary as the week progresses. The graph is in Figure 5. 0.016 Day Of Week Figure 5: Mean Arrival Rate and Mean Service Rate For Floor 2 19 ///. 1.6 QUEUEING MODELS AND SIMULA TION MODELS To determine the number of photocopiers that should be maintained by Koerner Library to meet peak demand, while still maintaining a respectable service level, a broad range of scenarios were analyzed. The methodology chosen to do this included both queueing theory and simulation, in conjunction with the "Worst Case" Distributions and "Mean Case" Distributions found previously. The exponential distributions for both the service time and the inter-arrival time allow the queueing system to be modelled as an M/M/c queue. The first M corresponds to a Markovian arrival process. That is, the queueing system has Poisson arrivals, or, equivalently, the queueing system has Exponential inter-arrivals. The second M corresponds to exponential service times. The letter c represents the number of parallel servers in the system. In this system, there were c photocopiers that service one user at a time with the same service rate. The M/M/c queue is one of the simpler systems in queueing theory. There are formulas available to calculate performance measures, such as mean number in queue and mean waiting time in system. These formulas help to reduce the time needed to calculate performance measures for the analysis of the queueing system. For the queueing analysis, the Queueing Tool-Pak was used. This is an add-in specifically built for Excel by Armann Ingolfsson, et al., at the University of Alberta and consists of all the formulas needed to do queueing analysis on an M/M/c type of queueing model. A description can be found on the website http://www.bus.ualberta.ca/aingolfsson/QTP/default.htm. IH.1.6.1 Assumptions for Queueing Model It was assumed the system is in steady state and the system operates on a first-come-first-served basis. A steady state system is a system in which the probability distribution for the number of users in the system at time t and the number of users in the queue at time t does not vary over time. That is, the probability distribution of the state of the system is independent of the initial state and does not vary over time. It was assumed the users arrive at a constant rate of X and a user was served at a constant rate of p. It was also assumed the arrival rate is stationary. It was assumed the inter-arrival times are independently and identically distributed (i.i.d), the inter-arrival times are exponentially distributed, the service times are i.i.d, and the service times are exponentially distributed. The assumption that the system was stationary is not a valid assumption since the photocopier system was non-stationary. The arrival rate does depend on time t and it would be more appropriate to use a Nonhomogeneous Poisson Process for the analysis. Doing so would increase the complexity of the analysis dramatically and the analysis would be 20 extremely excessive for the type of solutions U B C Library's management would like to receive. However, the system was in steady state for small intervals. The analysis was conducted on these small intervals and recommendations were made on a "worst case" situation. Any carry over of surplus demand from one period to another period will not excessively affect the mean waiting time in queue. In particular, any influence by this carry over would not change the recommendation. As a result, assuming the system was stationary was adequate for this analysis. 111.1.6.2 Notation for Queueing Model c = number of photocopiers X = mean arrival rate of a user to the system u, = mean service rate of a single server L =• mean number of users in the system L q = mean number of users waiting in the queue; mean queue length W = mean time a user is in the queueing system; mean time in system W q = mean time a user is in the queue; mean waiting time p = server utilization; mean time of work for each server P n = the equilibrium probability that there are n users in the system 111.1.6.3 Formulas for Queueing Model i . Photocopier Utilization, p P = — 11. Mean Queue Length, Lq Po{cp)c p c\{\-p)2 in. Mean Number of Users In System, L iv. Mean Time In Queue, Wq 21 L W =-*-q X Mean Time In System, W W = Wq + vi. Equilibrium Probabilities, Po, Pn P„ = C-] Z 1^1=0 + c\ \-p dc" Po 0<n<c n>c vii. Distribution Of The Waiting Time P(W >t)=e :l{l-p) c-\-cp Formulas /, ii, iv, and v were used for the charts found in Appendix X V and Appendix XVII . The formula found under vii were used for the matrices found in Appendix X V I and Appendix XVIII. III.1.6.4 Inputs for Queueing Model The distributions for service times and inter-arrival times for each floor found in the non-shifted exponential "Worst Case" distributions of Table 5 and Table 12 were used as inputs for the queueing model. For each floor, mean time in system, mean time in queue, mean queue length, and utilization were calculated for a range of numbers of photocopiers. Also for each floor, a matrix was developed that indicates the minimum number of photocopiers needed to ensure the probability that a user waits less than a certain length of time is greater than or equal to a certain service level. The distributions for service times and inter-arrival times for each floor found in the non-shifted exponential "Mean Case" distributions of Table 9 and Table 16 were used as the 22 inputs for the queueing model. For each floor, mean time in system, mean time in queue, mean queue length, and utilization were calculated for a range of numbers of photocopiers. Also for each floor, a matrix was developed that indicates the minimum number of photocopiers needed to ensure the probability a user waits less than a certain length of time is greater than or equal to a certain service level. III.1.6.5 Simulation The next step of the analysis was the development of a simulation. The simulation was used mainly to compare the shifted and non-shifted distributions to determine i f using the non-shifted distribution would affect the results of the analysis. Furthermore, the simulations were used for a visual presentation of the results to the client. For the comparison of the shifted and non-shifted service time distributions of the "Worst Case" scenario, two simulation models were developed. The first model uses the shifted distributions from table 4 as inputs for the service time with the corresponding non-shifted inter-arrival time from table 12. The second model uses the corresponding non-shifted distributions from table 4 as inputs for the service time, without change to the inter-arrival time. For example, the first model for the comparison for floor 1 uses the distribution 40 + exp(415) as the input for service time and exp(83.1) as the input for the inter-arrival time. The second model uses the distribution exp(455) as the input for service time with exp(83.1) as the input for the inter-arrival time. For the comparison of the shifted and non-shifted service time distributions of the "Mean Case" scenario, the first model uses the shifted distributions from table 8 as inputs for the service time and the second model uses the corresponding non-shifted distributions. The input for the inter-arrival time distribution is the corresponding non-shifted distributions in table 16. For the comparison of the shifted and non-shifted inter-arrival time distributions of the "Mean Case" scenario, the first model uses the shifted distributions from table 15 as inputs for inter-arrival time and the second model uses the corresponding non-shifted distributions. The input for the service time distribution is the corresponding non-shifted distributions in table 9. Four values were collected for each model and compared: mean queue length, mean time in queue, mean time in system, and mean number of users. 23 III. 2 RESULTS The results of the analysis supplied three types of information. The first was an analysis of the status quo and what service values the copy rooms was currently experiencing. The second was a time frame of peak and low hours. The third was a matrix or chart of the trade-off between the number of photocopiers in a room and the waiting time accrued with that number of photocopiers. IIL2.1 ANALYSIS OF STATUS QUO SYSTEM Both the event-driven graph and the time-driven graph show the copy room on floor 1 never reaches capacity, the copy room on floor 2 did at times reach and exceed capacity, and the copy room on floor 4 stayed mainly empty. This was in alignment with the contention there are too many photocopiers. The event-driven graph of Appendix I was more useful for depicting whether or not a copy room reaches full capacity, stays empty, or is somewhere in between. The time-driven graph of Appendix II was more useful in determining the length of time of peak periods and low periods. The chart on the utilization of each of the three copy rooms for the entire day in Table 17 shows the portion of time the copy room was at a particular usage level, rounded to two decimal places. This chart reveals that with the exception of Floor 2 on Friday the usage on each floor per day was light and rarely very busy. The only floor that was very busy was floor 2. Day Very Busy Moderate Light F1 Tues 0.01 0.31 0.68 F1 Thurs 0.00 0.20 0.80 F2 Tues 0.06 0.47 0.47 F2 Wed 0.21 0.37 0.42 F2 Thurs 0.08 0.43 0.50 F2 Fri 0.06 0.56 0.38 F2 Mon 0.02 0.35 0.63 F4 Wed 0.00 0.12 0.88 F4 Fri 0.00 0.11 0.89 Table 17: Fraction of Use of Each Usage Level for Each of the Three Copy Rooms The chart on the utilization for the time between 8:30 and 10:30 of each of the three copy rooms in Table 18 shows the fraction of time the copy room is in a particular usage level, rounded to two decimal places. 24 Day Very Busy Moderate Light F1 Tues 0.00 0.00 1.00 F2 Tues 0.00 0.00 1.00 F2 Wed 0.00 0.05 0.95 F2 Mon 0.00 0.03 0.97 F4 Wed 0.00 0.00 1.00 Table 18: Fraction of Use Between 8:30 and 10:30 of Each Usage Level for Each of the Three Copy Rooms This chart reveals usage during this time frame was very low in comparison to usage throughout the day. The fraction of use between 8:30 and 10:30 was almost always light, rarely moderate, and never busy. This comparison illustrates that even within the peak period throughout the year, utilization of the photocopiers during a particular portion of the day may be extremely low. The consideration of only floor 2 during the time period of 10:30 and 14:30 that is depicted in Figure 4 shows that there was a difference in the number of users during the week. Wednesday was the busiest day and Monday was the slowest day out of the days data collection took place. The observations made regarding the demand patterns on floor 2 are transferable to the library as a whole since time periods where parallel data collection took place showed demand patterns of the other copy rooms as similar to that of floor 2. Thus, the observations for floor 2 impart information about the general pattern of photocopier demand and are usable for other studies U B C Libraries may wish to conduct. The three analyses above show there is evidence to support the contention there are too many photocopiers housed in Koerner Library. The analyses also reveal Wednesday as the peak day throughout the week and 10:30 to 14:30 as the peak hours throughout each day. Figure 5 shows that for floor 2, Tuesday and Wednesday have the highest mean arrival rate and that the service rate remained relatively the same, regardless of the day of the week. These results are in agreement with expectations. The service rates were the same regardless of the day of the week. Arrival rates, on the other hand, were very dependent on the day of the week. This is due to the nature of the traffic within the library as a whole. The library tends to be busier earlier in the week, with usage tapering off as the week progresses. III.2.2 QUEUEING MODEL The "Worst Case" scenario is a very conservative scenario as the arrival rate of this capacity happens for only a small portion of time throughout the entire year. The "Mean Case" scenario is more reflective of the usage for the majority of the year. 25 111.2.2.1 Queueing Analysis of "Worst Case" The charts in Appendix X V show the mean time in system, mean time in queue, mean queue length, and utilization for a range of number of photocopiers for each floor. The matrices in Appendix X V I depict the minimum number of photocopiers needed to ensure that the probability a user waits less than a certain length of time is greater than or equal to a certain service level. The numbers in the table show the minimum number of photocopiers required at different service levels and different waiting times. Both the charts and the matrices illustrate some of the trade-offs of increasing or decreasing the number of photocopiers in a room. Utilization on floor 1 was 55%, on floor 2 was 80%, and the utilization on floor 4 was 94% for the current number of photocopiers. The mean time in queue for floor 1 was 6.07 seconds, for floor 2 was 87.26 seconds, and for floor 4 was 252.37 seconds. This illustrates that a reduction of photocopiers on floor 1 and floor 2 is feasible, while the number of photocopiers on floor 4 should remain the same. It is acceptable to U B C Library's management that at least 80% of the users wait no more than 300 seconds on any of the floors. The matrices from Appendix X V I indicate that seven photocopiers on floor 1, seven on floor 2, and one on floor 4 meet this standard. The utilization and mean time in queue in such a case is 78% and 101.47 seconds respectively (as apposed to 55% and 6.07 seconds) for floor 1, while floor 2 and floor 4 remain the same. 111.2.2.2 Queueing Analysis of "Mean Case" The charts in Appendix XVII indicate the mean time in system, mean time in queue, mean queue length, and utilization for a range of number of photocopiers for each floor. The matrices in Appendix XVIII depict the minimum number of photocopiers needed to ensure that the probability a user waits less than a certain length of time is greater than or equal to a certain service level. The numbers in the table show the minimum number of photocopiers required at different service levels and different waiting times. Both the charts and the matrices illustrate some of the trade-offs of increasing or decreasing the number of photocopiers in a room. Utilization on floor 1 was 23%, on floor 2 was 60%, and the utilization on floor 4 was 14% for the current number of photocopiers. The mean time in queue for floor 1 was 0.01 seconds, for floor 2 was 21.97 seconds, and for floor 4 was 32.75 seconds. Using the same criteria as the "Worst Case", this also illustrates that a reduction of photocopiers on floor 1 and floor 2 is feasible, while the number of photocopiers on floor 4 should remain the same. 111.23 SIMULATION For each comparison between the shifted and non-shifted distributions, mean time in system, mean time in queue, mean queue length, and mean number of users was simulated for both 26 distributions. The results are shown in Appendix X I X , as well as a screen-shot of the simulation of the status quo for floor 1, floor 2, and floor 4 and of the simulation for the recommendation for floor 1. The comparison shows that there is little effect on the model when using the corresponding non-shifted distribution for the input in the model. 27 3 INTERPRET A TION OF RESUL TS U B C Libraries only experience excessive demand for their photocopiers for a small portion of the entire year. Based on historical facts and figures, the period data collection took place is representative of the busiest demand throughout the year. The analyses revealed Wednesday as the peak day throughout the week and 10:30 to 14:30 as the peak hours throughout each day. The service rate remained the same, regardless of the day of the week or time of day. This analysis showed the number of photocopiers housed in Koerner Library is disproportionate to the actual demand. In particular, during the period in which data collection took place, floor 1 never experienced as much demand as there are photocopiers. The study indicated a reduction of photocopiers is feasible for floor 1 and the number of photocopiers on floor 2 and floor 4 should remain the same. The main results are depicted in Appendix X V and Appendix X V I for the "Worst Case" scenario and Appendix XVII and Appendix XVIII for the "Mean Case" scenario. The charts in Appendix X V and Appendix XVII can be used to quantify changes in the mean time in the system (i.e. the mean waiting time and service time), the mean time in the queue, the mean queue length, and the utilization of the photocopiers when the number of photocopiers on a floor is decreased (or increased for the case of floor 4) from the number of photocopiers that the floor currently houses. The matrices in Appendix X V I and Appendix XVIII can be used to determine how many photocopiers are needed to meet a certain level of customer service, for each floor. Note that for floor 4, there was not enough data to develop a matrix that was a good indication of reality. Downtimes were not considered in the analysis, as the actual downtimes would be difficult to calculate. One way to alleviate this exclusion was with the use of a "Worst Case" scenario for the analysis. For each floor each day, the service times and the inter-arrival times were considered separately to find "Worst Case" distributions and the resulting distributions were then combined to conduct the analysis. This inflated the reality of the peak period, ensuring peak demand could be met when using the outcome of the analysis. For each floor, the "Worst Case" scenarios were used so as to meet peak demand for all days of the week. The queueing theory and the simulation phase for the "Worst Case" scenario of the analysis suggests reducing the number of photocopiers down to seven would result in a mean queue length of 1.21, a mean time in queue of 101.47 seconds (or slightly more than one and a half minutes), and a utilization of 78%. For floor 2, the number of photocopiers should not change and would result in a mean queue length of 1.22, a mean time in queue of 87.26 seconds (or slighter less than one and a half minutes), and a utilization of 8%. There was not enough data to conduct a proper analysis for floor 4. Considering the data that was available, it seems acceptable to keep the number of photocopiers on floor 4 at one. U B C Library's management considered this recommendation to be reasonable. The occurrence of the "Worst Case" scenario is rare as the data showed the selected inter-arrival rates occurring for only a brief quantity of time. Furthermore, peak demand is only 28 experienced for a short period within the entire year. For decisions where the criterion is meeting peak demand, using the "Worst Case" results is adequate. However, it is recommended that any further decisions made using the charts and the matrices should take into account both the "Worst Case" scenario and the "Mean Case" scenario. That is, the decision maker should first determine what type of demand to capture and then consult the appropriate matrix and chart. For example, i f the decision maker wants to maximize photocopier utilization throughout the year, the "Mean Case" scenario matrices and charts will be more useful. 