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

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S I M U L A T I O N A N D QUEUEING M O D E L S FOR R E S O U R C E 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 P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR THE D E G R E E 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  ABSTRACT 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 U B C 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 selfcharge 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 A  B  S  T  R  T  A  B  L  E  OF A  C  T  O  F  L  I  S  T  O  F  T  L  I  S  T  O  F  F  A  C  K  N  O  I  N  I I  L  I  I I I  P  III. 1  C A G  L  O  L  E  N  T  S  I  S  V  M  D  E  U T  C  E  C  T  U O  N  I  R P  T  S  O  N  E I  R E  R  V  E  V A  I N  A  E  W  L  Y  S  I  I  I  S  METHODS  :  RESULTS  111.3  INTERPRETATION OF R E S U L T S  111.4  SUMMARY  Comments Further Work  .-  1 1  1 1 1 1 1 1 2 2 2 2 2 2 2  III. 2.1 Analysis of Status Quo system III. 2.2 Queueing Model 111.2.2.1 Queueing Analysis of "Worst Case" 111.2.2.2 Queueing Analysis of "Mean Case" III.2.3 Simulation  III. 4.1 III.4.2  I  .  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 III. 1.3.1 Outlier Test For Tuesday 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 III. 1.3.5 Determining Service Time "Mean Case" Distributions III. 1.4 Analysis Of Inter-Arrival Times III. 1.4.1 Determining Inter-Arrival Time Best-Fit Distributions III. 1.4.2 Determining Inter-Arrival Time "Worst Case" Distributions III. 1.4.3 Determining Inter-Arrival Time "Mean Case" Distributions ///. 7.5 Analysis Of Mean Arrival Rate and Mean Service Rate III. 1.6 Queueing Models and Simulation Models III. 1.6.1 Assumptions for Queueing Model III. 1.6.2 Notation for Queueing Model III. 1.6.3 Formulas for Queueing Model III. 1.6.4 Inputs for Queueing Model III. 1.6.5 Simulation 111.2  I V  E E  A O  T S  R  O R  T  N  G  R E  O  U  E  T T  H  B  I  W  I  CONTENTS  :. ;  2 2 2 2 2 2 3  :  3 3  IV  33  HUMAN RESOURCESANALYSIS IV. 1  METHODS  33  IV. 1.1 Data Collection I V . l . 1.1 Transformation of Data IV. 1.2 Analysis of Status Quo I V.l. 3 Development of Baseline IV.1.4 Period Formulation IV. 1.4.1 Grouping of Like Months IV. 1.4.2 Grouping of Like Days of the Week IV. 1.4.3 Grouping of Like Hours of the Day IV. 1.5 Calculation of Arrival Rates IV. 1.6 Development of Staffing Rules IV. 1.7 Comparison and Sensitivity Analysis IV. 1.8 Self-Charge Assessment IV.1.8.1 Data Collection for Self-Charge Machines......... IV. 1.8.2 Assessment of Reliability By Machine IV. 1.8.3 Assessment of Reliability By Scan Barcode Placement IV. 1.8.4 Determination of Interaction Effect IV.2  RESULTS  34 35 36 36 37 37 38 39 41 42 42 42 43 44 44 44 46  IV.2.1 Status Quo IV. 2.2 Baseline IV.2.3 Periods IV.2.3.1 Grouping of Like Months IV.2.3.2 Grouping of Like Days of the Week IV.2.3.3 Grouping of Like Hours of the Day IV.2.3.4 Summary of Groupings Per Branch IV. 2.4 Arrival Rates IV. 2.5 Results of Queueing Analysis IV.2.5.1 Staffing Rules IV. 2.6 Comparison and Sensitivity Analysis IV.2.6.1 Change in Percentage of Renewal Transactions IV.2.6.2 Change in Percentage of Money-Collected Transactions IV.2.6.3 Change in Average Number of Transactions Per Client IV.2.6.4 Change in Service Time IV. 2.7 Self-Charge Machines  46 46 46 .46 48 50 52 53 55 55 57 57 58 59 59 60  IV.3  INTERPRETATION OF RESULTS  62  IV.4  SUMMARY  64  IV.4.1 IV.4.2  Comments Further Work  REFERENCES  :  64 64 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  V  Probability Density Function for a Shifted Exponentional Distribution  86  VI  Chi Square Test  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  XIV  Inter-Arrival Time Graphs  94  XV  Queueing Analysis "Worst Case" Charts  96  XVI  Queueing Analysis "Worst Case" Matrices  97  XVII  Queueing Analysis "Mean Case" Charts  98  XVIII  Queueing Analysis "Mean Case" Matrices  99  XIX  Simulation comparison and screen shots  100  XX  Monthly Graphs  103  XXI  Daily Graphs  107  XXII  Comparison of Proportions of Two Independent Samples  110  XXIII  Binary Categorical Logistic Regression  111  XXIV  Minimum Number of Servers Charts For Baseline  112  XXV  Mean Waiting Time In Queue Charts For Baseline  115  XXVI  Minimum Number of Servers Charts For Period Combinations  121  XXVII  Mean Waiting Time in Queue Charts for Period Combinations  122  s  XXVIII Comparison of Minimum Number of Servers  ,. 