29 IIL4 SUMMARY The main focus of this study was to analyze the usage of the photocopiers in Koerner Library during a peak period. From this analysis, a "worst case" scenario was developed to determine how many photocopiers there should be in Koerner Library to meet peak demand. A list of alternatives, as well as any recommendations, was to be imparted as to what resources are needed for the library to meet peak demand. The key results were recommendations based on two types of charts for each floor analyzed of Koerner Library. The first type can be used to quantify changes in the mean time in the system (i.e. the mean waiting time and service time), the mean time in the queue, the mean queue length, and the utilization of the photocopiers when the number of photocopiers on a floor is decreased or increased from the number of photocopiers that the floor currently houses. Theisecond type can be used to determine how many photocopiers are needed to meet a certain level of customer service, for each floor. Use of the report and recommendations by U B C Library's management is in progress. The intention was to use the report and quantitative findings to determine the size of photocopier fleet they will need to meet demand over the next few years. Changes have already been made within Koerner Library, as well as other library branches, as a result of recommendations made. U B C Libraries now has signs within the library depicting where the photocopier rooms are located. They have taken actions with regard to improving daily maintenance of the photocopiers to reduce downtime within the photocopier fleet. U B C Libraries has also begun to decide how to reallocate machines from low use areas to high use areas and has already reduced the number of photocopiers in Koerner Library. As a result of this analysis and the recommendations made, UBC Library's management asked the Centre for Operations Excellence (COE) to analyze the photocopier fleet of two other library branches and to analyze staffing levels for the circulation and reference desk for three branches. III.4.1 COMMENTS The methodology used in this study is portable to studies for all other U B C Library branches. However, the results of the other branches cannot be extrapolated from this study based at Koerner Library. Because of location, type of resources available, hours, and other factors, demand and service time are not portable. If it is assumed that demand, service time, and all other factors of the other libraries have similar characteristics as Koerner Library, the results are transferable. An observation noted during data collection is that i f a user either needs a copy card or needs to put money on their copy card, a significant amount of time is spent doing so. If a user needs to do either of the two tasks and they are currently using a photocopier, they leave what they are photocopying on the photocopier while they performed the task so as not to 30 lose possession. This ties up a resource, decreasing the number of resources available, and also increases service time of that user. Currently, there is a copy card machine and a change machine on the third floor (which houses no photocopiers) and a copy card machine on the second floor, outside of the photocopy room. It is recommended that the copy card machine outside of the second floor copy room be moved inside the copy room and that the copy card machine on the third floor be moved into the copy room on the first floor. This would decrease the service time of a user who needs to use a copy card machine, thereby reducing the amount of time for a resource to become available. Another observation noted during data collection is that i f a copy room is full, a user always waits for a copier to become available rather than checking the availability on another floor. This may result from not knowing there is another copy room. In fact, not all of the copy rooms are well marked and in some cases, the user does not know of another copy room in the building. It is recommended the copy rooms on all floors be well marked and in each of the copy rooms posting a sign indicating i f the room is full to check other rooms for availability. This should help to spread high demand between the rooms and alleviate the necessity of increasing the number of photocopiers in any one room. Although downtimes are not considered within this analysis, they are a significant factor. The most significant factor in the downtime of a photocopier are minor incidences such as a paper jam in the photocopier, low toner, and the copier being out of paper. Downtimes caused by these occurrences could be reduced by having staff (or perhaps the manufacturer's technician) check the copiers every two hours between 10:30 and 16:30 and making repairs as needed (currently, copiers are being checked sporadically throughout the day). The regularity of such checks would not only reduce downtimes, but it would also shorten the amount of time needed to do such checks. Another factor in the downtime issue are incidences a bit more complex, where the manufacturer's technician needs to be called in. There is little delay between the technician being called in and his arrival. The significant time factor in this case is the time between the check of the copiers when no problem is noticed and the check when the problem is found. Again, utilizing a frequent and regular schedule for checking the copiers between 10:30 and 16:30 would greatly reduce the length of such a downtime. III.4.2 FURTHER WORK The analysis showed certain days of the week are busier than others, with Wednesday being the busiest and Monday being the least busy. It also showed certain times during a day are busier than others, with 8:30-10:30 and after 16:30 being slow periods and intermittent periods between 10:30 and 16:30 being busier. This indicates that variable pricing could be used to transfer demand from a peak period to a slower period. It is suggested that a pilot study be done offering reduced pricing on a certain floor before 10:30, after 16:30, on ^ Thursday, Friday, or Monday, or a combination of these options to determine i f variable pricing would transfer enough of the demand so that the number of photocopiers can be further reduced. 31 Since downtimes of the photocopiers are a considerable factor, U B C Library's management should consider studying the effect of this phenomenon on the queueing system. They may also want to consider a move to other multifunctional devices (printers, etc.). There is a noticeable trend of material being made available electronically. This material can be printed off as apposed to being photocopied, thereby reducing the need for photocopiers and increasing the need for printers. The library's management should employ forecasting methods to determine the magnitude of change to evaluate the adequacy of their current resources. 32 IV HUMAN RESOURCES ANAL YSIS Darrell Bailie and Dwight Tanner of U B C Library noted that utilization of the circulation desks in Koerner Library, Woodward Biomedical Library, and David Lam Library was extremely variable. They feel the current staffing rules may be inadequate for the variation in demand the branches experience. Anecdotal evidence also suggested the reliability of the self-charge machines located at Koerner Library and Woodward Biomedical Library is erratic. Recall, the primary function of staff manning the circulation desk in a U B C Library branch is to facilitate the checkout of allowable items, renew items already on charge, discharge material on loan, and collect money for fines. The self-charge machines allow for a patron to charge out his/her own material, thereby circumventing the circulation desk. Queueing theory was used to analyze the demand of the circulation desks located in Koerner Library, Woodward Biomedical Library, and David Lam Library and develop staffing rules for periods of different levels of demand to aid U B C Library's management with scheduling of the circulation desks. A rudimentary reliability study was also conducted on the self-charge machines located at Koerner Library and Woodward Biomedical Library for the examination by UBC Libraries to gauge the reliability of the machines. IV. 1 METHODS Similar to the photocopier queueing system, the queueing system for the circulation desks is non-stationary. The system is in steady state for small intervals and the analysis is conducted on these intervals. The development of a model of the queueing system for the circulation desks at Koerner Library, Woodward Biomedical Library, and David Lam Library required six main steps. First, the necessary data on the queueing system was identified and transformed into a usable format. Second, an understanding of the status quo was established from the existing data with a baseline to assess the recommendation. Third, different periods were developed based on activity level. Fourth, the arrival rates for the determined periods were computed. Fifth, queueing analysis was used to establish the minimum number of staff needed at the circulation desk to meet a particular service level. Finally, the recommendation of staffing rules was compared with the baseline and sensitivity analysis was done on the recommendation. For the analysis on the self-charge machines, the assessment on whether reliability varies by machine or by bar code placement was done by statistical tests concerning the difference between proportions of two independent samples. 33 IV. 1.1 DA TA COLLECTION The option to collect data on arrival times or service times at the circulation desks was not available. U B C Library management supplied the data for this study. There were five types of data needed. Four of the five types of data were used to determine the arrival rate at the circulation desk, while the fifth established the service rate of a server at the circulation desk. The first type of data was electronic data collected by the computers for the circulation desk. When a patron uses their library account at the circulation desk, the computer software used to conduct the procedure notes the number of each transaction type. At the end of each hour, the program tallies the number of each transaction type and stores this information for future reference. This data consisted of the total number of circulation transactions at each branch for each hour of the day and was available for the period of September 1, 2001 to May 31, 2002. The portion of the electronic data appropriate for this project contained charges, renewals, and money collected. These transaction types, or rather portions of them, were the only types of transactions that occur at the circulation desk while a client is at the desk. The second type of data needed was a percentage of clients that only do renewal transactions and no other type of transaction at the same time at the circulation desk. Data collection for this took place on Friday July 12, 2002 between the hours of 11:00 am and 3:00 pm at Koerner Library by tallying the number of clients that renew books only and tallying the number of clients that renew books and do another type of transaction at the same time. The third type of data needed for this project was the percentage of money-collected transactions that occur at the circulation desk while a client is at the desk. The value for this portion of the data came from anecdotal observations made by personnel at U B C Libraries. For each of the three branches studied, a total of the number of transactions throughout the year was calculated, as well as the total number of clients. From this, the average number of transactions conducted per client was calculated for each of the branches. This comprises the fourth data type. Interviewing personnel at U B C Libraries and taking an average of the responses determined the service time, the fifth data type, needed for this study. The service time was based on their approximations and estimated to be 60 seconds. Collection of the data for the reliability study occurred by personally doing charge transactions on the self-charge machines. Two data sets were collected on each self-charge machine for the analysis. The first data set was charge transactions of books with the bar code on the inside of the book. The second data set was charge transactions of books with the bar code on the outside of the book. A tally was kept for each data set of the number of self-charge transactions that were made and the number of self-charge transactions that actually worked. 34 One assumption made was that data could be transformed to give an adequate description on demand of each of the circulation desks experienced by the three branches. Another assumption made was that distributions for both the service time and the arrival rate were exponential. IV.1.1.1 Transformation of Data Since the option to collect the relevant data for the analysis was not available, an alternate approach needed to be taken. This involved transforming available data into a usable format. The first four types above were used to determine the arrival rates for the three branches. The electronic data collected by the computers for the circulation desk consists of the total number of circulation transactions, made up of charges, renewals, and money collected, at each branch for each hour of the day. Using only these hourly transaction totals is not justifiable in determining the arrival rate for each hour, as they over estimate demand. This is because every charge, renewal, and money collected incidence counted as one transaction, whether one or many clients did it. For example, ten clients can charge out one book each or one client can charge out ten books and the number of charge transactions would be 10. As a result, these totals needed to be scaled for use in the analysis. To do the scaling of the electronic data for use in the analysis, the three other types of data were needed. The client may do a renewal transaction and another type of transaction, or the client may only do a renewal transaction. This distinction is critical as the service time for a client who renews books while doing another transaction is negligible, whereas the service time for a client who only renews books is not. To deal with this complexity, a percentage of clients that only do renewal transactions and no other type of transaction at the same time at the circulation desk were needed. Data collected by U B C Library personnel showed that 75% of the clients do a renewal transaction while doing another transaction at the same time. This means that for 75% of the renewal transactions, the change in the service time for the corresponding clients was negligible. The remaining 25% of the renewal transactions will result in a sizable service time for a client. The electronic data hourly totals were multiplied by 1/4 to reflect this observation. Another critical anomaly of the electronic data is that it includes all money-collected transactions that happen at the circulation desk. This also causes an inflation in the calculation of the arrival rate of clients to the circulation desk as the money-collected remittance may have came via mail or some other form of delivery and the actual transaction input may occur at the circulation personnel's convenience when there is no client at the circulation desk. The percentage of money-collected transactions that occur at the circulation desk while a client is at the desk was 75%, as determined by UBC Library personnel. The electronic data hourly totals were multiplied by 3/4 to alleviate this anomaly. The electronic data is independent of the client. A client may do one of the