86  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 B y Day O f The Week For Koerner Library 38 Table 23: Total Arrivals B y Day O f The Week For Woodward Biomedical Library 39 Table 24: Total Arrivals B y Day O f The Week For David Lam Library 39 Table 25: Total Arrivals B y Hour Of The Day For Koerner Library 40 Table 26: Total Arrivals B y Hour Of The Day For Woodward Biomedical Library 40 Table 27: Total Arrivals B y Hour Of The Day For David Lam Library 41 Table 28: The Number O f Successes For The Self-Charge Machines 43 Table 29: The Number O f Successes For The Self-Charge Machines In Batches Of 10 44 Table 30: Breakdown O f Periods For Koerner Library 52 Table 31: Breakdown O f Periods For Woodward Biomedical Library 53 Table 32: Breakdown O f Periods For David Lam Library 53 Table 33: Summary O f Calculation Process For Koerner Library 54 Table 34: Summary O f Arrival Calculation Process For Woodward Biomedical Library... 54 Table 35: Summary O f 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  vi  LIST O F FIGURES Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure  1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15:  Current Set-up of the CopyRoom on Floor 1 at Koerner Library 6 Current Set-up of the Copy Room on Floor 2 at Koerner Library 6 Current Set-up of the Copy Room on Floor 4 at Koerner Library 7 Number of Users on Floor 2 10 Mean Arrival Rate and Mean Service Rate For Floor 2 19 Estimated Marginal Means Of The Probability Of Success 45 Total Arrivals By Month For Koerner Library 47 Total Arrivals By Month For Woodward Biomedical Library 47 Total Arrivals By Month For David Lam Library 48 Total Arrivals By Day O f The Week For Koerner Library 49 Total Arrivals By Day O f The Week For Woodward Biomedical Library .... 49 Total Arrivals By Day O f The Week For David Lam Library 50 Total Arrivals By Hour O f The Day For Koerner Library 50 Total Arrivals By Hour Of The Day For Woodward Biomedical Library 51 Total Arrivals By Hour O f The Day For David Lam Library 52  ACKNOWLEGEMENTS First and foremost, I would like to thank N S E R C 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 U B C Libraries. I would like to thank the C O E 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 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. 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 selfcharge 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. A n 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 if the situation calls for it. This study is similar to the circulation desk project done for U B C 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 o f copies o f 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.  III. I  METHODS  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 U B C 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.  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. Thefirsttype 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 oneflooron 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 U B C 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 1 2 4 Table 1:  # Copiers Very Busy >7 10 >5 7 >1 1  Moderate  Light  4-7 3-5 1  0-3 0-2 0  Classification of Usage Levels ofEach 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:  III.1.3  ANALYSIS  OF SERVICE  Number of Users on Floor 2  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. A n 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 Quartile Q,: st  c - np/100 = (386*25)/100 = 96.5 96.5 rounded up is 97. Thus, Qi corresponds to the value 120. ii)  3 Quartile Q : c = np/100 = (386*75)/100 = 289.5 289.5 rounded up is 290. Thus, Q3 corresponds to the value 510. rd  3  3. Find the inter-quartile range (IQR). IQR = Q - Qi = 510 - 120 = 390. 3  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 + S4 = 510 + 585 = 1095. 