transactions or ten of the transactions, and this differentiation is not reflected in the data. This will amplify the arrival rate, so the average number of transactions that are conducted by one client for each of the branches was needed. The Library's IT department maintains a database that 35 includes the average number of transactions per client for each branch. U B C Library's management conveyed these averages for use in this analysis. For Koerner Library and David Lam Library, the average number of transactions done by a client was three, while for Woodward Biomedical Library it was only two. For each branch, the electronic data was then scaled by its corresponding average factor. The transformed data gives a more accurate description of the number of arrivals per hour over the time period analyzed. The arrival rate for all hours each branch is open was then determined and assumed to be exponential. This was the arrival rate used in the analysis. IV.1.2 ANALYSIS OF STATUS QUO A representation of the status quo is given from two types of graphs for each of the branches. Both graph types indicate the number of clients serviced at different time-periods. The first graph set depicts the total arrivals for the period from September 1, 2001 to May 31, 2002 by month. Each point in the graph is the total number of arrivals for a day of the month. The day of the month runs along the x-axis with the total number of arrivals on the y-axis. The graphs for Koerner Library, Woodward Biomedical Library, and David Lam Library are in Appendix X X . The second set of graphs depict the total arrivals for the period from September 1, 2001 to May 31, 2002 by day. Each point in the graph is the total number of arrivals for an hour of the day. The hour of the day runs along the x-axis with the total number of arrivals on the y-axis. Excerpts of the graphs for Koerner Library, Woodward Biomedical Library, and David Lam Library are in Appendix X X I . IV.1.3 DEVELOPMENT OF BASELINE A baseline was needed for comparison with the recommended staffing rules. This was used to ensure that the recommendation meets the demand adequately and there was not an abundance of unsatisfied demand or an abundance of low utilization of the servers. Queueing theory was used with the assumption that at least one staff member will be available at the desk for all open time-periods. The scaled electronic data of arrival rates and the reported service times were inputs for the baseline queueing models. It was assumed both types of inputs are exponentially distributed. This assumption allowed the queueing system to be modelled as an M/M/c queue. An explanation of the theory, as well as any relevant formulas, was given in section III. 1.6. Notation and other assumptions stated previously remain the same. Again, the Queueing Tool-Pak was used for the analysis. The first chart set of the baseline used the formula for the distribution of the waiting times in section III. 1.6.3. The second chart set used the formula for the mean time in queue in section III. 1.6.3. 36 IV. 1.4 PERIOD FORMULA TION The circulation desks of U B C Libraries experience variable demand, not only throughout the year, but also throughout the week and throughout the day. Some time frames have similar demand, so it was not necessary to determine the minimum staff needed at each circulation desk for all time-periods of the year. Instead, comparable time-periods were grouped together to reduce the amount of unnecessary calculations when generating the staffing rules. There were three types of time frames considered for the groupings. The first grouping consists of similar months, the second grouping consists of similar days of the week, and the third grouping consists of similar hours of the day, for each of the libraries independently. IV.1.4.1 Grouping of Like Months For each branch, the scaled transactions (hereafter referred to as arrivals) were summed for all months and the totals for the months were considered for analysis. Charts of these totals are seen in Table 19, Table 20, and Table 21. Month Total # of Arrivals September 2001 12187 October 2001 18952 November 2001 20884 December 2001 5973 January 2002 13208 February 2002 15413 March 2002 22435 April 2002 11745 May 2002 7288 Table 19: Total Arrivals By Month For Koerner Library Month Total # of Arrivals September 2001 4194 October 2001 7342 November 2001 7371 December 2001 2371 January 2002 6502 . February 2002 5077 March 2002 6345 April 2002 3773 May 2002 2652 Table 20: Total Arrivals By Month For Woodward Biomedical Library 37 Month Total # of Arrivals September 2001 197 October 2001 260 November 2001 306 December 2001 111 January 2002 256 February 2002 279 March 2002 335 April 2002 229 May 2002 187 Table 21: Total Arrivals By Month For David Lam Library For the Koerner Library, any month with a monthly total less than or equal to 10,000 arrivals was grouped as a low month. Any month with a monthly total greater than 10,000 arrivals but less than or equal to 20,000 arrivals was grouped as a moderate month. Any month with a monthly total greater than 20,000 arrivals was grouped as a high month. For the Woodward Biomedical Library, any month with a monthly total less than or equal to 3,000 arrivals was grouped as a low month. Any month with a monthly total greater than 3,000 arrivals but less than or equal to 6,000 arrivals was grouped as a moderate month. Any month with a monthly total greater than 6,000 arrivals was grouped as a high month. For the David Lam Library, any month with a monthly total less than or equal to 190 arrivals was grouped as a low month. Any month with a monthly total greater than 190 arrivals but less than or equal to 280 arrivals was grouped as a moderate month. Any month with a monthly total greater than 280 arrivals was grouped as a high month. IV.1.4.2 Grouping of Like Days of the Week For each branch, the arrivals were summed for all days of the week and the totals for the days of the week were considered for analysis. Charts of these totals are seen in Table 22, Table 23, and Table 24. Day of the Week Total # of Arrivals Monday 21692 Tuesday 21231 Wednesday 23585 Thursday 21559 Friday 21743 Saturday 10413 Sunday 7861 Table 22: Total Arrivals By Day Of The Week For Koerner Library 38 Day of the Week Total # of Arrivals Monday 7865 Tuesday 8494 Wednesday 7944 Thursday 7570 Friday 7583 Saturday 3215 Sunday 2956 Table 23: Total Arrivals By Day Of The Week For Woodward Biomedical Library Day of the Week Total # of Arrivals Monday 375 Tuesday 378 Wednesday 361 Thursday 421 Friday 285 Saturday 224 Sunday 115 Table 24: Total Arrivals By Day Of The Week For David Lam Library For the Koerner Library, any day with a daily total less than or equal to 12,000 arrivals was grouped as a low day of the week, and any day with a daily total greater than 12,000 arrivals was grouped as a high day. For the Woodward Biomedical Library, any day with a daily total less than or equal to 4,000 arrivals was grouped as a low day of the week, and any day with a daily total greater than 4,000 arrivals was grouped as a high day. For the David Lam Library, any day with a daily total less than or equal to 300 arrivals was grouped as a low day of the week, and any day with a daily total greater than 300 arrivals was grouped as a high day. IV.1.4.3 Grouping of Like Hours of the Day For each branch, the arrivals were summed for all hours of the day and the totals for the hours of the day were considered for analysis. Charts of these totals are seen in Table 25, Table 26, and Table 27. 39 Hour of the Day Total # of Arrivals 8:00 1307 9:00 3984 10:00 7505 11:00 11128 12:00 13863 13:00 15184 14:00 14698 15:00 14907 16:00 14149 17:00 9509 18:00 6844 19:00 4325 20:00 3795 21:00 3488 22:00 3361 23:00 38 Table 25: Total Arrivals By Hour Of The Day For Koerner Library Hour of the Day Total # of Arrivals 8:00 1864 9:00 3036 10:00 3919 11:00 4408 12:00 4862 13:00 4623 14:00 4499 15:00 4599 16:00 5151 17:00 3617 18:00 1543 19:00 1225 20:00 901 21:00 893 22:00 472 23:00 14 Table 26: Total Arrivals By Hour Of The Day For Woodward Biomedical Library 40 Hour of the Day Total # of Arrivals 8:00 37 9:00 70 10:00 144 11:00 188 12:00 272 13:00 249 14:00 258 15:00 237 16:00 281 17:00 192 18:00 78 19:00 51 20:00 49 21:00 34 22:00 22 23:00 0 Table 27: Total Arrivals By Hour Of The Day For David Lam Library For the Koerner Library, any hour with an hourly total less than or equal to 6,000 arrivals was grouped as a low hour of the day. Any hour with an hourly total greater than 6,000 arrivals but less than or equal to 12,000 arrivals was grouped as a moderate hour. Any day with an hourly total greater than 12,000 arrivals was grouped as a high hour. For the Woodward Biomedical Library, any hour with an hourly total less than or equal to 1,500 arrivals was grouped as a low hour of the day. Any hour with an hourly total greater than 1,500 arrivals but less than or equal to 4,000 arrivals was grouped as a moderate hour. Any day with an hourly total greater than 4,000 arrivals was grouped as a high hour. For the David Lam Library, any hour with an hourly total less than or equal to 100 arrivals was grouped as a low hour of the day. Any hour with an hourly total greater than 100 arrivals but less than or equal to 200 arrivals was grouped as a moderate hour. Any day with an hourly total greater than 200 arrivals was grouped as a high hour. IV.1.5 CALCULATION OF ARRIVAL RATES The arrival rate was calculated for the period combinations determined in IV. 1.4 for each branch. First, all of the hourly data on arrivals was grouped according to the period combination in which it belongs. The average number of arrivals was determined by dividing the total number of arrivals for that period combination by the total number of hours for that period combination. The resulting value for each period combination was used as the number of arrivals for an hour of that type of period combination for the circulation desk at the corresponding branch. 41 IV.1.6 DEVELOPMENT OF STAFFING RULES Queueing Theory was used to determine the minimum number of staff needed at the circulation desk to meet a particular service level for each of the period combinations. The same service time used previously was also used for the input in this queueing model. Recall it was assumed the service times are exponentially distributed. It was also assumed that the inter-arrival times are exponentially distributed. These assumptions allow the queueing system to be modelled as an M/M/c queue. It was assumed that at least one staff member will be available at the desk for all open time-periods. An explanation of the theory, as well as any relevant formulas, is given in section III. 1.6. Notation and other assumptions used previously remain the same. Again, the Queueing Tool-Pak was used to do the analysis. The first chart set used the formula for the distribution of the waiting times found in section III. 1.6.3. The second chart set used the formula for the mean time in queue found in section III. 1.6.3. U B C Library's management considered the probability a client waits no more than 120 seconds is greater than 80% an acceptable service level for a circulation desk. This criterion was used to develop the staffing rules. IV. l. 7 COMPARISON AND SENSITIVITY ANALYSIS The staffing rules were compared with the baseline to determine how well the staffing rules perform over the entire academic year. This was to ensure that with the chosen format of periods, the staffing rules meet the demand adequately and there was not an abundance of unsatisfied demand or an abundance of low utilization of the servers. The sensitivity analysis portion of the project determined how changes in the figures used to scale the electronic data affected the recommendation of staffing rules for the circulation desk for each branch. There were four areas considered: changes in the percentage of renewals that occur as the only transaction type done by a client, changes in the percentage of money-collected transactions that occur at the circulation desk while a client is there, changes in the average number of transactions for each client, and changes in service time. A range of allowable change for each of the four types was determined. IV1.8 SELF-CHARGE ASSESSMENT The final portion of the project covered an assessment on the self-charge machines located at Koerner Library and Woodward Biomedical Library. A self-charge machine allows for a patron to charge out his/her own material, thereby circumventing the circulation desk. 42 A l l books made available for use outside of the library have a bar code affixed to it for scanning into the computer. There are two areas the bar codes are placed on books. They can either be placed on the inside of the book cover or on the outside of the book cover. As well, there are two self-charge machines located at Koerner Library. This thesis analyzed the reliability of both barcode placement and machine selection. To do so, there were two data sets collected on each self-charge machine. The first data set was charge transactions of books with the bar code on the inside of the book. The second data set was charge transactions of books with the bar code on the outside of the book. Both types of data sets were collected on each machine. The self-charge machine located at Woodward Biomedical Library is the same type of machine as those located at Koerner Library. The assessment was only done on the machines located at Koerner Library and, because of the similarity of the machines, it was assumed the results are transferable to the machine at Woodward Biomedical Library. Note that Machine 1 is the self-charge machine closest to the exit doors and machine 2 is the self-charge machine closest to the elevator in Koerner Library. IV.1.8.1 Data Collection for Self-Charge Machines 50 books (in batches of 10) where the bar code is placed on the outside of the book and 50 books (in batches of 10) where the bar code is placed on the inside of the book were charged out on each of the self-charge machines. There were a total of 200 observations and 20 batches. The data is shown in Table 28 and Table 29. The values in the Table 28 are the number of successes out of 50 trials on the corresponding machine with the corresponding bar code placement. The values in Table 29 are the number of successes in each batch of 10. A success was when a book can be charged (checked out) at the self-charge machine. Bar Code Placement inside cover outside cover hine Machine 1 38 39 Mac Machine 2 43 38 Table 28: The Number Of Successes For The Self-Charge Machines 4 3 Batch Successes Batch Successes 1 7 11 8 2 9 12 10 3 8 13 8 4 7 14 9 5 7 15 8 6 10 16 7 7 8 17 8 8 5 18 7 9 8 19 8 10 8 20 8 Table 29: The Number Of Successes For The Self-Charge Machines In Batches Of 10 IV.1.8.2 Assessment of Reliability By Machine The first assessment of reliability determined i f the selection of self-charge machine had an effect on the number of successful charge transactions. The statistical test in Appendix XXII was used to determine if the difference between proportions of the two samples is significant. Sample 1 corresponds to self-charge machine 1 and sample 2 corresponds to self-charge machine 2. IV.1.8.3 Assessment of Reliability By Scan Barcode Placement The second assessment of reliability determined i f the placement of the bar code had an effect on the number of successful charge transactions. The same statistical test from Appendix XXII was used to determine i f the difference between proportions of the two samples is significant. Sample 1 corresponds to a bar code placement on the inside of the cover and sample 2 corresponds to a bar code placement on the outside of the cover. IV.1.8.4 Determination of Interaction Effect To determine if there was an interaction effect between machine and bar code placement, two analyses were carried out. First, a graph of the estimated marginal means of the probability of success was developed. Second, a binary categorical logistic regression was done. The graph in Figure 6 shows the estimated marginal means of the probability of success. 44 # 0.88 a> u 3 0.84 co ° 0.8 3 0.76 JO o a. 0.72 Inside Outside Machine 1 Machine 2 Machine Figure 6: Estimated Marginal Means Of The Probability Of Success The binary categorical logistic regression was done using SPSS, a statistical software package, to determine i f there was an interaction effect of the machine choice and bar code choice on the probability of a successful charge transaction. An explanation of this type of regression is in Appendix XXIII. 