3  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 o f 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 X I 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 C h i Square Test and the Kolmogorov-Smimov Test were used to test this hypothesis. If the selection had a small pvalue for either o f the tests, the next longest service time was chosen, until there was a pvalue 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  1/Tues 2/Tues 2/Wed 4/Wed 1/Thurs 2/Thurs 2/Fri 4/Fri 2/Mon  443 385 331 327 455 401 325 322 395  Distribution  p-value of Chi-Square test  p-value of KolmogorovSmirnov tests  10 + exp(433) exp(385) 2 + exp(329) 2 + exp(325) 40 + exp(415) 10 + exp(391) 5 + exp(320) 5 + exp(317) exp(395)  0.545 0.688 0.116 0.207 0.624 >0.75 0.486 0.506 0.208  >0.15 >0.15 >0.15 >0.15 >0.15 >0.15 >0.15 >0.15 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 4 2 1  Mean Service Time  Distribution  327  2 + exp(325)  Table 3:  401  10 + exp(391)  455  40 + exp(415)  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 ~ / f(y+9) = — e , for > 0, where 9 is some constant. y  /  p  The probability density function, as illustrated in Appendix IV, of the corresponding nonshifted 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 nonshifted 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 4 2 1 Table 5:  III.1.3.5  Mean Service Time  Distribution  327 401 455  exp(327) exp(401) exp(455)  Service Time Non-Shifted  Exponential  "Worst Case" Distributions  •  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 KolmogorovSmirnov 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  1/Tues 2/Tues 2/Wed 4/Wed 1/Thurs 2/Thurs 2/Fri 4/Fri 2/Mon  443 342 331 249 455 367 325 257 395 Table 6:  Distribution 10 + exp(433) exp (342)  2 +exp (329) 30 +exp (219)  40 +  exp  (415)  10 +exp (357)  5 + exp (320) 49 + exp (208) exp(395)  p-value of Chi-Square test 0.545 0.348 0.116 N/A  0.624 0.149 0.486 N/A  0.208  Floor/Day Service Time  p-value of KolmogorovSmirnov tests >0.15 >0.15 >0!l5 >0.15 >0.15 >0.15 >0.15 >0.15 0.0815  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  4 2 1  Mean Service Time  Distribution  257  49 + exp(208)  395  exp(395)  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  4 2 1  Table 9:  Mean Service Time  Distribution  257  exp(257)  395  exp(395)  455  exp(455)  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 Day Floor Day Day Hour Day Bi-hour Floor Day Hour Floor Day Bi-hour  Table 10:  Description of Data Set each day each floor each day each day each hour each day each two hours each floor each day each hour each floor each day each two hours  Scenario Sets for Analysis ofInter-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 A n 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. Mean Floor/Day Inter-arrival Time 83.1 1/Tues 77.5 2/Tues 69.3 2/Wed 501 4/Wed 129 1/Thurs 120 2/Thurs 96.7 2/Fri 96.7 4/Fri 132 2/Mon Table 11:  III.1.4.2  Distribution  p-value of Chi-Square test  p-value KolmogorovSmirnov tests  exp(83.1)  0.69  >0.15  exp(77.5)  0.393  >0.15  exp(69.3)  0.21  >0.15  260+exp(241)  N/A  >0.15  exp(129)  0.728  >0.15 -  exp(120)  0.169  >0.15  exp(96.7)  0.156  >0.15  exp(96.7)  0.156  >0.15  4+exp(128)  0.0327  >0.15  Inter-arrival Time Best-Fit Distributions  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 4 2 1 Table 12:  III.1.4.3  Mean Inter-arrival Time 96.7  Distribution '  exp(96.7)  69.3  exp(69.3)  83.1  exp(83.1)  Inter-arrival Time "Worst Case" Distributions  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.  Mean Floor/Day Inter-arrival Time 167 1/Tues 124 2/Tues 93.1 2/Wed 3900 4/Wed 202 1/Thurs 122 2/Thurs 103 2/Fri 1760 4/Fri 175 2/Mon Table 13:  Distribution  p-value Chi-Square test  p-value KolmogorovSmirnov tests  exp(167)  0.138  0.0454  exp(124)  0.163  0.0858  exp(93.1)  0.159  0.146  208 + exp(3700)  N/A  <0.01  exp(202)  0.284  >0.15  exp(122)  0.617  >0.15  exp(103)  0.546  >0.15  196+ exp( 1570)  N/A  >0.15  exp(175)  0.127  0.0815  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 4 2 1 Table 14:  Mean Inter-arrival Time  Distribution  1760  196 + exp(1570)  93.1  exp(93.1)  202  exp(202)  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 nonshifted exponential distribution. Shifted Distribution  Non-Shifted Distribution  e  196+ exp( 1570)  exp(1760)  196  Table 15:  D(0 < v < 6) 0.1054  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.  Table 16:  Floor  Mean Inter-arrival Time  Distribution  4 2 1  1760  exp(1760)  93.1  exp(93.1)  202  exp(202)  Inter-arrival Time Non-Shifted Exponential "Mean Case" Distributions  III.1.5 ANALYSIS RATE  OF MEAN  ARRIVAL  RATE  AND MEAN  SERVICE  Using the Service Time Best-Fit Distribution from Table 2 and the Inter-arrival Time BestFit 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-firstserved 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 X u,  = number of photocopiers = mean arrival rate of a user to the system = mean service rate of a single server  L L W W p P  =• = = = = =  q  q  n  mean number of users in the system mean number of users waiting in the queue; mean queue length mean time a user is in the queueing system; mean time in system mean time a user is in the queue; mean waiting time server utilization; mean time of work for each server the equilibrium probability that there are n users in the system  111.1.6.3 i.  Formulas for Queueing Model  Photocopier Utilization, p P = —  11.  Mean Queue Length, L  q  Po{cp) p c  c\{\-p)  2  in.  Mean Number of Users In System, L  iv.  Mean Time In Queue, W  q  21  Wq  L  =-*-  X  Mean Time In System, W W =W + q  vi.  Equilibrium Probabilities, Po, P  n  C-]  P„ =  Z  +  c\  ^11=0  \-p  0<n<c  dc" vii.  Po  n>c  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 nonshifted 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 nonshifted 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 nonshifted 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  IIL2.1  ANALYSIS OF STATUS QUO SYSTEM  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 tradeoff between the number of photocopiers in a room and the waiting time accrued with that number of photocopiers.  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 0.00 0.06 0.21 0.08 0.06 0.02 0.00 0.00  0.31 0.20 0.47 0.37 0.43 0.56 0.35 0.12 0.11  0.68 0.80 0.47 0.42 0.50 0.38 0.63 0.88 0.89  F1 Thurs F2 Tues F2 Wed F2 Thurs F2 Fri F2 Mon F4 Wed F4 Fri  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  F1 Tues F2 Tues F2 Wed F2 Mon F4 Wed  Table 18:  0.00 0.00 0.00 0.00 0.00  0.00 0.00 0.05 0.03 0.00  Light  1.00 1.00 0.95 0.97 1.00  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 X V I I 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 interarrival 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, U B C 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 U B C 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 U B C 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 yaxis. 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 yaxis. 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. A n 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:  Table 20:  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  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.  Table 22:  Day of the Week  Total # of Arrivals  Monday  21692  Tuesday  21231  Wednesday  23585  Thursday  21559  Friday  21743  Saturday  10413  Sunday  7861  Total Arrivals By Day Of The Week For Koerner Library 38  Table 23:  Day of the Week  Total # of Arrivals  Monday  7865  Tuesday  8494  Wednesday  7944  Thursday  7570  Friday  7583  Saturday  3215  Sunday  2956  Total Arrivals By Day Of The Week For Woodward Biomedical Library  Table 24:  Day of the Week  Total # of Arrivals  Monday  375  Tuesday  378  Wednesday  361  Thursday  421  Friday  285  Saturday  224  Sunday  115  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  Table 25:  Table 26:  Hour of the Day  Total # of Arrivals  8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00  1307 3984 7505 11128 13863 15184 14698 14907 14149 9509 6844 4325 3795 3488 3361 38  Total Arrivals By Hour Of The Day For Koerner Library Hour of the Day  Total # of Arrivals  8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00  1864 3036 3919 4408 4862 4623 4499 4599 5151 3617 1543 1225 901 893 472 14  Total Arrivals By Hour Of The Day For Woodward Biomedical Library  40  Table 27:  Hour of the Day  Total # of Arrivals  8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00  37 70 144 188 272 249 258 237 281 192 78 51 49 34 22 0  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.  