45 IV.2 RESULTS The results of the analyses supply four types of information. The first was an understanding of the status quo and what activities the circulation desks were currently experiencing. The second was a time frame of peak and low hours. The third was a set of staffing rules, with limits of allowable change. The fourth was an assessment on the reliability of the self-charge machines. IV.2.1 STATUS QUO The monthly graphs and the daily graphs of transactions of which a subset is depicted in Appendix X X and Appendix X X I , show that as the semester elapses there was an increasing trend of activity for all three branches. There was also a dramatic reduction of demand in all three branches at the end of the semester. The monthly graphs are useful for determining periods of increase and decrease of activity throughout the year. The daily graphs are useful for establishing peak and idle periods throughout the day. IV.2.2 BASELINE The baseline contains two types of charts. A subset of the first type is in Appendix X X I V . The charts indicate the minimum number of staff needed to ensure that the probability a client waits less than a certain length of time is greater than or equal to a certain service level, in this case 80%, for each branch across all hours, days, and months. The charts contain a range of waiting times for examination. Recall, it was assumed at least one staff member would be available at the desk for all open time periods. A subset of the second type of charts is in Appendix X X V . The charts indicate the mean time in queue for a range of number of staff at the circulation desk, for each branch across all hours, days, and months. The charts contain a variety of values for the number of staff available at the circulation desk. IV.2.3 PERIODS There are three sets of period groupings for each library: like months, like days of the week, and like hours of the day. Analysis was conducted on these grouping as apposed to every period, thereby eliminating redundant calculations. IV.2.3.1 Grouping of Like Months Table 19 shows that for Koerner Library, December and May were similar, September, October, January, February, and April were similar, and November and March were similar months, with respect to total number of arrivals for the month. 46 The total graphs and groupings can be seen in Figure 7. The light coloured bars are months grouped with a low number of arrivals; the slightly darker coloured bars are months grouped with a moderate number of arrivals; the black coloured bars are months grouped with a high number of arrivals. 25000.00 20000.00 - 15000.00 fl 10000.00 5000.00 0.00 Monthly Totals n Is I I O CM O I 5 CD O O CN CM O o CN CO CM 3 O CNJ o o CNJ o o < Month Figure 7: Total Arrivals By Month For Koerner Library Table 20 shows that for Woodward Biomedical Library, December and May were similar, September, February, and April were similar, and October, November January, and March were similar months, with respect to total number of arrivals for the month. The total graphs and groupings can be seen in Figure 8. The light coloured bars are months grouped with a low number of arrivals; the slightly darker coloured bars are months grouped with a moderate number of arrivals; the black coloured bars are months grouped with a high number of arrivals. 8000 Figure 8: Total Arrivals By Month For Woodward Biomedical Library 47 Table 21 shows that for David Lam Library, December and May were similar, September, October, January, February, and April were similar, and November and March were similar months, with respect to total number of arrivals for the month. The total graphs and groupings can be seen in Figure 9. The light coloured bars are months grouped with a low number of arrivals; the slightly darker coloured bars are months grouped with a moderate number of arrivals; the black coloured bars are months grouped with a high number of arrivals. Figure 9: Total Arrivals By Month For David Lam Library IV.2.3.2 Grouping of Like Days of the Week Table 22 shows that for Koerner Library, Monday, Tuesday, Wednesday, Thursday, and Friday were similar days and Saturday and Sunday were similar days, with respect to total number of arrivals for the week. The total graphs and groupings can be seen in Figure 10. The light coloured bars are days grouped with a low number of arrivals; the black coloured bars are days grouped with a high number of arrivals. 48 Figure 10: Total Arrivals By Day Of The Week For Koerner Library Table 23 shows that for Woodward Biomedical Library, Monday, Tuesday, Wednesday, Thursday, and Friday were similar days and Saturday and Sunday were similar days, with respect to total number of arrivals for the week. The total graphs and groupings can be seen in Figure 11. The light coloured bars are days grouped with a low number of arrivals; the black coloured bars are days grouped with a high number of arrivals. Figure 11: Total Arrivals By Day Of The Week For Woodward Biomedical Library Table 24 shows that for David Lam Library, Monday, Tuesday, Wednesday, and Thursday were similar days and Friday, Saturday, and Sunday were similar days, with respect to total number of arrivals for the week. 49 The total graphs and groupings can be seen in Figure 12. The light coloured bars are days grouped with a low number of arrivals; the black coloured bars are days grouped with a high number of arrivals. Figure 12: Total Arrivals By Day Of The Week For David Lam Library IV.2.3.3 Grouping of Like Hours of the Day Table 25 shows that for Koerner Library, 8:00, 9:00, 19:00, 20:00, 21:00, 22:00, and 23:00 were similar hours, 10:00, 11:00, 17:00, and 18:00 were similar hours, and 12:00, 13:00, 14:00, 15:00, and 16:00 are similar hours, with respect to total number of arrivals for the hour. The total graphs and groupings can be seen in Figure 13. The light coloured bars are hours grouped with a low number of arrivals; the slightly darker coloured bars are hours grouped with a moderate number of arrivals; the black coloured bars are hours grouped with a high number of arrivals. o ro o 16000 14000 12000 10000 8000 \ 6000 4000 2000 0 -I Hour Of The Day C M C M C M eg Figure 13: Total Arrivals By Hour Of The Day For Koerner Library 50 Table 26 shows that for Woodward Biomedical Library, 19:00, 20:00, 21:00, 22:00, and 23:00 were similar hours, 8:00, 9:00, 10:00, 17:00, and 18:00 were similar hours, and 11:00, 12:00, 13:00, 14:00, 15:00, and 16:00 were similar hours, with respect to total number of arrivals for the hour. The total graphs and groupings can be seen in Figure 14. The light coloured bars are hours grouped with a low number of arrivals; the slightly darker coloured bars are hours grouped with a moderate number of arrivals; the black coloured bars are hours grouped with a high number of arrivals. finnn Figure 14: Total Arrivals By Hour Of The Day For Woodward Biomedical Library Table 27 shows that for David Lam Library, 8:00, 9:00, 18:00, 19:00, 20:00, 21:00, 22:00, and 23:00 were similar hours, 10:00, 11:00, and 17:00 were similar hours, and 12:00, 13:00, 14:00, 15:00, and 16:00 were similar hours, with respect to total number of arrivals for the hour. The total graphs and groupings can be seen in Figure 15. The light coloured bars are hours grouped with a low number of arrivals; the slightly darker coloured bars are hours grouped with a moderate number of arrivals; the black coloured bars are hours grouped with a high number of arrivals. 51 Hour Of The Day Figure 15: Total Arrivals By Hour Of The Day For David Lam Library IV.2.3.4 Summary of Groupings Per Branch A summary of the period groupings for Koerner Library, Woodward Biomedical Library, and David Lam Library are depicted in Table 30, Table 31, and Table 32. For all three branches, there were eighteen different combinations of period types. Low Moderate High December September November .c May October March c o January S February April Saturday Not Applicable Monday >. Sunday Tuesday re Q Wednesday Thursday Friday 8:00 10:00 12:00 9:00 11:00 13:00 u 19:00 17:00 14:00 —i O -T-20:00 18:00 15:00 21:00 16:00 22:00 23:00 Table 30: Breakdown Of Periods For Koerner Library 52. Low Moderate High December September October •«-» C May February November o 5 April January March Saturday Not Applicable Monday >, Sunday Tuesday. (0 Q Wednesday Thursday Friday 19:00 8:00 11:00 20:00 9:00 12:00 3 21:00 10:00 ,13:00 O T. 22:00 17:00 14:00 23:00 18:00 15:00 16:00 Table 31: Breakdown Of Periods For Woodward Biomedical Library Low Moderate High December September November May October March c o January S February April Friday Not Applicable Monday ns Saturday Tuesday Q Sunday Wednesday Thursday 8:00 10:00 12:00 9:00 11:00 13:00 18:00 17:00 14:00 3 19:00 15:00 O i 20:00 16:00 21:00 22:00 23:00 Table 32: Breakdown Of Periods For David Lam Library IV.2.4 ARRIVAL RATES Using the calculation described in section IV. 1.5, the arrival rate for each of the combinations was calculated for all three branches. Table 33> Table 34, and Table 35 show the values determined. The final column in the table, labelled 'Average # Arrivals' (which is 53 the average number of arrivals for an hour of that type of period combination), was used as input for arrivals for the corresponding period combination in the queueing analysis. Note that the values in the table are rounded to two decimal places. Month Day Hour # Of Arrivals # Of Days Average # Arrivals Low Low Low 314.25 126 2.49 Low Low Moderate 441.42 72 6.13 Low Low Peak 1190.00 90 13.22 Low Peak Low 1513.75 308 4.91 Low Peak Moderate 2961.67 176 16.83 Low Peak Peak 6839.92 220 . 31.09 Moderate Low Low 1492.83 294 5.08 Moderate Low Moderate 2369.75 168 14.11 Moderate Low Peak 5023.58 210 23.92 Moderate Peak Low 9385.25 756 12.41 Moderate Peak Moderate 17388.00 432 40.25 Moderate Peak Peak 35844.75 540 66.38 Peak Low Low 1723.25 126 13.68 Peak Low Moderate 1922.92 72 26.71 Peak Low Peak 3796.67 90 42.19 Peak Peak Low 5868.75 301 19.50 Peak Peak Moderate 9902.42 172 57.57 Peak Peak Peak 20104.92 215 93.51 Table 33: Summary Of Calculation Process For Koerner Library Month Day Hour # Of Arrivals # Of Days Average # Arrivals Low Low Low 0.00 90 0.00 Low Low ' Moderate 94.38 90 1.05 Low Low Peak 437.50 108 4.05 Low Peak Low 378.25 220 1.72 Low Peak Moderate 1404.38 220 6.38 Low Peak Peak 2707.50 264 10.26 Moderate Low Low 0.00 130 0.00 Moderate Low Moderate 405.38 130 3.12 Moderate Low Peak 1238.63 156 7.94 Moderate Peak Low 943.00 310 3.04 Moderate Peak Moderate 3543.50 310 . 11.43 Moderate Peak Peak 6913.63 372 18.59 Peak Low Low 0.50 170 0.00 Peak Low Moderate 891.50 170 5.24 Peak Low Peak 3103.00 204 15.21 Peak Peak Low 2183.25 445 4.91 Peak Peak Moderate 7639.50 445 17.17 Peak Peak Peak 13742.25 534 25.73 Table 34: Summary Of Arrival Calculation Process For Woodward Biomedical Library 54 Month Day Hour # Of Arrivals # Of Days Average # Arrivals Low Low Low 2.42 216 0.01 Low Low Moderate 6.00 81 0.07 Low Low Peak 76.67 135 0.57 Low Peak Low 43.83 280 0.16 Low Peak Moderate 50.33 105 0.48 Low Peak Peak 118.58 175 0.68 Moderate Low Low 11.42 496 0.02 Moderate Low Moderate 65.00 186 0.35 Moderate Low Peak 259.75 310 0.84 Moderate Peak Low 187.83 704 0.27 Moderate Peak Moderate 240.92 264 0.91 Moderate Peak Peak 455.92 440 1.04 Peak Low Low 7.67 224 0.03 Peak Low Moderate 52.33 84 0.62 Peak Low Peak 142.67 140 1.02 Peak Peak Low 86.50 264 0.33 Peak Peak Moderate 110.00 99 1.11 Peak Peak Peak 241.75 165 1.47 Table 35: Summary Of Arrival Calculation Process For David Lam Library IV.2.5 RESULTS OF QUEUEING ANALYSIS Two types of charts were developed for each branch to determine staffing rules that will meet demand, while ensuring utilization is still acceptable. The first set of charts indicate the minimum number of staff needed at each branch across all period combinations to ensure the probability a client waits less than a certain length of time is greater than or equal to a certain service level, in this case 80%. The charts contain a range of waiting times for examination. These charts are in Appendix X X V I . The second set of charts indicate the mean time in queue for a range of number of staff at the circulation desk for each branch across all period combinations. The charts contain a variety of values for the number of staff available at the circulation desk. These charts are in Appendix X X V I I . IV.2.5.1 Staffing Rules U B C Library's management considered the probability a client waits no more than 120 seconds is greater than 80% an acceptable service level for a circulation desk. Using this criterion, the staffing rules were determined from Appendix X X V I and Appendix X X V I I . The staffing rules for each branch are depicted in Table 36, Table 37, and Table 38. 55 # Of Staff Needed At Month Day Hour Circulation Desk Low Low Low 1 Low Low Moderate 1 Low Low Peak 2 Low Peak Low 1 Low Peak Moderate 2 Low Peak Peak 2 Moderate Low .Low 1 Moderate Low Moderate 2 Moderate Low Peak 2 Moderate Peak Low 2 Moderate Peak Moderate 2 Moderate Peak Peak 2 Peak Low Low 2 Peak Low Moderate 2 Peak Low Peak 2 Peak Peak Low 2 Peak Peak Moderate 2 Peak Peak Peak 3 Table 36: Staffing Rules For Koerner Library #Of Staff Needed At Month Day Hour Circulation Desk Low Low Low 1 Low Low Moderate 1 Low Low Peak 1 Low Peak Low 1 Low Peak Moderate 1 Low Peak Peak 1 Moderate Low Low 1 Moderate Low Moderate 1 Moderate Low Peak 1 Moderate Peak Low 1 Moderate Peak Moderate 1 Moderate Peak Peak Peak Low Low 1 Peak Low Moderate 1 Peak Low Peak 2 Peak Peak Low 1 Peak Peak Moderate 2 Peak Peak Peak 2 Table 37: Staffing Rules For Woodward Biomedical Library # Of Staff Needed At Month Day Hour Circulation Desk Low Low Low 1 Low Low Moderate 1 Low Low Peak 1 Low Peak Low 1 Low Peak Moderate 1 Low Peak Peak 1 Moderate Low Low 1 Moderate Low Moderate 1 Moderate Low Peak 1 Moderate Peak Low 1 Moderate Peak Moderate 1 Moderate Peak Peak 1 Peak Low Low 1 Peak Low Moderate 1 Peak Low Peak 1 Peak Peak Low 1 Peak Peak Moderate 1 Peak Peak Peak 1 Table 38: Staffing Rules For David Lam Library To develop a schedule that can be implemented, the staffing rules can be used as constraints in a linear program or in scheduling software for each day. One only needs to note the activity level for that period combination to determine the number of staff required for that hour as input for the constraint. Note that there may be occasions where there are more personnel scheduled than is required. IV.2.6 COMPARISON AND SENSITIVITY ANALYSIS The comparison between the baseline determined earlier in the thesis and the staffing rules used the criterion that the probability a client waits no more than 120 seconds is greater than 80%. The staffing rules were compared to the baseline with respect to the minimum number of servers needed at the circulation desk over all open hours within a branch. A subset of graphs illustrating the comparison is in Appendix XXVIII . The comparison indicated the staffing rules perform adequately. Four things were considered for the sensitivity analysis: changes in the percentage of renewals that occur as the only transaction type done by a client, changes in the percentage of money-collected transactions that occur at the circulation desk while a client is there, changes in the average number of transactions for each client, and changes in service time. IV.2.6.1 Change in Percentage of Renewal Transactions There were two types of changes considered for the percentage of renewal transactions that occur at the circulation desk, where this is the sole transaction. The first was a reduction of 57 the percentage of renewal transactions. The second was an increase of the percentage of renewal transactions. A summary of the results is in Table 39. Recall, the percentage of renewal transactions as the sole transaction was 25%. Koerner Library Woodward Biomedical Library David Lam Library Lower Limit 24% 20% None Upper Limit 32% 25% None Table 39: Range Of Allowable Change In Percentage Of Renewal Transactions An increase of staff needed at the circulation desk for any of the different period combinations for Koerner Library is realized when the percentage of renewal transactions decreases beyond 24%. A decrease of staff is realized when the percentage of renewal transactions increases beyond 32%. An increase of staff needed at the circulation desk for any of the different period combinations for Woodward Biomedical Library is realized when the percentage of renewal transactions decreases beyond 20%. A decrease of staff is realized when the percentage of renewal transactions increases beyond 25%. There is no change in the staffing rules for David Lam Library when the percentage of renewal transactions decreases or increases. IV.2.6.2 Change in Percentage of Money-Collected Transactions There were two types of changes considered for the percentage of money-collected transactions that occur at the circulation desk while a client is there. The first was a reduction of the percentage of money-collected. The second was an increase of the percentage of money-collected transactions. A summary of the results is in Table 40. Recall, the percentage of money-collected transactions was 75%. Koerner Library Woodward Biomedical Library David Lam Library Lower Limit 71% 58% None Upper Limit 97% 76% None Table 40: Range OfAllowable Change In Percentage Of Money-Collected Transactions An increase of staff needed at the circulation desk for any of the different period combinations for Koerner Library is realized when the percentage of money-collected transactions decreases beyond 71%. A decrease of staff is realized when the percentage of money-collected transactions increases beyond 97%. 58 An increase of staff needed at the circulation desk for any of the different period combinations for Woodward Biomedical Library is realized when the percentage of money-collected transactions decreases beyond 58%. A decrease of staff is realized when the percentage of money-collected transactions increases beyond 76%. There is no change in the staffing rules for David Lam Library when the percentage of money-collected transactions decreases or increases. IV.2.6.3 Change in Average Number of Transactions Per Client Two types of changes were considered for the average number of transactions for a client. The first was a reduction of the average number of transactions for a client. The second was an increase of the average number of transactions for a client. A summary of the results is in Table 41. Recall, the average number of transactions per client for Koerner Library and David Lam Library was 3 and the average number of transactions per client for Woodward Biomedical Library was 2. Koerner Library Woodward Biomedical Library David Lam Library Lower Limit 2.3 2.0 0.4 Upper Limit 3.1 2.5 None Table 41: Range Of A llowable Change In Average Number Of Transactions For A Client An increase of staff needed at the circulation desk for any of the different period combinations for Koerner Library is realized when the average number of transactions for a client decreases beyond 2.3. A decrease of staff is realized when the average number of transactions for a client increases beyond 3.1. An increase of staff needed at the circulation desk for any of the different period combinations for Woodward Biomedical Library is realized when the average number of transactions for a client decreases beyond 2.0. A decrease of staff is realized when the average number of transactions for a client increases beyond 2.5. There is no change in the staffing rules for David Lam Library when the average number of transactions for a client increases beyond 2, but an increase of staff is realized when the average number of transactions for a client decreases beyond 0.4. IV.2.6.4 Change in Service Time There were two types of changes considered for service time. The first was a reduction of time needed to serve a client at the circulation desk. The second was an increase of time needed to serve a client at the circulation desk. A summary of the results is in Table 42. Recall, The service time for all three branches was 60 seconds. 59 Koerner Library Woodward Biomedical Library David Lam Library Lower Limit 60 57 None Upper Limit 66 60 72 Table 42: Range Of Allowable Change In Service Time A decrease of staff needed at the circulation desk for any of the different period combinations for Koerner Library is realized when the service time decreases beyond 60 seconds. A n increase of staff is realized when the service time for a client increases beyond 66 seconds. A decrease of staff needed at the circulation desk for any of the different period combinations for Woodward Biomedical Library is realized when the service time decreases beyond 57 seconds. An increase of staff needed is realized when the service time for a client increases beyond 60 seconds. There is no change in the staffing rules for David Lam Library when the service time for a client decreases beyond 60 seconds, but an increase of staff is realized when the service time for a client increases beyond 72 seconds. IV.2.7 SELF-CHARGE MACHINES The first hypothesis tested was that the proportion of successful charge transactions on machine 1 is the same as the proportion of successful charge transactions on machine 2. The alternative was that there is a noticeable difference. The p-value for this test was 0.4902, which is rather large. Thus, there was not enough evidence to reject the null hypothesis. Therefore, the reliability of a self-charge machine does not depend on which machine is used. The second hypothesis tested was that the proportion of successful charge transactions with the bar code placed on the inside of the front cover is the same as the proportion of successful charge transactions with the bar code placed on the outside of the front cover. The alternative was that there is a noticeable difference. The p-value for this test was 0.4902. Thus, there was not enough evidence to reject the null hypothesis. Therefore, the reliability of a self-charge machine does not depend on where the bar code is placed on the book. The graph in Figure 6, the estimated marginal means of the probability of success, indicated there is some interaction effect. However, the binary categorical logistic regression shows that for / = 0, 1,2, and 3, Pi was not significantly different from 0 ((30 = 1.153 with P = 0.000, Bi = 0.662 with P = 0.207, p 2 = 0.113 with P = 0.812, and p 3 = -0.775 with P = 0.274). There was not enough data to indicate that this effect is significant. This interaction effect showing in Figure 6 is likely due to the fact that the set up of machine 2 corresponds to the placement of the bar code on the inside of the cover. With adjustment, machine 2 can also be used when the bar code is on the outside of the cover. This indicates that machine choice and bar 60 code placement are not significant, corroborating previous findings. This also indicates that interaction between self-charge machine choice and bar code placement is not significant The data in Table 29 shows that in 18 out of 20 (90%) of the trials, at least one book is unable to be processed by the self-charge machine. A client will then have to carry on their transaction at the circulation desk. 90% of the clients would have to use the circulation desk after using the self-charge machine. In fact, the proportion of successful charge transactions on the machines found with the data overall is 0.79 or 79%. Both of these percentages of unsuccessful transactions that occur at the self-charge machines are quite high. When you consider the large volume of transactions that occur at the self-charge machines, the increase in the number of clients at the circulation desk as carry over from the self-charge machines due to this unreliability is dramatic. 61 IV. 3 INTERPRETA TION OF RESUL TS Utilization of the circulation desks in Koerner Library, Woodward Biomedical Library, and David Lam Library was found to be extremely variable. The activity levels varied not only within the year, but also within the week and the day. For Koerner Library, the analyses revealed that December and May were light months, September, October, January, February, and April were average months, and November and March are heavy months, with respect to total number of arrivals for the month; that Monday, Tuesday, Wednesday, Thursday, and Friday are heavy days and Saturday and Sunday are light days; and that 8:00, 9:00, 19:00, 20:00, 21:00, 22:00, and 23:00 are light hours, 10:00, 11:00, 17:00, and 18:00 are average hours, and 12:00, 13:00, 14:00, 15:00, and 16:00 are heavy hours. For Woodward Biomedical Library, the analyses reveal that December and May are light months, September, February, and April are average months, and October, November January, and March are heavy months, with respect to total number of arrivals for the month; that Monday, Tuesday, Wednesday, Thursday, and Friday are heavy days and Saturday and Sunday are light days; and that 19:00, 20:00, 21:00, 22:00, and 23:00 are light hours, 8:00, 9:00, 10:00, 17:00, and 18:00 are average hours, and 11:00, 12:00, 13:00, 14:00, 15:00, and 16:00 are heavy hours. For David Lam Library, the analyses reveal that December and May are light months, September, October, January, February, and April are average months, and November and March are heavy months, with respect to total number of arrivals for the month; that Monday, Tuesday, Wednesday, and Thursday are heavy days and Friday, Saturday, and Sunday are light days; and that 8:00, 9:00, 18:00, 19:00, 20:00, 21:00, 22:00, and 23:00 are light hours, 10:00, 11:00, and 17:00 are average hours, and 12:00, 13:00, 14:00, 15:00, and 16:00 are heavy hours. The main results are depicted in Table 36, Table 37, and Table 38. The tables are the recommended staffing rules for all 18 period combinations for each of the branches. These staffing rules indicate the minimum number of staff needed on a circulation desk at different time periods to meet the respective time period's demand. The number of staff needed at the circulation desk in Koerner Library is 3 for a peak month, day, and hour, 1 for a low month, day, and hour, 1 for a low month and day and a moderate hour, 1 for a low month and hour and a peak day, 1 for a low day and hour and a moderate month, and 2 for all other combinations. The number of staff needed at the circulation desk in Woodward Biomedical Library is 2 for a peak day and hour and a moderate month, 2 for a peak month and hour and a low day, 2 for a peak month and day and a moderate hour, 2 for a peak month, day, and hour, and 1 for all other combinations. The number of staff needed at the circulation desk in David Lam Library is 1 for all periods. A l l three branches had a large range for allowable change for percentage of renewals that occur as the only transaction type done by a client at the circulation desk, changes in the 62 percentage of money-collected transactions that occur at the circulation desk while a client is there, and changes in the average number of transactions for each client. David Lam Library had the largest range, with minimal limits to the allowable change. The only limit that David Lam Library did have is that the average number of transactions for a client could not go below 0.4 without a change occurring in the staffing rules. Koerner Library and Woodward Biomedical Library are more limited than David Lam Library, though they both have significant ranges of allowable change. For changes in the percentage of renewals that occur as the only transaction type done by a client at the circulation desk and changes in the percentage of money-collected transactions that occur at the circulation desk while a client is there, Koerner Library had more freedom in the increase of both. On the other hand, Woodward Biomedical Library had more freedom in the reduction of both. For changes in the average number of transactions for each client, a decrease is preferable for Koerner Library, while a decrease would result in staffing rule changes for Woodward Biomedical Library. An increase in the average number of transactions for each client is preferable for Woodward Biomedical Library. The change in service rate analysis indicates that changes in the arrival rate input of the queueing system will have a slight effect on the staffing rules for all three of the branches. Small changes in the service time for Koerner Library and Woodward Biomedical Library will affect the staffing rules, but there is minimal effect for David Lam Library. In summary, the staffing rules are not susceptible to changes in the percentage of renewals that occur as the only transaction type done by a client at the circulation desk, changes in the percentage of money-collected transactions that occur at the circulation desk while a client is there, and changes in the average number of transactions for each client. The staffing rules are, however, dependent on the service time of the staff. The reliability of the self-charge machines is limited and the analysis strongly indicates that the majority of clients using the machines must continue their transactions, at the circulation desk. There is no discernable difference between machines or bar code placement with respect to reliability. 63 IV.4 SUMMARY The main focus of this study is to analyze the demand o f the circulation desks located in Koerner Library, Woodward Biomedical Library, and David Lam Library. From this analysis, staffing rules (the number of staff required for the circulation desk) for periods of different levels o f demand are developed to aid U B C Library's management with scheduling of the circulation desks. A rudimentary reliability study on the self-charge machines located at Koerner Library and Woodward Biomedical Library is conducted to gauge the reliability o f the machines. The key results are a set o f staffing rules for the circulation desks o f Koerner Library, Woodward Biomedical Library, and David Lam Library and a summary of the reliability of the self-charge machines for the use of U B C Library's management. The staffing rules consist of three matrices, one for each branch, that specify the minimum number of staff that need to be offering service at each circulation desk to ensure that a client waits no more than 120 seconds for any of the different periods of activity levels. To accompany these staffing rules, there are three matrices, one for each branch, that give the mean waiting time in the queue. Due to mitigating circumstances, no changes have been made to staffing levels for the circulation desks as a result o f this report. However, U B C Library's management has ascertained the need for action on improving or changing the self-charge machines from the reliability analysis. A s a result of this project and the type of recommendations that are made, U B C Libraries has again employed the Centre for Operations Excellence (COE) to perform other projects. IV.4.1 COMMENTS A n observation noted during data collection on the reliability of the self-charge machines is that machine 2 is set up for books in which the barcode is on the inside cover. If the barcode is on the outside o f the cover o f the book, the machine w i l l not work without modifications in the scanning method. Another book has to be placed in a way that acted as the open cover of the book being scanned before the machine wi l l work. It is recommended that the machines be modified or upgraded to improve reliability of self-charge and to capture this incongruity. IV.4.2 FURTHER WORK The more appropriate tool for the analysis is the use o f a nonhomogeneous Poisson Process as the system is non-stationary. In doing so, the complexity of the analysis increases dramatically. The advantage o f using such a process is to ensure that carry over of unsatisfied demand from one period to the next is accounted for in the analysis. Since this option is not used in this analysis, it is suggested that a simulation be done to check the validity of the staffing rules when there is a carry over o f unsatisfied demand. 64 The electronic data is not a direct reflection of what really happens in the queueing system, though it is accurate enough for this analysis. For a more in-depth study on the queueing system and the development of staffing rules that better closely follow demand patterns, it is recommended that data be observed at each of the branches to use as input in the analysis. 65 V REFERENCES Alston, R. "Why Are We Waiting." Public Library Journal, Volume 11 Number 2 March/April 1996: 37-43. Ashley, D.W. " A Spreadsheet Optimization System for Library Staff Scheduling." Computers Operations Research, Volume 22, Number 6, 1995: 615-624. Bluman, A . G . Elementary Statistics: A Step By Step Approach. Third Edition. WCB/McGraw Hi l l , Boston, 1997. Guo, Y . "Staff Scheduling and Workstation Allocation at UBC Libraries." Unpublished Thesis. UBC March 2003. Hall, R.W. Queueing Methods. Prentice Hall, Englewood Cliffs, 1991. Hillier, F.S., Lieberman, G.J. Introduction To Operations Research. Seventh Edition. McGraw Hil l , Boston, 2001. Kelton W.D., Sadowski R.P, Sadowski D.A. Simulation With Arena. WCB/McGraw Hi l l , Boston, 1998. Mansfield, J.W. "Human Factors of Queuing: A Library Circulation Model." The Journal of Academic Librarianship, Volume 6. Number 6, 1981: 342-344. McKern, D. "Copiers for Bound Volumes: A Survey of Available Equipment." Library Technology Reports. 25 (6) Nov/Dec 89: 863-867. Mendenhall W., Wackerly D., Scheaffer R. Mathematical Statistics With Applications. Fourth Edition. PWS-Kent Publishing Company, Boston, 1990. Morse, P .M. Library Effectiveness: A Systems Approach. M.I.T. Press, Massachusetts, 1968. Morton, E. " C L A Defends Copyright Amendments." Feliciter, 42 (11/12) Nov/Dec 1996: 26-29. Rasmussen, M . "Copyright Police: Why Libraries?" Canadian Library Journal. 47 (2) Apr 90, 1999: 77-79. Smith, J .M. "The Use Of Queuing Networks and Mixed Integer Programming to Allocate Resources Optimally within a Library Layout." Journal Of The American Society For Information Science, 32 (1) Jan 81: 33-42. Swart, I. "Choosing a Photocopier". Cape Librarian, 39(1) Jan 95: 45. 66 Warwick, J. " A Queuing Theory Model for Book Reservations and Circulation." Collection Management. Volume 23 Numbers 1/2 1998: 125-137. Wilson, B. "The Utilisation and Potential of Audiovisual and Computing Facilities in a Resource Centre." Audiovisual Librarian, 16 (1) Feb 90: 12-17. Website for Arena Simulation Software http://www.arenasimulation.com/. Website of Rockwell Software for Arena http://www.software.rockwell.com/. Website for Queueing ToolPak Software developed at University of Alberta http://www.bus.ualberta.ca/aingolfsson/QTP/default.htm 67 A P P E N D I X I EVENT-DRIVEN GRAPHS Every change in the graph represents either a user arriving or a user departing the system and the graph only changes when one of these two events occurs. Each graph is a representative of the photocopying activity of a floor on a particular day. Time is along the horizontal axis and the number of users in the copy room is along the vertical axis. The bold line indicates the number of photocopiers in the room. Floor 1 Tuesday 12 CD LO co CN LO O CO LO LO LO o o LO o LO LO o o CO •st co CN N - o CN LO CO CN CO o LO •vt o N— CO o CO o T — T — CN CN CO CD T — CD LO o cb cb LO co 1^  o CN in 6) •it -<t O CO LO CN •vt o CN co LO o T — CN •st LO CO •vt LO o CN od O o N1"' CN CN CO CO CO CO co •vt •st •vf •st LO LO ib ib cb cb Time Floor 1 Thursday 12 o p o o o O I V -CN o co LO r-•vt |s- CO CN IV- IO •vt LO O co O co LO •st LO T— in o o o co m •st o o LO T — in .