Mac hine  Bar Code Placement  Table 28:  inside cover  outside cover  Machine 1  38  39  Machine 2  43  38  The Number Of Successes For The Self-Charge Machines  43  Batch 1 2 3 4 5 6 7 8 9 10  Table 29:  IV.1.8.2  Successes  7 9 8 7 7 10 8 5 8 8  Batch 11 12 13 14 15 16 17 18 19 20  Successes  8 10 8 9 8 7 8 7 8 8  The Number Of Successes For The Self-Charge Machines In Batches Of 10  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  3 co  0.84  °  0.8  a> u  3  Inside  Outside  0.76  JO  o  a.  0.72  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. A n 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. Monthly Totals 25000.00 20000.00 -  fl  15000.00  n  10000.00 5000.00 0.00  Is  II O CM  O  I 5 CD O O CN  CM O  o  CNJ CO  3  CM O  o o  o o  CNJ  CN  < 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: IV.2.3.2  Total Arrivals By Month For David Lam Library  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: IV.2.3.3  Total Arrivals By Day Of The Week For David Lam Library  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.  16000 14000 12000 10000 o ro o  8000 \ 6000 4000 2000 0  -I  CM  CM  CM  eg  Hour Of The Day  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:  IV.2.3.4  Total Arrivals By Hour Of The Day For David Lam Library  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.  .c c  Low  Moderate  High  December  September  November  May  October  March  January  So  February April Saturday  >. re Q  Not Applicable  Sunday  Monday Tuesday Wednesday Thursday Friday  u —i  O  -T-  8:00  10:00  12:00  9:00  11:00  13:00  19:00  17:00  14:00  20:00  18:00  15:00  21:00  16:00  22:00 23:00  Table 30:  Breakdown Of Periods For Koerner Library  52.  •«-» C  o  Low  Moderate  High  December  September  October  May  February  November  April  January  5  March Saturday  Not Applicable  Sunday  >, (0  Monday Tuesday. Wednesday  Q  Thursday Friday 19:00 3 O  T.  8:00  11:00  20:00  9:00  12:00  21:00  10:00  ,13:00  22:00  17:00  14:00  23:00  18:00  15:00 16:00  Table 31:  Breakdown OfPeriods For Woodward Biomedical Library  Low  Moderate  High  December  September  November  October  March  May c  January  S  February  o  April Friday ns Q  Not Applicable  Monday  Saturday  Tuesday  Sunday  Wednesday Thursday  i  3 O  8:00  10:00  12:00  9:00  11:00  13:00  18:00  17:00  14:00  19:00  15:00  20:00  16:00  21:00 22:00 23:00  Table 32: IV.2.4  ARRIVAL  Breakdown OfPeriods For David Lam Library  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  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  Average # Arrivals  Low  Peak  Peak  6839.92  220  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:  .  31.09  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 4.05  Low  Low  Peak  437.50  108  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  Moderate  Peak  Moderate  3543.50  310  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:  3.04 .  11.43  Summary OfArrival Calculation Process For Woodward Biomedical Library  54  Month  Day  Low Low Low  Low Low Low  Low Low Low  Peak Peak Peak  Moderate Moderate Moderate Moderate Moderate Moderate Peak Peak Peak Peak Peak Peak  Low Low Low Peak Peak Peak Low Low Low Peak Peak Peak  Table 35:  Hour  # Of Arrivals  # Of Days  Average # Arrivals  Low  2.42 6.00 76.67  216 81 135 280 105 175 496 186 310 704 264 440 224 84 140 264 99 165  0.01 0.07 0.57  Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak  43.83 50.33 118.58 11.42 65.00 259.75 187.83 240.92 455.92 7.67 52.33 142.67 86.50 110.00 241.75  0.16 0.48 0.68 0.02 0.35 0.84 0.27 0.91 1.04 0.03 0.62 1.02 0.33 1.11 1.47  Summary OfArrival 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  Month  Day  Hour  Low Low Low Low Low Low Moderate Moderate Moderate Moderate Moderate Moderate Peak Peak Peak Peak Peak Peak  Low Low Low Peak Peak Peak Low Low Low Peak Peak Peak Low Low Low Peak Peak Peak  Low Moderate Peak Low Moderate Peak .Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak  Table 36:  1 1  2 1  2 2 1  2 2 2 2 2 2 2 2 2 2 3  Staffing Rules For Koerner Library  Month  Day  Hour  Low Low Low Low Low Low Moderate Moderate Moderate Moderate Moderate Moderate Peak Peak Peak Peak Peak Peak  Low Low Low Peak Peak Peak Low Low Low Peak Peak Peak Low Low Low Peak Peak Peak  Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak  Table 37:  # Of Staff Needed At Circulation Desk  #Of Staff Needed At Circulation Desk  1 1 1 1  1 1 1  1 1 1 1 1  1 2 1  2 2  Staffing Rules For Woodward Biomedical Library  Month  Day  Hour  Low Low Low Low Low Low Moderate Moderate Moderate Moderate Moderate Moderate Peak Peak Peak Peak Peak Peak  Low Low Low Peak Peak Peak Low Low Low Peak Peak Peak Low Low Low Peak Peak Peak  Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak  Table 38:  # Of Staff Needed At Circulation Desk 1  1 1 1  1 1 1 1 1 1 1 1 1 1 1  1 1 1  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 OfAllowable Change In Percentage Of Renewal Transactions  A n 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 OfMoney-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 moneycollected 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  Lower Limit  2.3  2.0  0.4  Upper Limit  3.1  2.5  None  Table 41:  David Lam Library  Range OfA 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  Lower Limit Upper Limit  Koerner Library  Woodward Biomedical Library  David Lam Library  60  57  None  66  60  72  Table 42:  Range OfAllowable 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. A n 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 ((3 = 1.153 with P = 0.000, Bi = 0.662 with P = 0.207, p = 0.113 with P = 0.812, and p = -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 0  2  3  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 selfcharge 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 o f this study is to analyze the demand o f the circulation desks located in Koerner Library, Woodward Biomedical Library, and David L a m Library. From this analysis, staffing rules (the number o f staff required for the circulation desk) for periods o f 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 L a m Library and a summary o f the reliability o f the self-charge machines for the use o f U B C Library's management. The staffing rules consist o f three matrices, one for each branch, that specify the minimum number o f staff that need to be offering service at each circulation desk to ensure that a client waits no more than 120 seconds for any o f the different periods o f 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 o f this project and the type o f recommendations that are made, U B C Libraries has again employed the Centre for Operations Excellence ( C O E ) to perform other projects.  IV.4.1  COMMENTS  A n observation noted during data collection on the reliability o f 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 i n a way that acted as the open cover o f the book being scanned before the machine w i l l work. It is recommended that the machines be modified or upgraded to improve reliability o f 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 o f the analysis increases dramatically. The advantage o f using such a process is to ensure that carry over o f 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 o f 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 Hill, 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 Hill, Boston, 2001. Kelton W.D., Sadowski R.P, Sadowski D.A. Simulation With Arena. WCB/McGraw Hill, 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 O f 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  APPENDIX 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 CN  N  -  LO  CN  od O o -<t  O  CN o cCOCOo CDLO CN  N "' 1  LO LO  T—  CN CN  LO •vt CO CO o o CN c O CNLO COCN o o o o LO CN 6) •ito o o o LO CN •st LO CO •vt LO CN o c b c b c o 1 ^ o i n CO CO CO CO •vt •st •vf •st LO LO o co co o o ib ib cb cb Time CO CD •vt CN  CO  LO  LO  LO  LO  N—  LO  LO  T—  CO T—  •st  o o  T—  LO  in — .  cn  CO  T—  Floor 1 Thursday 12  o p  o o  co  CN  o  o  o  O  s  IV-  CN  o cp  co LO CO co  r•vt O •?  |scb  -?t  CO CN  IV-  o  co  CN  CN  IO  •vt  CO CN CN  LO  s CN  O co CN  O co 66  LO  -Nt CN  cb  cb  o  •st •st  LO T— h-' CN CO  in cb co CO  o o  cb co CO  o  co CN cb  m •st  $  cb  ib LO cb  LO cb  o •it  in CN  •it  Time  68  Floor 2 Tuesday  ®  3 o  12 10 8 6  HI .  4 2 0  0 0 1 •tf^ CmN s  66  I; 1  —1-171  C CD O CN 1 -dtf^ 0 -tf CD in in cri 0 cb CO CN 6 cb m 0 CN csi cri Lf) dco CN CO 1—  CO 0 in 0 0 c•tfb ib 0 CN co 5O cb cb csi C cb in c 0b •tf  in 0 0 0 to ^ mcb in 1 0 CO C O cb  •tf  -tf  co -tf 0 -tf cri cb s CO in in in cb cb cCN b  m cco b  00 co CN O  •tf  •tf  co -tf  •tf  CN 0 cinb cin 0b  Time  Floor 2 Wednesday 12 10 i_ 8 OJ 6  3  o  ___L ! I  4 2 0  c-tfo m 0 m 0 0 0 ccCOb in o cCb in ib O d d CN 06 cCNri T—  T—  CSI  in 0 in  0 in in 0 0 0 c0 0si 0 0 0 0 •tf 0 •tf 0 0 0 ci—ri inCOccosi CO0 oi 1 ^ CN in c o • t f 0 s s CO csi cSi CM in cinri cino inin csi cCOsi csi cbTimecb cb CN cb cb  O O d  in in CO  CN  •tf  •tf  T—  \—  T—  •tf  •tf  •tf  •tf  T—  Floor 2 Thursday 12 10 ID k_ 8 0) 3 6  III  **- 4 o 2 0  in 0 0 0 cb  m in 0 co in 0 0 m 0 in 0 •tf 0 0 cb cinb in 0 cb cb CO in  iTT[tnTnTinmTM''imTirirviTrHin'mmHmTri7mTiiT^  co •tf d  d  d  CN  CM  CO  HI"  m 0 in 0 0 co 0 co 0 0 •tf 0 cri in c b d d cb co •tf 0 ib co CN •tf (Si CSI cSi csi (Si cb CO cb cb  0 T—  Time  in 0 in 0 cb •tf x— csi CO cb 0 •tf  •tf  0 0 1^ •tf  69  .Floor 2 Friday 12 10  (0 •  3 o *  8 6  4 2 0 If) m ib o  in o di m o  o CN CO CN  o  m C O o CO CO  o m oi CO  o o in CO o <Si CN  s  in  m o o ai CM CN oi CN  o O O o in co CO in CN CN CN  in CO CO o co  m o CO co  o CN CN co  5-  m  in  in  in  o o o lO CO  CO cb CO co cd  o  m in o  o  o <> j T—  o in in CN  Time  Floor 2 Monday 12 -r  o cSi •<J00  o CN ai cn T—  o in co  in o cb CN o  o oo m ai  CO  •st cb d CN  •M-  in in d  o o CO  p  in m ai p  o  CO  ib CN  o m ib co  m CO ai -?  m ai o CN  cSi CN CN  co CN  o  oo  T—  cb  o o cb  CO  CN CO  CO  CO  o cb  CN  in  o  in  in co  o •st  cb  CN  CO  CN  Time  Floor 4 Wednesday  Floor 4 Friday  V)  2  1  Z> o r'i""T T o o o CM O cb cb CO in o o csi csi CO ir  CO m cb o  in CM cb in  in o ai CN CO  CN CN CO CO  Time  o d  a> o in CO •tf  r  mm  T—  1^ T—  o CO ai CN CO  \ 1 o cb CSI  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 o  o o o CO oi  o OI CO oi  o o Tt CO oi  o o CD CO 6S  o o CO CO oi  o o o ci  o o oi T* cn  o o _r 4 di  o o cb" CO  o p T? cn  o p o m oi  o o OI cn  o o o o Tf , (O V) o> oi  o o CO oi  tr o o o p d  K, o p oi o o  o o o d  o o cb o d  o o cb o o  o o o  o o oi  o o  o o CD  o o 00  o  o  o  O  d  o o o OI o  o o oi OI o  o o Tf OI d  o o cb' OI d  o o 00 OI o  Time  12 10 8 6 4 2  18:00 (  o p tb  o p 00  oi  oi  oi  oi  oi  28:00  16:00  TT  26:00  14:00 o o  24:00  12:00 o p oi  22:00  10:00 o o d  20:00  1mm- — 11-  0 •  o p d OI oi  o o oi CN oi  o o  o p CD CN oi  o p 00 CN oi  Time  12 »  5  1 •R  10 8 6 4 2  o  o o o CO  o o oi CO  o o TJ"  CO  o o CD CO  o p cb CO  o o d TT  o o oi Tf  o o 4  o p <o Tf  o p •5  o p d in  o o oi p oi  o p oi in  o o  Tf O oi  o o u> o oi  o o cb p oi  Tf OI oi  Time  71  # of Arrivals  # of A r r i v a l s  # of A r r i v a l s  o to -f^ 0> oo o to  o to 4>> o> oo o ro  o ro 4k o> oo o ro  # of A r r i v a l s  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"  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  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  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  # 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  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  ro .u cn co o NJ  o  # of Arrivals  fo 6 oi co o ro  12:30:00  11 30:00  10:30:00  12:32:00  11 32:00  10:32:00  12:34:00  11 34:00  10:34:00  12:36:00  11 36:00  10:36:00  12:38:00  11 38:00  10:38:00  12:40:00  11 40:00  10:40:00  12:42:00  11 42:00  10:42:00  12:44:00  11 44:00  10:44:00  12:46:00  11 46:00  10:46:00  12:48:00  11 48:00  10:48:00  12:50:00  11 50:00  10:50:00  12:52:00  11 52:00  10:52:00  12:54:00  11 54:00  10:54:00  12:56:00  11 56:00  10:56:00  12:58:00  11 58:00  10:58:00  13:00:00  Time  13:58:00  o  