— in co CN s cp CO co O •? cb -?t o co CO CN s CN -Nt 66 o •st h-' CN cb co cb co CN $ ib LO cn LO CO o CN o o CN CN CN CN CN cb cb CO CO CO cb cb cb cb •it •it Time 68 Floor 2 Tuesday ® 3 o 12 10 8 6 4 2 0 HI . —1-171 I; 1 0 0 1^  •tf CN m CD CO CN 1— 1^  -tf 0 -tf CD m CO 0 CO in CN 00 in 0 00 CO in CO 0 m m 00 CO co CN co -tf 0 CN in co -tf 0 -tf s cb Lf) CO co in CN 60 CO d in cri 0 cb CN cb •tf ib 0 •tf co 5- to in 1^  0 cb cb co • t f O s cb in cb 0 cri cb CN 66 cri d CN csi csi cb cb cb cb Time cb •tf -tf •tf •tf in in in cb cb cb Floor 2 Wednesday i_ OJ 3 o 12 10 8 6 4 2 0 ___L ! I 0 CO 0 CO m 0 m 0 co -tf CSI O O in CN 0 in 0 0 T— in CO 0 CO 0 0 0 0 0 0 CM in •tf 0 0 0 •tf 0 0 in •tf 0 CO cb co cb CN in in T— ib CN in CO d in •tf csi CO cri i— csi co in s cSi CN csi co oi •tf 1^  0 cri T— co CN in s \— 06 cri d d T— T— csi csi csi cb cb cb •tf • tf •tf • t f in in in in cb cb Time Floor 2 Thursday ID k_ 0) 3 **-o 12 10 8 6 4 2 0 HI" III iTT[tnTnTinmTM''imTirirviTrHin'mmHmTri7mTiiT^ in 0 0 0 m •tf 0 co in 0 0 m 0 in in 0 0 0 0 T— m in 0 0 •tf 0 co 0 co 0 CN 0 0 in 0 in 0 CO 0 0 co cb •tf in cb 0 cb CN cb CM CO in CO cb in cb CSI cb co cri •tf in 0 ib d co d •tf cb •tf x— 0 csi 1^  d d d (Si cSi csi (Si cb CO cb cb cb •tf • t f • t f Time 69 .Floor 2 Friday 12 10 (0 • 8 3 6 o 4 * 2 0 If) in o o m o m o CN m ib di CO o CO oi m CN CO CO CO o o s o o in m o O O in m o in in o o m o o in o o in CO o CN o in in CO o ai co CO CO CO cb o o <j> in o CM CN CO in o CN CO lO o T — CN <Si CN oi CN CN CN CN co co co co cd CO Time Floor 2 Monday 12 -r o o CN o o in o •st CN • M -o o in m o CO o m m CO m 5- m in o CO oo o o in CN o in CO cSi •<J- ai T — in co oo m cb CN cb in in CO p ai p ib CN ib co ai -? ai o cSi CN CO co T — o cb cb CN in o cb CN 00 cn CO ai o d d CN CN CN CN cb CO CO co •st Time Floor 4 Wednesday Time CO in in CN o a> m CM o CN o cb cb ai CO d in o in CN CO CO CO •tf Floor 4 Friday V) Z> o 2 1 r'iri"r"TmmT o o o o \ 1 o CM O T— CO cb cb CO 1^  ai cb in o o T— CN CSI csi csi CO CO Time 70 II TIME-DRIVEN GRAPHS Every change in the graph is a moment in time, with the number of users in the room at that point in time. Each graph is a representative of the photocopying activity of a floor on a particular day. Time is along the horizontal axis and the number of users in the copy room is along the vertical axis. The bold line indicates the number of photocopiers in the copy room. Floor 1 Tuesday 1 9 to tr K, o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o p p o o Tf o o o p o o o o o o o o o o o o o o OI Tt CD CO o oi _r cb" o OI (O CO o oi cb cb o oi CD 00 o oi Tf cb' 00 CO CO CO CO CO T* 4 T? m , V) p o o o o OI OI OI OI OI oi oi oi 6S oi ci cn di CO cn oi cn o> oi oi d o d d o o o o O d o o d d o Time 12 10 8 6 4 2 0 • 1mm - — 11-10:00 12:00 14:00 16:00 18:00 ( 20:00 22:00 24:00 26:00 28:00 Time 12 » 10 5 8 1 6 •R 4 2 o o o o o o o o o o p o o o o o o o p o p o p o p o oi TJ" CD cb d oi <o d oi CO CO CO CO CO TT Tf 4 Tf •5 in in o o o o o o o o o o o o o o o o o o o p o p p p o o p p oi Tf u> cb d oi TT tb 00 d oi Tf CD 00 p O o p OI CN OI CN CN oi oi oi oi oi oi oi oi oi oi oi oi oi oi Time 71 # of Arrivals o ro o> oo o ro 15:30:00 ' - • • 15:32:00 15:34:00 15:36:00 15:38:00 15:40:00 15:42:00 15:44:00 15:46:00 15:48:00 15:50:00 15:52:00 15:54:00 15:56:00 15:58:00 H 3 16:00:00 ro 16:02:00 16:04:00 16:06:00 16:08:00 16:10:00 16:12:00 16:14:00 16:16:00 16:18:00 16:20:00 16:22:00 16:24:00 16:26:00 16:28:00 ,cf1 T" # of Arrivals o to -f^  0> oo o to 14:30:00 14:32:00 14:34:00 14:36:00 14:38:00 14:40:00 14:42:00 14:44:00 14:46:00 14:48:00 14:50:00 14:52:00 14:54:00 i; 14:56:00 14:58:00 15:00:00 15:02:00 15:04:00 15:06:00 15:08:00 15:10:00 15:12:00 15:14:00 15:16:00 15:18:00 15:20:00 15:22:00 15:24:00 15:26:00 15:28:00 # of Arrivals o to 4>> o> oo o ro 13:30:00 13:32:00 13:34:00 13:36:00 13:38:00 13:40:00 13:42:00 13:44:00 13:46:00 13:48:00 13:50:00 13:52:00 13:54:00 13:56:00 13:58:00 3' 14:00:00 CD 14:02:00 14:04:00 14:06:00 14:08:00 14:10:00 14:12:00 14:14:00 14:16:00 14:18:00 14:20:00 14:22:00 14:24:00 14:26:00 14:28:00 am # of Arr ivals o ro 4k o> oo o ro 12:30:00 12:32:00 12:34:00 12:36:00 12:38:00 12:40:00 12:42:00 12:44:00 12:46:00 12:48:00 12:50:00 12:52:00 12:54:00 12:56:00 12:58:00 13:00:00 13:02:00 13:04:00 13:06:00 13:08:00 13:10:00 13:12:00 13:14:00 13:16:00 13:18:00 13:20:00 13:22:00 13:24:00 13:26:00 13:28:00 o © c n </> a o s 3' s n a # of Arrivals o 10 4^  o oo o ro 13:30:00 * 13:32:00 13:34:00 13:36:00 13:38:00 13:40:00 13:42:00 13:44:00 13:46:00 13:48:00 13:50:00 13:52:00 13:54:00 13:56:00 13:58:00 14:00:00 14:02:00 14:04:00 14:06:00 14:08:00 14:10:00 14:12:00 14:14:00 14:16:00 14:18:00 14:20:00 14:22:00 14:24:00 14:26:00 14:28:00 # of Arrivals o ro .u cn co o NJ 12:30:00 12:32:00 12:34:00 12:36:00 12:38:00 12:40:00 12:42:00 12:44:00 12:46:00 12:48:00 12:50:00 12:52:00 12:54:00 12:56:00 12:58:00 13:00:00 13:02:00 13:04:00 13:06:00 13:08:00 13:10:00 13:12:00 13:14:00 13:16:00 13:18:00 13:20:00 13:22:00 13:24:00 13:26:00 13:28:00 # of Arrivals o fo 6 oi co o ro 11 30:00 11 32:00 11 34:00 11 36:00 11 38:00 11 40:00 11 42:00 11 44:00 11 46:00 11 48:00 11 50:00 11 52:00 11 54:00 11 56:00 Time 11 12 12 58:00 00:00 02:00 12 04:00 12 06:00 12 08:00 12 10:00 12 12:00 12 14:00 12 16:00 12 18:00 12:20:00 12 22:00 12 24:00 12 26:00 12 28:00 I ip # of Arrivals o ro O) co o ro 10:30:00 10:32:00 10:34:00 10:36:00 10:38:00 10:40:00 10:42:00 10:44:00 10:46:00 10:48:00 10:50:00 10:52:00 10:54:00 10:56:00 10:58:00 H 3' 11:00:00 <B 11:02:00 11:04:00 11:06:00 11:08:00 11:10:00 11:12:00 11:14:00 11:16:00 11:18:00 11:20:00 11:22:00 11:24:00 11:26:00 11:28:00 © o Floor 2 Tuesday 12 -I 10 in £ 8 3 6 o 4 * 2 n -P i ! B illlll|jPW^ P!p| j||j|%|||Ji|||||^ ^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^ j^ ^^ ^^ S^^ a^ ^^ ^^ ^^ .ljtj j rjmg 9:30:00 9:32:00 9:34:00 9:36:00 9:38:00 9:40:00 9:42:00 9:44:00 9:46:00 9:48:00 9:50:00 9:52:00 9:54:00 9:56:00 9:58:00 0:00:00 0:02:00 0:04:00 0:06:00 0:08:00 0:10:00 0:12:00 j 0:14:00 0:16:00 0:18:00 0:20:00 0:22:00 0:24:00 0:26:00 0:28:00 Time Time Time 74 Floor 2 Tuesday continued 12 10 o 8 3 6 4 j—sip mmw^Smmrm^s ~m I P P ^ i i ^ f c z B s i M t : | — P 2 j j t S l B I I H l ^ ^ 0 % r sli_^_l_J. ^ ^ v ^ _ - = = ^ . . = ^ . . = S ^ . - t ^ . ™ J J = S . - ^ i : i L o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 0 4 - ^ < J C > W O C N J ^ < i > C O O C ^ ^ ( J > C O O ( o o c o c o c o ^ j ' T r T r ^ ' T r w > i o i o » o m o o o o o t - f - - « - T - - . - r s i c N i c M Time Time Time 12 10 4 8 6 4 2 •vt CO •vt io io in m u> o T- T- T- V— T-N N N N N U ) I O U ) I O I O C 0 ( 0 l 0 t 0 ( 0 C 0 < 0 ( 0 C 0 < 0 ( 0 < 0 ( O C 0 ( 0 Time 75 Floor 2 Wednesday 1 2 j 1 0 • 8 . 3 6 o 4 -0 ""1 o o o o o o o o o o o o o o CD o o o o CD o o o o o o o o o o o o CD CD CD CD CD o o o o o o o o o CD o CD o CD o CD CD CD CD CD o o o CO csi CO Tf CO CO CO CO CO o Tf o i Tf Tf Tf CD Tf CO Tf o cvi IO Tf i n CD CO CO CO CD o o i CD Tf o CO o CO o CD CN Tf CD CO CD CN o i CN CN CO CN cb CN CO CO CO CO CO CO c b c b c b c b CO CO CO c b CO CT) c n 6S o i c n cn cn c n c n cri CT) cn o> CT) Time 76 Floor 2 Wednesday continued 12 10 8 6 4 2 CN|CN|CMC^CNC^CNCN|C>ICs|C»4Cg(N|CMr>4COOCOe->COCOOOeOeOCOCOOCOe-> Time Time Time o 4 • i f c ^ a ^ ^ f c * • —III —! IlilllBllilll o o o o o o o o o o o o o o o o o o o o o o o o o o o p o o o p o p o • 9? O O O O O O O O O O O O O O O O O O O O O O O O o CO CO CO CO CO CO CO o Tf csi Tf Tf Tf CD Tf CO Tf o UT) CN to Tf in CD in oo o ci in p p Tf p CD O CO O O CN Tf CD CO O CN CN CN Tf CN CO CN CO CN <o in u> u> u> u> u> to in in  i  IO CD CD Time CD CD CD CD CD CD CD CD CD CO CD CD CD 77 # of Users o ro ^ cn co o N 13:30:00 13:32:00 13:34:00 13:36:00 13:38:00 13:40:00 13:42:00 13:44:00 13:46:00 13:48:00 13:50:00 .] 13:52:00 13:54:00 13:56:00 13:58:00 14:00:00 14:02:00 14:04:00 14:06:00 14:08:00 14:10:00 14:12:00 14:14:00 14:16:00 14:18:00 14:20:00 14:22:00 14:24:00 14:26:00 14:28:00 oo #of Users o ro ^  o) oo o ro 12:30:00 12:32:00 12:34:00 12:36:00 12:38:00 12:40:00 12:42:00 12:44:00 12:46:00 12:48:00 12:50:00 12:52:00 12:54:00 12:56:00 12:58:00 - i 3 13:00:00 CD 13:02:00 13:04:00 13:06:00 13:08:00 13:10:00 13:12:00 13:14:00 13:16:00 13:18:00 13:20:00 13:22:00 13:24:00 13:26:00 13:28:00 #of Users o ro m co o ro 11:30:00 11:32:00 11:34:00 11:36:00 11:38:00 11:40:00 11:42:00 11:44:00 11:46:00 11:48:00 11:50:00 11:52:00 11:54:00 11:56:00 11:58:00 12:00:00 12:02:00 12:04:00 12:06:00 12:08:00 12:10:00 12:12:00 12:14:00 12:16:00 12:18:00 12:20:00 12:22:00 12:24:00 12:26:00 12:28:00 lilu • ;pJ| # of Users o ro ^  cn oo o ro O O 1 10:30:00 10:32:00 10:34:00 10:36:00 10:38:00 10:40:00 10:42:00 10:44:00 10:46:00 10:48:00 10:50:00 10:52:00 10:54:00 10:56:00 10:58:00 -1 3 11:00:00 (D 11:02:00 11:04:00 11:06:00 11:08:00 11:10:00 11:12:00 11:14:00 11:16:00 11:18:00 11:20:00 11:22:00 11:24:00 11:26:00 11:28:00 H i . Floor 2 Friday Time 12 O N ' ^ c o o o o c v i ^ t o o o o c N i ^ c o o o o c N i T r c D o o o r s l ^ - c o o o o r N i ^ r c o o o Time 12 j 10 w S 8 •o 4 o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p o p oCO CM CO TT CO u> CO CO CO d T* CM Tf Tf Tf CO Tf CO Tf o CM up Tf u> CO CO CO CO o  CM p Tf p CO p CO p o CM Tf CO CO oCM CM CM Tf CM CO CM CO CM CO CO CO CO CO CO CO CO CO CO CO Tf Tf" Tf" Tf Tf" Tf" Tf Tf" Tf" Tf Tf Tf" Tf" Tf Tf" Time 79 # of Arrivals o ro 4* o> oo © 10 11 30:00 : 11 32:00 : 11 34:00 | 11 36:00 ; 11 38:00 11 40:00 i 11 42:00 j 11 44:00 ; 11 46:00 j 11 48:00 I 11 50:00 ) 11 52:00 j 11 54:00 ! 11 56:00 : 11 58:00 : 3' 12 00:00 \ CD 12 02:00 | 12 04:00 12 06:00 12 08:00 12 10:00 12 12:00 12 14:00 ! 12 16:00 j 12 18:00 • 12 20:00 12 22:00 12 24:00 i 12 26:00 12 28:00 # of Arrivals o M J» cn CD o w 10:30:00 10:32:00 10:34:00 10:36:00 10:38:00 10:40:00 10:42:00 10:44:00 10:46:00 10:48:00 10:50:00 10:52:00 10:54:00 10:56:00 10:58:00 11:00:00 11:02:00 11:04:00 11:06:00 11:08:00 11:10:00 11:12:00 11:14:00 11:16:00 11:18:00 11:20:00 11:22:00 11:24:00 11:26:00 11:28:00 # of Arrivals O M O) CO O M 9:30:00 9:32:00 9:34:00 9:36:00 9:38:00 9:40:00 9:42:00 9:44:00 9:46:00 9:48:00 9:50:00 9:52:00 9:54:00 9:56:00 9:58:00 3 10:00:00 CD 10:02:00 10:04:00 10:06:00 10:08:00 10:10:00 10:12:00 10:14:00 10:16:00 10:18:00 10:20:00 10:22:00 10:24:00 10:26:00 10:28:00 # of Arrivals o ro cn co o ro 8:30:00 8:32:00 8:34:00 8:36:00 8:38:00 8:40:00 8:42:00 8:44:00 8:46:00 8:48:00 8:50:00 8:52:00 8:54:00 8:56:00 8:58:00 H 3 9:00:00 CD 9:02:00 9:04:00 9:06:00 9:08:00 9:10:00 9:12:00 9:14:00 9:16:00 9:18.00 9:20:00 9:22:00 9:24:00 9:26:00 9:28:00 Floor 2 Monday continued i 2 -.'= 0 o O O O O o o o O O O O o O o o o o o o o o o o o o o o o o p O O O O p p o O O O O p O p p p p p p p p o p p p p o p p d CN T f CO OO o Tt CO 00 o CSI Tf CO CO o CN Tf CO CO o CN T f CD CO d CN Tf" CO 00 r o co CO CO CO T f T f Tf Tf T f *p. CO CO CO CO p p p p p CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CO CO CO CO co co CO CO CO CO CO CO CO CO CO Time 12 o p o p o o o p o p o p o p o o o p o p o p o p o o o p o p o p o p o o o p o p o p o p o o o p o p o p o p o o o p o p o CN T f cb OO o CN Tf cb OO d> CN T f cb CO o CN Tf cb cb o CN T f CO 00 o CN Tf CO CO CO CO CO CO CO T f T f Tf T f T f CO CO CO co CO p o o o o T— T— T - T - CN CN CM CN CM CO CO CO CO CO CO CO CO CO CO cb CO cb CO CO T f T f Tf T f Tf" Tf T f T f Tf Tf" Tf" Tf Tf" Tf Tf Time 81 # of Users 11 :30:00 11 :32:00 11 :34:00 11 :36:00 11 :38:00 11 :40:00 11 :42:00 11 :44:00 11 :46:00 11 :48:00 11 :50:00 11 :52:00 11 :54:00 11 :56:00 11 .58:00 -I 3 12:00:00 , ro I 12:02:00 12:04:00 12:06:00 12:08:00 12:10:00 12:12:00 12:14:00 12:16:00 12:18:00 12:20:00 12:22:00 12:24:00 12:26:00 12:28:00 # of Users o -fc ro 10:30:00 j j 10:32:00 10:34:00 | 10:36:00 10:38:00 | 10:40:00 | 10:42:00 I 10:44:00 10:46:00 I 10:48:00 ( 10:50:00 10:52:00 10:54:00 10:56:00 10:58:00 Time 11 11 :00:00 ( :02:00 11 :04:00 | 11 :06:00 j 11 :08:00 11 :10:00 11 :12:00 ; 11 :14:00 11 :16:00 ) 11 :18:00 11 :20:00 11 :22:00 | 11 :24:00 \ 11 26:00 11 •28:00 # of Users 9:30:00 9:32:00 9:34:00 9:36:00 9:38:00 9:40:00 9:42:00 9:44:00 9:46:00 9:48:00 9:50:00 9:52:00 9:54:00 9:56:00 9:58:00 -t 3 10:00:00 to 10:02:00 10:04:00 10:06:00 10:08:00 10:10:00 10:12:00 10:14:00 10:16:00 10:18:00 10:20:00 10:22:00 10:24:00 10:26:00 10:28:00 #of Users 8:30:00 8:32:00 8:34:00 8:36:00 8:38:00 8:40:00 8:42:00 8:44:00 8:46:00 8:48:00 8:50:00 8:52:00 8:54:00 8:56:00 8:58:00 H 3' 9:00:00 ro 9:02:00 9:04:00 9:06:00 9:08:00 9:10:00 9:12:00 9:14:00 9:16:00 9:18:00 9:20:00 9:22:00 9:24:00 9:26:00 9:28:00 Floor 4 Wednesday continued Floor 4 Friday 84 Ill SERVICE TIME GRAPHS For each floor each day, service times are put in bins of one minute. The corresponding frequencies of the bins are calculated and then graphed. The bins are the service times (in minutes) and are along the horizontal axis. The frequencies are along the vertical axis. Floor 1 Tuesday March 13, 2001 Floor 1 Thursday March 15, 2001 20 o o o o o p p p o o o o o p p p p O S o " ^ ( N (N (N CO CO 5 T? IO O O o 0 0 0 0 0 " 0 0 0 0 0 0 0 0 o ^ ^ ' Service Time (minutes) •-pll f • IT l fh m 11 m 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 o ^ 2 ^ C ^ ^ r ^ S f N C N ( N f O S to ro ro o o o o o b o o b d o 0 0 b o b b o d o Service Time (minutes) Floor 2 Tuesday March 13, 2001 g 30 o cr £ 20 u. 10 R • R i S o o o o o o o o o o o o o Service Time (minutes) Floor 2 Wednesday March 14, 2001 * - IO <7> T- LO cn Service Time (minutes) Floor 2 Thursday March 15, 2001 0 0 0 0 0 0 0 0 0 0 N CM Service Time (minutes) Floor 2 Friday March 16, 2001 O O O O O O O O O O Service Time (minutes) 85 Floor 2 Monday March 19, 2001 40 -g 30 -a 3 CT © 20 U -10 • 1 B P „ ra „ m i 0:03:00 ; 0:05:00 ; 0:07:00 ; 0:09:00 ; 0:11:00 ; GO • 0:13:00 g 0:15:00 ; |! 0:17:00 ; Z. 0:19:00 " 3 5' 0:21:00 Sf 0:23:00 ' 0:25:00 ; 0:27:00 ; 0:29:00 ; 0:31:00 ~_ 0:33:00 ; 0:35:00 ; Floor 4 Wednesday March 14, 2001 p p p p o q q o o p o p p q q o o o O Q O O O O O O O O j o o o o o o o Service Time (minutes) Floor 4 Friday March 16, 2001 IV PROBABILITY DENSITY FUNCTION FOR AN EXPONENTIONAL DISTRIBUTION J 6>0 0 < y < oo fiy) = • otherwise V PROBABILITY DENSITY FUNCTION FOR A SHIFTED EXPONENTIONAL DISTRIBUTION 1 ~yA — e / h P B>0 6<y otherwise 86 VI CHI SQUARE TEST The Chi Square Test is used as a test for goodness-of-fit. The test determines if a particular type of distribution is a good fit for some observed data. That is, the tests are used to determine if a sample comes from a population with a certain type of distribution. i. ASSUMPTIONS The data is collected from a random sample. The expected frequency for each category is five or more. ii. T E S T STATISTIC with degrees of freedom (df) equal to the number of cells less 1. Where: O is the observed frequency E is the expected frequency k is the number of cells ni is the cell count of cell i n is the sample size np; is the expected values of cell i iii. HYPOTHESIS H 0 : the data follows a specific distribution > H a : the data do not follow a specific distribution iv. PASS/FAIL CRITERIA The hypothesis is rejected when the p-value is of a small value. 87 VII KOLMOGOROV-SMIRNOV TEST The Kolmogorov-Smirnov Test is used as a test for goodness-of-fit. The test determines i f a particular type of distribution is a good fit for some observed data. That is, the tests are used to determine i f a sample comes from a population with a certain type of distribution. i. A S S U M P T I O N S The data is collected from a random sample. The data is continuous, interval, or ratio. The theoretical distribution is fully specified at the onset of analysis. ii. T E S T ST A T I S T I C 1<1<N Where: F is the theoretical cumulative distribution N is the sample size Yj is one of N ordered data points Equivalently: D = max\F'(x)-F(x)} Where: F is the hypothesized continuous distribution function of the samples F' is the empirical distribution function of the samples and / \ _ numberofsamples < x ~ N iii. HY P O T H E S I S H 0 : the data followsa specific distribution H a : the data do not follow a specific distribution iv. P A S S / F A I L C R I T E R I A The hypothesis is rejected when the p-value is of a small value. 88 VIII OUTLIER TEST 1 Arrange the data in order from lowest to highest. 2. For Qi and Q3, compute: c = np/100 Where n = total number of values p = percentile Qi corresponds to the 25th percentile and Q3 corresponds to the 75th percentile. So, for Qi, the formula is c = 25n/100 and for Q3 the formula is c = 75n/100. If c is not a whole number, round it up to the next whole number. Start at the lowest value and count over to the rounded cl value. If c is a whole number, use the value halfway between the c and c+1 values (find it by adding c and c+1, then divide by two) when counting up from the lowest value. 3. Find the inter-quartile range (IQR). IQR = Q3 - Q, 4. Multiply the inter-quartile range by 1.5. S4= 1.5*IQR 5. Subtract the value obtained in Step 4 from Q3 - Qi and add the value to Q3 - Q|. S5- = Qj - S4 S5+ = Q3 + S4 Check the data set for any data values that fall outside the range of S5- to S5+. These values are outliers. 89 IX SERVICE TIME DATA FOR TUESDAY 0 2 4 5 5 10 10 10 12 13 15 15 18 20 22 22 28 30 30 30 30 30 31 32 37 40 40 40 40 41 43 43 45 45 47 49 49 50 50 52 53 53 54 55 55 57 58 58 59 60 60 60 61 62 65 70 70 70 71 72 75 75 75 76 78 78 80 80 81 82 85 88 88 90 90 90 90 90 92 95 95 96 97 97 101 102 105 105 106 108 110 111 112 114 118 118 120 120 120 120 121 126 127 128 128 130 130 130 130 131 134 134 135 137 138 138 140 141 144 145 145 146 148 152 152 154 155 155 156 156 157 158 160 163 165 165 166 168 168 170 170 170 170 172 173 173 177 178 180 184 185 186 186 188 188 188 190 190 190 195 195 197 197 201 206 209 210 215 218 220 220 223 225 228 235 235 235 235 240 243 244 249 250 252 257 259 260 270 275 277 277 278 278 279 280 280 288 289 290 295 296 297 300 300 302 305 305 307 308 308 310 310 315 315 315 320 321 325 332 332 335 338 339 340 340 343 345 347 350 353 355 355 357 358 359 360 360 360 368 370 370 372 375 382 383 385 385 386 388 390 390 400 400 400 402 402 405 410 410 422 425 429 430 433 435 440 448 448 448 450 452 453 454 455 457 457 460 461 471 472 475 480 488 488 493 495 502 505 507 510 510 513 518 520 523 525 536 538 545 545 550 575 580 581 594 603 610 610 610 617 618 620 625 625 655 658 664 665 668 671 675 688 698 707 720 720 720 724 726 729 735 750 757 763 764 764 781 790 810 814 820 830 845 850 853 862 866 888 892 915 918 920 935 935 950 985 992 997 1006 1020 1035 1052 1081 1103 1125 1147 1173 1181 1190 1200 1211 1285 1330 1383 1474 1477 1500 1517 1720 1750 1875 1960 2397 2401 2500 4079 / 90 X BOX PLOT FOR TUESDAY 0 1000 2000 3000 4000 Tuesday XI SERVICE TIME DATA FOR FLOOR 1 TUESDAY 10 12 13 20 30 30 40 40 53 55 55 60 61 62 65 70 72 75 75 80 81 90 90 92 97 105 105 110 120 120 120 126 127 130 131 135 138 140 145 145 146 152 154 156 160 163 170 173 180 185 186 188 195 195 197 197 201 220 223 235 249 250 257 260 277 278 279 280 288 290 295 296 300 302 307 308 310 315 315 315 320 325 335 338 345 355 355 357 358 360 370 372 382 385 386 390 390 400 400 402 405 410 410 430 435 448 450 454 455 461 471 480 493 495 502 507 513 520 525 536 545 550 575 580 581 594 617 620 664 675 720 720 720 726 729 735 781 790 810 814 820 830 845 850 853 888 892 918 920 935 950 985 997 1103 1125 1200 1285 1383 1517 1750 1875 1960 2397 2401 4079 I 91 XII BOX PLOT FOR FLOOR 1 TUESDAY 1000 2000 3000 4000 Floor 1 Tuesday XIII ARRIVAL RATE GRAPHS For each floor for each day, the number of arrivals in each half hour is tallied. For each half hour, the arrival rate is determined. The arrival rate bins for the half hours are then grouped into hour bins with 8:30-9:00 and 9:00-9:30 in the 8:30-9:30 bin, and so on. The frequency for each bin of arrival rates is the number of arrivals that were assigned to that particular bin. The resulting bins and respective frequencies are depicted in these graphs. The bins are the hour blocks and are along the horizontal axis. The frequencies are along the vertical axis. Floor 1 Tuesday March 13, 2001 Floor 1 Thursday March 15, 2001 92 Floor 2 Tuesday March 13, 2001 120 100 tn ro 80 > Arri 60 o 40 20 0 Time Floor 2 Wednesday March 14, 2001 < O 40 Time Floor 2 Thursday March 15, 2001 Floor 2 Friday March 16, 2001 Floor 2 Monday March 19, 2001 120 100 tn ta 80 > 60 < o 40 * 20 0 O ,_ -•- CM Time Floor 4 Wednesday March 14, 2001 Floor 4 Friday March 16, 2001 93 XIV INTER-ARRIVAL TIME GRAPHS Inter-arrival times are calculated by subtracting the arrival time of a user from the arrival time o f the next user. Inter-arrival times are put in bins o f one minute. The corresponding frequencies of the bins are calculated and then graphed. The inter-arrival times also seem to follow an exponential distribution. The bins are the inter-arrival times and are along the horizontal axis. The frequencies are along the vertical axis. Floor 1 Tuesday March 13, 2001 Floor 1 Thursday March 15, 2001 Inter-arrival Time (minutes) Floor 2 Tuesday March 13, 2001 Floor 2 Wednesday March 14, 2001 Floor 2 Thursday March 15, 2001 Inter-arrival Time {minutes) Floor 2 Friday March 16, 2001 O -r- T- T- T-Inter-arrival Time (minutes) 94 Floor 2 Monday March 19, 2001 u c CD 3 CT g) LL o o o o o o o o o o o o o p o o o o o o o o o o o o O O O O O O O T - " ~ I - I - T - T -b b b b b b b b b b b b b Inter-arrival Time (minute) Floor 4 Wednesday March 14, 2001 o o o o o o o o o o o o o o o o o o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p p O y ^ ^ c ^ r j C N r o c O ' ^ ^ ' ^ - L n i n o o o i - ' -Inter-arrival Time (minutes) Floor 4 Friday March 16, 2001 o o o o o o o o o o o o o o o o o p p p o o o o o o o o o o o o o o V i s ' d r i o o N i i i i i j r V S O M O c i f i oo^-T- i-T-cMtNCMtonn-^--^-TfTr io b b b b b b b b b b b b b b b b b Inter-arrival Time (minutes) 95 X V Q U E U E I N G A N A L Y S I S " W O R S T C A S E " C H A R T S The charts depict the theoretical mean time in system (the mean time in queue and time spent photocopying), the mean time in queue, the mean queue length, and utilization (the fraction o f time that the resources are in use) rounded to two decimal places, for various numbers of photocopiers on each floor. The units are all in seconds. Floor 1: #Of Copiers Mean Time In System Mean Time In Queue Mean Queue Length Utilization 1 4903.33 4448.33 9.78 1.00 2 2599.11 2144.11 9.42 1.00 3 1789.45 1334.45 8.80 1.00 4 1316.82 861.82 7.53 0.99 5 955.15 500.15 5.28 0.96 6 696.04 241.04 2.79 0.88 7 556.47 101.47 1.21 0.78 8 495.50 40.50 0.49 0.68 9 470.84 15.84 0.19 0.61 10 461.07 6.07 0.07 0.55 Floor 2: #Of Copiers Mean Time In System Mean Time In Queue Mean Queue Length Utilization 1 3124.22 2723.22 6.79 1.00 2 1698.84 1297.84 6.47 1.00 . 3 1196.79 795.79 5.94 1.00 4 913.23 512.23 5.05 0.99 5 715.20 314.20 3.74 0.96 6 575.43 174.43 2.32 0.89 7 488.26 87.26 1.22 0.80 Floor 4: #Of Copiers Mean Time In System Mean Time In Queue Mean Queue Length . Utilization 1 579.37 252.37 0.72 0.94' 2 419.56 92.56 0.49 0.86 3 369.46 42.46 0.31 0.78 4 347.25 20.25 0.17 0.70 96 XVI QUEUEING ANALYSIS "WORST CASE" MATRICES The numbers in the matrix are defined to be the minimum number of photocopiers needed to ensure that the probability that a user waits less than the Waiting Time units is greater than or equal to the Service Level. The numbers in the table shows the minimum number of photocopiers required for different service levels and different waiting times. For example, i f you want at least 75 % of the users to wait no more than 180 seconds on floor 1, then at least seven photocopiers would be required. Floor 1: Waiting Time (in seconds) 0 30 60 90 120 150 180 210 240 270 300 100% 17 16 16 15 15 15 14 14 13 13 13 95% 11 10 10 10 10 9 9 9 9 9 8 90% 10 10 9 9 9 9 8 8 8 8 8 CD U evel 85% 9 9 9 8 8 8 8 8 8 8 8 Servi evel 80% 9 9 8 8 8 8 8 8 7 7 7 Servi 75% 8 8 8 8 8 8 7 7 7 7 7 70% 8 8 8 8 7 7 7 \ 7 7 7 7 65% 8 8 8 7 7 7 7 \ 7 7 7 7 60% 8 8 7 7 7 7 7 \ s7 7 7 7 Example Floor 4: Waiting Time (in seconds) 0 30 60 90 120 150 180 210 240 270 300 100% 13 12 12 11 11 10 j 10 9 9 8 8 95% 8 7 7 6 6 5 5 5 4 4 4 90% 7 6 6 5 5 .5 4 4 4 4 3 o evel 85% 6 6 5 5 4 4 4 4 3 3 3 Servi evel 80% 6 5 5 4 4 4 4 3 3 3 3 Servi _i 75% 5 5 4 4 4 4 3 3 3 3 3 70% 5 5 4 4 4 3 3 3 3 3 2 65% 5 • 4 4 4 3 3 3 3 2 2 2 60% 4 4 4 3 3 3 3 2 2 2 2 Floor 2: Waiting Time (in seconds) 0 30 60 90 120 150 180 210 240 270 300 100% 18 17 16 16 15 15 14 14 13 13 13 95% 11 11 10 10 10 9 9 9 9 ' 9 9 90% 10 10 10 9 9 9 9 8 8 8 8 o evel 85% 10 9 9 9 9 8 8 8 8 8 8 Servi evel 80% 9 9 9 8 8 8 8 8 8 8 7 Servi 75% 9 9 8 8 8 8 8 8 7 7 7 70% 9 8 8 8 8 8 7 7 7 7 7 65% 8 8 8 8 8 7 7 7 7 7 7 60% 8 8 8 7 7 7 7 7 7 7 7 97 X V I I Q U E U E I N G A N A L Y S I S " M E A N C A S E " C H A R T S The charts depict the theoretical mean time in system (the mean time in queue and time spent photocopying), the mean time in queue, the mean queue length, and utilization (the fraction of time that the resources are in use) rounded to two decimal places, for various numbers o f photocopiers on each floor. The units are all in seconds. Floor 1: #Of Copiers Mean Time In System Mean Time In Queue Mean Queue Length Utilization 1 4642.38 4187.38 9.20 1.00 2 1826.99 1371.99 5.81 0.96 3 741.79 286.79 1.41 0.74 4 516.95 61.95 0.31 0.56 5 470.06 15.06 0.07 0.45 6 458.69 3.69 0.02 0.38 7 455.87 0.87 0.00 0.32 8 455.19 0.19 0.00 0.28 9 455.04 0.04 0.00 0.25 10 455.01 0.01 0.00 0.23 Floor 2: #Of Copiers Mean Time In System Mean Time In Queue Mean Queue Length Utilization 1 3038.22 2643.22 6.69 1.00 2 1604.46 1209.46 6.12 1.00 3 1058.36 663.36 4.97 0.99 4 732.94 337.94 3.17 0.93 5 543.75 148.75 1.54 0.82 6 453.86 58.86 0.63 0.70 7 416.97 21.97 0.24 0.60 Floor 4: #Of Mean Time In Mean Time In Mean Queue Copiers System Queue Length Utilization 1 289.75 32.75 0.02 0.14 2 258.18 1.18 0.00 0.07 3 257.04 0.04 0.00 0.05 4 257.00 0.00 0.00 0.04 98 XVIII QUEUEING ANALYSIS "MEAN CASE" MATRICES The numbers in the matrix are defined to be the minimum number of photocopiers needed to ensure that the probability that a user waits less than the Waiting Time units is greater than or equal to the Service Level. The numbers in the table shows the minimum number of photocopiers required for different service levels and different waiting times. For example, i f you want at least 75 % of the users to wait no more than 180 seconds on floor 1, then at least four photocopiers would be required. Floor 1: Waiting Time (in seconds) 0 30 60 90 120 150 180 210 240 270 300 100% 11 10 10 10 9 9 9 9 9 8 8 95% 6 6 6 6 5 5 5 5 5 5 5 90% 5 5 5 5 5 5 5 5 4 4 4 CO u evel 85% 5 5 5 5 5 4 4 4 4 4 4 Servi evel 80% 5 5 4 4 4 4 4 4 4 4 4 Servi 75% 4 4 4 4 4 4 4\ 4 4 4 4 70% 4 4 4 4 4 4 4 \ 4 4 4 4 65% 4 4 4 4 4 4 - 4 \ 4 4 4 3 60% 4 4 4 4 4 4 4 \4 3 3 3 Example Floor 2: Waiting Time (in seconds) 0 30 60 90 120 150 180 210 240 270 300 100% 15 14 14 13 13 12 12 12 11 11 11 95% 9 9 8 8 8 8 7 7 7 7 7 90% 8 8 8 7 7 7 7 7 7 7 6 cu o evel 85% 8 7 7 7 7 7 7 6 6 6 6 Servi evel 80% 7 7 7 7 7 6 6 6 6 6 6 Servi _i 75% 7 7 . 7 6 6 6 6 6 6 6 6 70% 7 7 6 6 6 6 6 6 6 6 6 65% 6 6 6 6 6 6 6 6 6 5 5 60% 6 6 6 6 6 6 6 5 5 5 5 Floor 4: Waiting Time (in seconds) 0 30 60 90 120 150 180 210 240 270 300 100% 4 4 4 4 4 3 3 3 3 3 3 95% 2 2 2 2 2 2 2 2 2 2 2 90% 2 2 2 2 1 1 1 1 1 1 1 i rvice o > ID 85% 80% ] } ] 1 1 1 1 ] w to 75% 70% 65% 60% 1 1 \ i i i ; i 1 i 99 XIX SIMULATION COMPARISON AND SCREEN SHOTS For each comparison between the shifted and non-shifted distributions, mean time in system, mean time in queue, mean queue length, and mean number of users was simulated for both distributions. The mean time in queue and mean time in system are in seconds. Service Time Distribution Inter-Arrival Time Distribution Mean Mean T . . , ^ Mean Time Mean Time Number of Queue . ~ . 0 . .. . in Queue in System Users Length ' 2 + exp(325) ' exp(327) exp(96.7) exp(96.7) 299.26 0.72147 251.93 581.75 298.96 0.71985 251.05 579.63 10 + exp(391) exp(401) exp(69.3) exp(69.3) 416.79 1.163 81.979 477.5 414.14 1.1506 81.623 479.91 40 + exp(415) exp(455) exp(83.1) exp(83.1) 344.71 0.0616 5.0661 456.72 343.09 .0.05145 4.294 451.62 Service Time Distribution Inter-Arrival Time Distribution Mean Mean .. Mean Time Mean Time Number of Queue . ~ . 0 . ,. . in Queue in System Users Length 3 49 + exp(208) exp(257) exp(1760) exp(1760) 16.4 0.01352 22.452 272.71 16.27 0.0129 22.351 273.57 40 + exp(415) exp(455) exp(202) exp(202) 142.27 0.000017535 0.00353 449.01 142.27 0.000036178 0.00729 447.65 Inter-Arrival Time Distribution Service Time Distribution Mean Mean Mean Time Mean Time Number of Queue . ^ . 0 . .. . in Queue in System Users Length ' 196 + exp(1570) exp(1760) exp(257) exp(257) 17.15 0.01263 21.723 287.58 16.27 0.0129 22.351 273.57 100 ST A T U S Q U O } Afra - |F1 10 - Run Mode] 3 File £drt ifew loot* Aitange fibject Run Wirtdow y«slp o Common o Advanced Transfer o Advanced PTOCPS-. 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View loots Arrange Object Bur. Window Help -|t?j x| • H >• 14 • |T;? o Common o Advanced Transfer <L Advanced Process o Basic Process O Reports "Si Navigate 4- Top-Levsl Model . T F1Tues(1] : T F1 Tues Detab (2) T F1 Tues Overview (3) For Help, press F1 1/1IM 1495(1863 Usei interrupted (1128.37) Page 39 Sec 2 44/44 At 5.6" Ln 6 Coll XX MONTHLY GRAPHS Each point in the graph is the total number of arrivals for a day of the month. The day of the month runs along the x-axis with the total number of arrivals on the y-axis. K O E R N E R L I B R A R Y 1200 • September 2001 October 2001 1200 November 2001 1200 December 2001 o I * 300 -)-January 2002 103 1200 March 2002 April 2002 1200 900 600 300 B . W O O D W A R D B I O M E D I C A L L I B R A R Y 500 September 2001 500 October 2001 500 November 2001 Day 500 December 2001 co Day 104 January 2002 Day February 2002 500 400 jfl re > 300 k_ < ». 200 o 100 0 Day March 2002 Day April 2002 Day May 2002 C. D A V I D L A M L I B R A R Y 50 • 40 • re > 30 -^ 20 -o 10 0 -September 2001 Day October 2001 Day 105 March 2002 April 2002 T r ^ o c o c D O c s j i n c o T - T - ^ h - o c o c o c r j c N i n a o ' - T - T - T - C M C N J C N I C O T - 1 - T - T - C S I C M C N J Day Day 50 May 2002 40 w TO > -. _ >3U 20 o * 10-n u T - T - T - T - C \ ] ( M C \ i n Day 106 XXI DAILY GRAPHS Each point in the graph is the total number of arrivals for an hour of the day. The hour of the day runs along the x-axis with the total number of arrivals on the y-axis. Note that there is ah abundance of these graphs, so only excerpts of the graphs are given. The time frames chosen will also be chosen for any other excerpts needed. A. K O E R N E R L I B R A R Y 107 B. W O O D W A R D B I O M E D I C A L L I B R A R Y 1 0 8 C . D A V I D L A M L I B R A R Y 10 2 4-o 26 September 2001 CM CM CM CM Hour 10 > 6 ! 4 o 29 September 2001 co O) o CM ro Hour 109 XXII COMPARISON OF PROPORTIONS OF TWO INDEPENDENT SAMPLES This test is used to determine if the difference between proportion of successes of two samples drawn from different populations is significant or not. That is, this test is used to determine if the choice of one machine or bar code placement faired better than the other did. v. N O T A T I O N a. Ti] and 712 are the population proportions for population 1 and 2, respectively. b. ni and r\2 are the sample sizes for sample 1 and 2, respectively. c. xi and X2 are the observed successes in ni, and n2 for sample 1 and 2, respectively. d. pi and p2 are the probability of success for sample 1 and 2, respectively. vi. F O R M U L A S XI X2 P\=— P2= — n\ m vii. AS S U M P T I O N S a. data is collected from a random sample. b. the two proportions are independent. c. Tt) and 7t 2 are not close to 0 or 1. d. np > 5 and nq > 5. viii. T E S T STATISTIC ix. H Y P O T H E S I S H0: 7i] - 712 = 0 Ha: Ti] - 7 i 2 < 0 or 7 i i - 7t 2 > 0 x. P A S S / F A I L C R I T E R I A The hypothesis is rejected when the p-value is of a small value. PP-X1 + X2 n\ + m qP = 1 - pP 110 XXIII BINARY CATEGORICAL LOGISTIC REGRESSION Binary Logistic Regression is used when the dependent variable is binary. That is, the dependent variable is only one of two possible values. Binary Logistic Regression also allows the independent variables to all be categorical, or predictors. Analysis done via Logistic Regression describes how a binary response variable is associated to the explanatory variables. i. DEFINITIONS Ttjj = the probability of a successful charge transaction when machine i is used and the bar code is placed on area j . f 0 i f machine 1 is used [ 1 i f machine 2 is used 0 i f bar code is on inside of book 1 i f bar code is on outside of book ii. M O D E L l ° g i t ( ^ ) = log = B0 +/?,x, + B2x2 + /? 3x,x 2 + e •J J where • Po is the log odds of a successful charge transaction on machine 1 when the bar code is on the inside of the cover. • Pi is the increment in log odds when using machine 2. • p2 is the increment in log odds when the bar code is on the outside of the cover. • P3 is the increment in log odds when using machine 2 and the bar code is on the outside of the cover (interaction). • £ is the error term. I l l XXIVMINIMUM NUMBER OF SERVERS CHARTS FOR BASELINE The charts indicate the minimum number of staff needed at the circulation desk to ensure that the probability that a client waits less than a certain length of time is greater than or equal to a certain service level, in this case 80%. The date is across the top of the chart, while the hour of the day and waiting time (in seconds) are along the left hand side. The value in the chart is the minimum number of staff needed to meet the corresponding waiting time. A. K O E R N E R L I B R A R Y September 26 September 29 March 20 March 23 May 14 May 17 0.80 2001 2001 2002 2002 2002 2002 o 60 2 1 2 1 1 1 o 120 1 1 1 1 1 1 00 180 1 1 1 1 1 1 o 60 2 1 2 1 2 2 o 120 2 1 2 1 1 1 Oi 180 1 1 2 1 1 1 o 60 2 2 2 2 o © 120 2 1 2 1 2 2 180 2 1 2 1 1 1 o 60 3 2 3 2 2 2 o 120 3 2 3 2 2 2 180 3 2 2 1 1 2 o 60 2 2 3 2 2 2 o CN 120 2 2 3 2 2 2 180 2 2 3 2 1 2 o 60 2 2 3 2 2 2 o CO 120 2 2 3 2 2 2 180 2 2 3 2 2 2 o 60 3 2 3 2 2 2 o Tt 120 2 2 3 2 2 2 - 180 2 1 2 2 2 2 o 60 2 2 3 2 2 2 o uS 120 2 2 3 2 2 2 T- 180 2 2 3 2 1 1 o 60 2 * 2 3 2 2 3 o CD 120 2 2 3 2 2 2 180 2 1 2 2 2 2 o 60 2 2 3 2 2 2 o 1^  120 2 2 2 2 2 T— 180 2 2 2 2 1 1 o 60 4 2 2 2 2 1 o 00 120 3 1 2 2 2 1 180 3 1 2 2 1 1 o 60 2 1 2 2 2 1 o ai 120 2 1 2 2 1 1 T- 180 1 1 2 1 1 1 o 60 2 1 2 2 2 o o 120 2 1 2 2 1 1 CM 180 1 1 2 2 1 1 o 60 2 1 2 2 2 •i o 120 1 1 2 2 2 1 CM 180 1 1 1 2 1 1 o 60 2 1 2 2 2 1 o csi 120 1 1 2 2 1 1 CM 180 1 1 1 2 1 1 o 60 1 1 1 2 1 1 o CO 120 1 1 1 1 1 1 M 180 1 1 1 1 1 1 112 B. W O O D W A R D B I O M E D I C A L L I B R A R Y September 26 September 29 March 20 March 23 May 14 May 17 0.80 2001 2001 2002 2002 2002 2002 o 60 2 1 2 1 2 2 o 120 2 1 1 1 1 i 00 180 2 1 1 1 1 1 o 60 2 1 2 1 o 120 1 1 1 1 1 1 ai 180 1 1 1 -'\ 1 1 1 o 60 2 2 2 o o 120 2 1 2 1 1 1 180 1 1 1 1 1 1 o 60 2 2 2 o 120 2 1 2 2 1 1 *- 180 1 1 1 1 1 1 o 60 2 2 2 o CM 120 2 1 2 1 1 1 -180 1 1 1 1 1 1 o 60 2 2 2 o CO 120 2 2 1 1 180 1 1 2 1 1 1 o 60 2 2 2 o Tj" 120 2 1 1 1 1 1 180 1 1 1 1 1 1 o 60 2 2 2 o in 120 2 2 2 1 1 T- 180 1 1 1 1 1 1 o 60 2 2 2 o CO 120 2 1 2 2 1 1 T— 180 1 1 1 1 1 1 o 60 2 2 2 o 120 2 1 2 2 1 1 180 1 1 2 1 1 1 o 60 2 2 1 1 o CO 120 1 1 2 1 1 1 180 1 1 1 1 1 1 o 60 2 1 2 1 1 o ay 120 2 1 1 1 1 1 T- 180 1 1 1 1 1 1 O 60 2 1 2 1 1 o o 120 1 1 2 1 - 1 1 CM 180 1 1 1 1 1 1 o 60 2 1 1 1 1 1 o 120 1 1 1 1 1 1 CN 180 1 1 1 1 1 1 O 60 2 1 2 1 1 O CN 120 1 1 1 1 1 1 CN 180 1 1 1 1 1 1 O 60 1 1 1 1 1 o CO 120 1 1 1 1 1 1 N 180 1 1 1 1 1 1 113 C . D A V I D L A M L I B R A R Y 0.80 September 26 September 29 March 20 March 23 May 14 May 17 2001 2001 2002 2002 2002 2002 8:00 60 120 180 j 1 ] 3 3 ] 9:00 60 120 180 j 3 ] 3 3 10:00 60 120 180 1 \ ] 3 11:00 60 120 180 ] \ 3 3 j 12:00 60 120 180 ] \ J 13:00 60 120 180 J \ 1 j 3 14:00 60 120 180 1 3 3 3 15:00 60 120 180 \ j j 16:00 60 120 180 | \ 3 3 j 17:00 60 120 180 | j j ] 18:00 60 120 180 3 \ J j ] 19:00 60 120 180 3 3 ] j ] 20:00 60 120 180 j \ 3 3 3 21:00 60 120 180 3 ] j 3 22:00 60 120 180 3 3 j 3 23:00 60 120 180 j 114 XXV MEAN WAITING TIME IN QUEUE CHARTS FOR BASELINE The values in the charts indicate the average waiting time (in seconds) of a client in the queue for the circulation desk with respect to a particular number of servers. The date is across the top of the chart, while the hour of the day and the number of staff available at the circulation desk are along the left hand side. A. K O E R N E R L I B R A R Y September 26 September 29 March 20 March 23 May 14 May 17 0.80 2001 2001 2002 2002 2002 2002 1 12.24 0.00 8.25 0.00 0.00 0.00 2 0.43 0.00 0.22 0.00 0.00 0.00 o 3 0.02 0.00 0.01 0.00 0.00 0.00 o 4 0.00 0.00 0.00 0.00 0.00 0.00 CO 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 43.10 0.00 180.00 0.00 4.19 13.10 2 2.74 0.00 9.82 0.00 0.06 0.49 o 3 0.22 0.00 1.18 0.00 0.00 0.02 o 4 0.02 0.00 0.14 0.00 0.00 0.00 O) 5 0.00 0.00 0.02 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 840.00 3.44 00 5.06 22.60 41.17 2 16.70 0.04 25.28 0.09 1.14 2.59 o 3 2.23 0.00 3.51 0.00 0.06 0.20 o o 4 0.32 0.00 0.56 0.00 0.00 0.01 5 0.04 0.00 0.08 0.00 0.00 0.00 6 0.01 0.00 0.01 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 00 149.71 00 28.34 48.54 98.24 2 225.88 8.76 73.05 1.58 3.16 6.40 o 3 16.95 1.02 9.11 0.10 0.26 0.68 o V" 4 3.35 0.12 1.71 0.01 0.02 0.07 5 0.72 0.01 0.33 0.00 0.00 0.01 6 0.15 0.00 0.06 0.00 0.00 0.00 7 0.03 0.00 0.01 0.00 0.00 0.00 1 00 61.69 00 660.00 45.37 51.92 2 20.45 4.12 387.54 15.96 2.92 3.41 o 3 2.80 0.38 20.09 2.11 0.24 0.29 o CM 4 0.42 0.03 3.97 0.30 0.02 0.02 5 0.06 0.00 0.88 0.04 0.00 0.00 6 0.01 0.00 0.19 0.00 0.00 0.00 7 0.00 0.00 0.04 0.00 0.00 0.00 1 00 237.93 00 174.78 100.00 73.33 2 25.00 11.38 786.05 9.65 • 6.49 4'.91 o 3 3.47 1.41 23.00 1.15 0.69 0.48 o CO 4 0.55 0.18 4.54 0.14 0.07 0.04 5 0.08 0.02 1.02 0.01 0.01 0.00 6 0.01 0.00 0.22 0.00 0.00 0.00 7 0.00 0.00 0.05 0.00 0.00 0.00 1 00 30.19 00 00 103.02 119.25 2 35.76 1.73 100.27 26.59 6.65 7.46 o 3 4.98 0.11 11.29 3.70 0.72 0.83 o 4 0.85 0.01 2.17 0.59 0.07 0.09 5 0.14 0.00 0.43 0.09 0.01 0.01 6 0.02 0.00 0.08 0.01 0.00 0.00 7 0.00 0.00 0.01 0.00 0.00 0.00 115 September 26 September 29 March 20 March 23 May 14 May 17 0.80 2001 2001 2002 2002 2002 2002 1 00 184.07 00 879.13 51.05 32.70 2 24.11 9.95 1035.21 16.83 3.35 1.93 o 3 3.34 1.20 23.80 2.25 0.29 0.13 o in 4 0.52 0.14 4.69 0.32 0.02 0.01 5 0.08 0.02 1.06 0.04 0.00 0.00 6 0.01 0.00 0.23 0.01 0.00 0.00 7 0.00 0.00 • 0.05 0.00 0.00 0.00 1 351.43 47.20 00 00 740.00 00 2 13.39 3.06 82.26 32.20 16.33 34.87 o 3 1.72 0.25 9.91 4.49 2.17 4.86 o cb 4 0.23 0.02 1.88 0.75 0.31 0.82 5 0.03 0.00 0.37 0.12 0.04 0.14 6 0.00 0.00 0.07 0.02 0.00 0.02 7 0.00 0.00 0.01 0.00 0.00 0.00 1 315.65 161.54 00 157.09 48.54 0.93 2 12.86 9.20 34.37 9.04 3.16 0.00 o 3 1.64 1.08 4.79 1.06 0.26 0.00 o t^! 4 0.22 0.13 0.81 0.12 0.02 0.00 5 0.03 0.01 0.13 0.01 0.00 0.00 6 0.00 0.00 0.02 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 00 2.79 557.14 75.00 43.35 0.00 2 00 ' 0.03 15.35 5.02 2.76 0.00 o 3 36.28 0.00 2.02 0.49 0.22 0.00 o CO 4 6.87 0.00 0.28 0.05 0.02 0.00 5 1.62 0.00 0.04 0.00 0.00 0.00 6 0.38 0.00 0.00 0.00 0.00 0.00 7 0.09 0.00 0.00 0.00 0.00 0.00 1 16.19 0.00 557.14 49.09 5.06 0.00 2 0.69 0.00 15.35 3.20 0.09 0.00 o 3 0.03 0.00 2.02 0.27 0.00 0.00 o d> 4 0.00 0.00 0.28 0.02 0.00 0.00 5 0.00 0.00 0.04 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 23.40 0.00 140.00 75.42 10.47 0.00 2 . 1.20 0.00 8.38 5.04 0.33 0.00 o 3 0.07 0.00 0.96 0.50 0.01 0.00 o © 4 0.00 0.00 0.11 0.05 0.00 0.00 CN 5 0.00 0.00 0.01 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 12.36 0.00 16.46 82.57 37.30 0.00 2 0.44 0.00 0.70 5.49 2.29 0.00 o 3 0.02 0.00 0.03 0.56 0.17 0.00 o 4 0.00 0.00 0.00 0.05 0.01 0.00 CM 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 ' 9.68 0.00 49.92 84.00 0.34 0.00 2 0.29 0.00 3.26 5.58 . 0.00 0.00 O 3 0.01 0.00 0.28 0.57 0.00 0.00 O CM 4 0.00 0.00 0.02 0.06 0.00 0.00 CM 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 0.00 0.00 0.00 1.36 0.00 0.00 2 0.00 0.00 0.00 0.01 0.00 0.00 O 3 0.00 0.00 0.00 0.00 0.00 0.00 O CO 4 0.00 0.00 0.00 0.00 0.00 0.00 CM 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 116 B. W O O D W A R D B I O M E D I C A L L I B R A R Y September 26 September 29 March 20 March 23 May 14 May 17 0.80 2001 2001 2002 2002 2002 2002 1 340.00 0.00 12.18 0.00 1.54 5.01 2 13.23 0.00 0.43 0.00 0.01 0.09 o 3 1.69 0.00 0.02 0.00 0.00 0.00 o 4 0.22 0.00 0.00 0.00 0.00 0.00 CO 5 •0.03 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 9.23 0.00 9.23 0.00 12.91 3.02 2 0.27 0.00 0.27 0.00 0.47 0.03 . o 3 0.01 0.00 0.01 0.00 0.02 0.00 o 4 0.00 0.00 0.00 0.00 0.00 0.00 O) • 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 23.48 5.16 32.01 8.74 5.31 13.66 2 1.21 0.09 1.87 0.24 0.10 0.52 o 3 0.07 0.00 0.13 0.01 0.00 0.02 o o 4 0.00 0.00 0.01 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 31.14 5.90 15.99 42.13 12.73 7.92 2 1.80 0.12 0.67 2.67 0.46 0.20 o 3 0.12 0.00 0.03 0.21 0.02 0.01 o 4 0.01 0.00 0.00 0.01 0.00 0.00 T _ 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 41.77 7.61 48.27 10.24 3.58 11.29 2 2.64 0.19 3.14 0.32 0.05 0.38 o 3 0.21 0.00 0.26 0.01 0.00 0.01 o c\i 4 0.01 0.00 0.02 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 49.51 16.60 161.54 2.07 12.18 15.59 2 3:23 0.71 9.20 0.02 0.43 0.64 o 3 0.27 0.03 1.08 0.00 0.02 0.03 o CO 4 0.02 0.00 0.13 0.00 0.00 0.00 5 0.00 0.00 0.01 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 51.20 5.01 14.23 10.59 7.92 11.64 2 3.36 0.09 0.56 0.34 0.20 0.40 o 3 0.29 0.00 0.02 0.01 0.01 0.01 ' o 4 0.02 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 38.97 15.00 18.90 14.61 6.21 3.72 2 2.42 0.61 0.87 0.58 0.13 0.05 o 3 0.18 0.03 0.04 0.02 0.00 0.00 o u> 4 0.01 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 117 September 26 September 29 March 20 March 23 May 14 May 17 0.80 2001 2001 2002 2002 2002 2002 1 24.71 9.90 30.57 16.39 11.46 5.45 2 1.30 0.30 1.76 0.70 0.39 0.10 o 3 0.08 0.01 0.12 0.03 0.01 0.00 o CO 4 0.00 0.00 0.01 0.00 0.00 0.00 T- 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 17.21 9.73 56.13 33.20 9.90 7.13 2 0.75 0.29 3.72 1.97 0.30 0.17 o 3 0.03 0.01 0.33 0.14 0.01 0.00 o 1^  4 0.00 0.00 0.03 0.01 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 13.47 2.61 29.44 0.00 11.29 0.00 2 0.51 0.03 1.67 0.00 0.38 0.00 o 3 0.02 0.00 0.11 0.00 0.01 0.00 o 00 4 0.00 0.00 0.01 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 18.47 0.00 11.11 0.00 1.54 0.00 2 0.84 0.00 0.37 0.00 0.01 0.00 o 3 0.04 0.00 0.01 0.00 0.00 0.00 o 6) 4 0.00 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 7.92 0.00 25.71 0.00 1.02 0.00 2 0.20 0.00 1.38 0.00 0.00 0.00 o 3 0.01 0.00 0.08 0.00 0.00 0.00 o © 4 0.00 0.00 0.00 0.00 0.00 0.00 CN 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 3.72 0.00 0.00 0.00 0.00 0.00 2 0.05 0.00 0.00 0.00 0.00 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o 4 0.00 0.00 0.00 0.00 0.00 0.00 CN 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 1.54 0.00 13.10 0.00 0.00 0.00 2 0.01 0.00 0.49 0.00 0.00 0.00 o 3 0.00 0.00 0.02 0.00 0.00 0.00 o CN 4 0.00 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 0.00 0.00 0.00 0.00 0.00 0.00 2 0.00 0.00 0.00 0.00 0.00 0.00 O 3 0.00 0.00 0.00 0.00 0.00 0.00 o CO 4 0.00 0.00 0.00 0.00 0.00 0.00 N 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 118 C . D A V I D L A M L I B R A R Y September 26 September 29 March 20 March 23 May 14 May 17 0.80 2001 2001 2002 2002 2002 2002 1 0.00 0.00 0.00 0.00 0.00 0.00 2 0.00 0.00 0.00 0.00 0.00 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o 4 0.00 0.00 0.00 0.00 0.00 0.00 00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 1.28 0.00 0.34 0.00 0.00 0.00 2 0.01 0.00 0.00 0.00 0.00 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o 4 0.00 0.00 0.00 0.00 0.00 0.00 o> 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 1.80 0.00 0.67 0.00 1.71 0.00 2 0.01 0.00 0.00 0.00 0.01 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o o 4 0.00 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 1.54 0.00 0.67 0.00 0.00 0.00 2 0.01 0.00 0.00 0.00 0.00 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o 4 0.00 0.00 0.00 0.00 0.00 0.00 T ~ 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 0.00 0.93 1.02 2.97 1.36 0.00 2 0.00 0.00 0.00 0.03 0.01 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o c\i 4 0.00 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 • 0.00 0.00 0.00 0.00 1 1.02 0.00 0.76 0.00 2.07 0.00 2 0.00 0.00 0.00 0.00 0.02 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o CO 4 0.00 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 0.34 0.00 4.29 0.00 0.00 0.00 2 0.00 0.00 0.07 0.00 0.00 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o Tt 4 0.00 0.00 0.00 0.00 0.00 0.00 T— 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 1.02 0.00 1.36 0.00 0.67 0.67 2 0.00 0.00 0.01 0.00 0.00 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o IO 4 0.00 0.00 0.00 0.00 0.00 0.00 T— 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 119 September 26 September 29 March 20 March 23 May 14 May 17 0.80 2001 2001 2002 2002 2002 2002 1 0.00 0.67 5.06 0.00 0.00 2.07 2 0.00 0.00 0.09 0.00 0.00 0.02 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o CD 4 0.00 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 • 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 0.00. 0.00 0.34 1.54 0.34 0.00 2 0.00 0.00 0.00 0.01 0.00 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o 1^  4 0.00 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 0.00 0.00 0.00 0.34 0.34 0.00 2 0.00 0.00 0.00 0.00 0.00 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o oo 4 0.00 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 0.00 0.00 0.34 0.00 0.00 0.00 2 0.00 0.00 0.00 0.00 0.00 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o cn 4 0.00 0.00 0.00 0.00 0.00 0.00 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 0.00 0.00 0.00 0.00 0.34 0.00 2 0.00 0.00 0.00 0.00 0.00 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o o 4 0.00 0.00 0.00 0.00 0.00 0.00 CM 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 0.00 0.00 0.67 0.00 0.50 0.00 2 0.00 0.00 0.00 0.00 0.00 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o 4 0.00 0.00 0.00 0.00 0.00 0.00 CM 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 0.00 0.00 0.00 0.00 0.00 0.00 2 0.00 0.00 0.00 0.00 0.00 0.00 o 3 0.00 0.00 0.00 0.00 0.00 0.00 o CM 4 0.00 0.00 0.00 0.00 0.00 0.00 CM 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 1 0.00 0.00 0.00 0.00 0.00 0.00 2 0.00 0.00 0.00 0.00 0.00 0.00 O 3 0.00 0.00 0.00 0.00 0.00 0.00 O CO 4 0.00 0.00 0.00 0.00 0.00 0.00 CM 5 0.00 0.00 0.00 0.00 0.00 0.00 6 0.00 0.00 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 0.00 0.00 120 XXVIMINIMUM NUMBER OF SERVERS CHARTS FOR PERIOD COMBINATIONS The charts indicate the minimum number of staff needed at the circulation desk to ensure that the probability that a client waits less than a certain length of time is greater than or equal to a certain service level, in this case 80%. The waiting time (in seconds) is across the top of the chart, while the period combinations are along the left hand side. The value in the chart is the minimum number of staff needed to meet the corresponding waiting time. It is assumed that at least one staff member would be available at the desk for all open time-periods. A. K O E R N E R L I B R A R Y Waiting Time Month Day Hour 60 120 180 Low Low Low 2 1 1 Low Low Moderate 2 1 1 Low Low Peak 2 2 1 Low Peak Low 2 1 1 Low Peak Moderate 2 2 1 Low Peak Peak 2 2 2 Moderate Low Low 2 1 1 Moderate Low Moderate 2 2 1 Moderate Low Peak 2 2 1 Moderate Peak Low 2 2 1 Moderate Peak Moderate 2 2 2 Moderate Peak Peak 2 2 2 Peak Low Low 2 2 1 Peak Low Moderate 2 2 1 Peak Low Peak 2 2 2 Peak Peak Low 2 2 1 Peak Peak Moderate 2 2 2 Peak Peak Peak 3 3 2 B. W O O D W A R D B I O M E D I C A L L I B R A R Y Waiting Time Month Day Hour 60 120 180 Low Low Low 1 1 1 Low Low Moderate 2 1 1 Low Low Peak 2 1 1 Low Peak Low 2 1 1 Low Peak Moderate 2 1 1 Low Peak Peak 2 1 1 Moderate Low Low 1 1 1 Moderate Low Moderate 2 1 1 Moderate Low Peak 2 1 1 Moderate Peak Low 2 1 1 Moderate Peak Moderate 2 1 1 Moderate Peak Peak 2 1 Peak Low Low 2 1 1 Peak Low Moderate 2 1 1 Peak Low Peak 2 2 1 Peak Peak Low 2 1 1 Peak Peak Moderate 2 2 1 Peak Peak Peak 2 2 1 121 C. D A V I D L A M L I B R A R Y Waiting Time Month Day Hour 60 120 180 Low Low Low 2 1 1 • Low Low Moderate 2 1 1 Low Low Peak 2 1 1 Low Peak Low 2 1 1 Low Peak Moderate 2 1 1 Low Peak Peak 2 1 1 Moderate Low Low 2 1 1 Moderate Low Moderate 2 1 1 Moderate Low Peak 2 1 1 Moderate Peak Low 2 1 1 Moderate Peak Moderate 2 1 1 Moderate Peak Peak 2 1 1 Peak Low Low 2 1 1 Peak Low Moderate 2 1 1 Peak Low Peak 2 1 1 Peak Peak Low 2 1 . 1 Peak Peak Moderate 2 1 1 Peak Peak Peak 2 1 1 XXVII MEAN WAITING TIME IN QUEUE CHARTS FOR PERIOD COMBINATIONS The charts indicate the average waiting time (in seconds) of a client in the queue for the circulation desk with respect to a particular number of servers. The number of staff available at the circulation desk is across the top of the chart, while the period combinations are along the left hand side. A. K O E R N E R L I B R A R Y Number Of Servers At Circulation Desk Month Day Hour 1 2 3 4 Low Low Low 2.60 0.03 0.00 0.00 Low Low Moderate 6.83 0.16 0.00 0.00 Low Low Peak 16.96 0.74 0.03 0.00 Low Peak Low 5.35 0.10 0.00 0.00 Low Peak Moderate 23.39 1.20 0.07 0.00 Low Peak Peak 64.53 4.32 0.40 0.04 Moderate Low Low 5.55 0.11 0.00 0.00 Moderate Low Moderate 18.44 0.84 0.04 0.00 Moderate Low Peak 39.78 2.48 0.19 0.01 Moderate Peak Low 15.65 0.65 0.03 0.00 Moderate Peak Moderate 122.28 7.61 0.85 0.09 Moderate Peak Peak 00 26.45 3.68 0.59 Peak Low Low 17.71 0.79 0.04 0.00 Peak Low Moderate 48.13 3.13 0.26 0.02 Peak Low Peak 142.08 8.46 0.98 0.11 Peak Peak Low 28.88 1.63 0.10 0.01 Peak Peak Moderate 1422.82 17.94 2.41 0.35 Peak Peak Peak CO 92.77 10.75 2.06 122 B. WOODWARD BIOMEDICAL LIBRARY Number Of Servers At Circulation Desk Month Day H o u r 1 2 3 4 L o w L o w L o w 0.00 0.00 0.00 0.00 L o w Low Moderate 1.07 0.00 0.00 0.00 L o w L o w Peak 4.34 0.07 0.00 0.00 L o w Peak L o w 1.77 0.01 0.00 0.00 L o w Peak Moderate 7.14 0.17 0.00 0.00 L o w Peak Peak 12.37 0.44 0.02 0.00 Moderate Low L o w 0.00 0.00 0.00 0.00 Moderate Low Moderate 3.29 0.04 0.00 0.00 Moderate Low Peak 9.15 0.26 0.01 0.00 Moderate Peak L o w 3.20 0.04 0.00 0.00 Moderate Peak Moderate 14.12 0.55 0.02 0.00 Moderate Peak Peak 26.93 1.47 0.09 0.00 Peak L o w L o w 0.00 0.00 0.00 0.00 Peak Low Moderate 5.75 0.11 0.00 0.00 Peak Low Peak 20.38 0.98 0.05 0.00 Peak Peak L o w 5.34 0.10 0.00 0.00 Peak Peak Moderate 24.05 1.25 0.07 0.00 Peak Peak Peak 45.06 2.89 0.23 0.02 C . DAVID L A M LIBRARY Number Of Servers At Circulation Desk Month Day H o u r 1 2 3 4 L o w L o w L o w 0.01 0.00 0.00 0.00 L o w L o w Moderate 0.07 0.00 0.00 0.00 L o w L o w Peak 0.57 0.00 0.00 0.00 L o w Peak L o w 0.16 0.00 0.00 0.00 L o w Peak Moderate 0.48 0.00 0.00 0.00 L o w Peak Peak 0.69 0.00 0.00 0.00 Moderate L o w L o w 0.02 0.00 0.00 0.00 Moderate L o w Moderate 0.35 0.00 0.00 0.00 Moderate L o w Peak 0.85 0.00 0.00 0.00 Moderate Peak L o w 0.27 0.00 0.00 0.00 Moderate Peak Moderate 0.93 0.00 0.00 0.00 Moderate Peak Peak 1.05 0.00 o.oo • 0.00 Peak L o w L o w 0.03 0.00 0.00 0.00 Peak L o w Moderate 0.63 0.00 0.00 0.00 Peak L o w Peak 1.04 0.00 0.00 0.00 Peak Peak L o w 0.33 0.00 0.00 0.00 Peak Peak Moderate 1.13 0.01 0.00 0.00 Peak Peak Peak 1.50 0.01 0.00 0.00 XXVIII COMPARISON OF MINIMUM NUMBER OF SERVERS This subset of graphs give a representation of the comparison between the baseline and the staffing rules to determine how well the staffing rules perform over the entire academic year with the criteria that the probability a client waits no more than 120 seconds is greater than 80%. The solid bars in the graphs are the minimum number of staff needed on the circulation desk according to the baseline calculations. The lines with markers in the graphs are the minimum number of staff needed at the circulation desk according to the staffing rules. The time of the day runs along the x-axis with the total number of servers needed on the y-axis. A . KOERNER LIBRARY 124 B. WO O D W A R D B IOMEDICAL L I B R A R Y September 26 2001 (A <D t <•> „ 2 -« o >_ a> a> z< E Time September 29 2001 Time C . DAVID L A M LIBRARY September 26 2001 O i - CM CO Time at t O a i - C O 0) z E September 29 2001 CO OJ o *— CM CM CM CM Time ' -a 2 -May 14 2002 co cn o i— CM co CM CM CM CM Time May 17 2002 C O > « T, 2 05 u O co 1— co C O Z 1 E CO O) O f-Time 126 

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