12 00:00  I  H 3' 11:00:00 <B 11:02:00  13:02:00  12 02:00  13:04:00  12 04:00  11:04:00  13:06:00  12 06:00  11:06:00  13:08:00  12 08:00  11:08:00  13:10:00  12 10:00  11:10:00  13:12:00  12 12:00  11:12:00  13:14:00  12 14:00  11:14:00  13:16:00  12 16:00  11:16:00  13:18:00  12 18:00  11:18:00  13:20:00  12:20:00  11:20:00  13:22:00  12 22:00  11:22:00  13:24:00  12 24:00  11:24:00  13:26:00  12 26:00  11:26:00  13:28:00  12 28:00  11:28:00  ip  o  ro  O) co o ro  ©  o  8 6 o 4 * 2 illlll|jPW^P!p| n! B ^^^^^^^^^^^^^^^^^^^^^^^^^^^  0:12:00 j  0:26:00 0:28:00  0:24:00  0:18:00 0:20:00 0:22:00  0:16:00  0:14:00  Pi  0:10:00  j||j|%||Ji|||^^  0:04:00 0:06:00 0:08:00  0:02:00  0:00:00  3  9:56:00 9:58:00  9:52:00 9:54:00  9:46:00 9:48:00 9:50:00  9:44:00  9:42:00  9:40:00  9:38:00  9:34:00 9:36:00  9:32:00  9:30:00  Floor 2 Tuesday  12 -I 10  in  £  j^^^^^^S^a^^^^^^^.ljtjjm rj g  Time  Time  Time  74  Floor 2 Tuesday continued 12 10 o 8 3 6  4 j—sip  mmw^Smmrm^s  IPP^ii^fczBsiMt:  ~m  | —  P  2 jjtSlBIIHl^^ 0  % r _^_ _ . ^ ^ v ^ _ - = = ^ . . = ^ . . = ^ . - ^ . ™ . - ^ : o o o o o o o o o o o o o o o o o o oo oo oo oo oo oo oo oo oo oo o o o o o o o o o o sli  0  0  4  -  ^  <  J  C  >  W  O  C  N  J  ^  <  i  >  C  O  O  C  ^  ^  (  J  l  >  C  J  O  S  t  J  J  =  S  i  i  L  O  (oocococo^j'TrTr^'Trw>ioio»omooooot-f--«-T--.-rsicNicM Time  Time  Time  12 10 4 8 6 4 2 •vt CO  •vt  U)  IO  U)  io io in ImO I uO>C  0  (  0  l  0  t  0  (  o0 C TT- T- V — T-N 0 < 0 ( 0 C 0 < 0 ( 0  <  N 0 (  N O C  N 0 (  N 0  Time  75  Floor 2 Wednesday j  12  •  10  3 o  .  8  6 4  -  ""1  0 o o o CO CO  o o csi CO CO  o o Tf CO  o o CO CO  CO  CO  o o CO CO CO  o o o Tf CO  o o oi Tf cb  CD Tf Tf cb  o o CD Tf cb  o o  CD o  CO Tf cb  o CO  o o cvi IO CO  o o Tf in CO  o o  o o  CD CO  CO CO  cb  CO  o o CD o CT)  o CD oi CD cn  CD CD  CD CD  o o  Tf o 6S  CO o oi  CO o cn  o o CD cn  o o CN cn  o o Tf  o CD CD  cn  cn  o CD CO cri  o CD CD CN CT)  o CD oi CN cn  CD CD  CD CD  CN  CO CN  o>  CT)  o o cb CN  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 o o o CO < o  IlilllBllilll  o o o o o o o o C O o o CO CO CO C CO O o Tf in u> u> u>  i f c ^ a ^ ^ f c * • —III —  !  o o cTsfi  o o o o o o o o T f C D C O o Tf Tf in Tf UT) u> u> to  o o CN in to  o o Tf in in  o p oo p o o p o o o ci 9f? CD o in p p •T in in IO C pD D CDC  O O CD O CD  O O C OO CD  O O O CD  O O CN CD  O O Tf CD  O O CD CD  O O CO CD  O O O CN CD  O O C CN N  O O T f CN  O O C CO N CO CD CD  O O CO C N CD  Time  77  ro ^ cn co o N  o ro ^ o) oo o ro  o ro 11:30:00  # of Users  m co o ro 10:30:00  13:30:00  12:30:00  13:32:00  12:32:00  11:32:00  13:34:00  12:34:00  11:34:00  13:36:00  12:36:00  11:36:00  10:36:00  13:38:00  12:38:00  11:38:00  10:38:00  13:40:00  12:40:00  11:40:00  10:40:00  13:42:00  12:42:00  11:42:00  10:42:00  13:44:00  12:44:00  11:44:00  10:44:00  13:46:00  12:46:00  11:46:00  10:46:00  13:48:00  12:48:00  11:48:00  10:48:00  13:50:00 .]  12:50:00  11:50:00  10:50:00  13:52:00  12:52:00  11:52:00  10:52:00  13:54:00  12:54:00  11:54:00  10:54:00  13:56:00  12:56:00  11:56:00  10:56:00  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  lilu  • ;pJ|  10:32:00  12:58:00  11:58:00 12:00:00  3 11:00:00  CD  12:02:00  11:02:00  -i  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 ro ^ cn oo o ro  10:34:00  3 13:00:00  13:58:00  oo  #of Users  #of Users  # of Users o  10:58:00  -(D1  12:04:00  11:04:00  12:06:00  11:06:00  12:08:00  11:08:00  12:10:00  11:10:00  12:12:00  11:12:00  12:14:00  11:14:00  12:16:00  11:16:00  12:18:00  11:18:00  12:20:00  11:20:00  12:22:00  11:22:00  12:24:00  11:24:00  12:26:00  11:26:00  12:28:00  11:28:00  Hi.  O O  1  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  w S  12 j 10 8  •o 4  o p o CO  o p CM CO  o p TT CO  o p u> CO  o p o o o o p o C p pO CM p Tf CO C CO O d T* p TO f C TO f C TO f T f CO CO C CO  o p o CO  o o o o p o o o p o p C p pO pM Tf CO Tf C CO O o uM p p C O C p C u> pf" p CO C O CO CO Tf p Tf" T Tf  o p CO p Tf"  o p o o p o p pM Tf CO o C Tf" Tf Tf" Tf"  o p o o o p o p O p C pM CO o C M Tf C CM CM CM Tf T f Tf" T f" Tf  o p C O CM  Tf"  Time  79  # o f A r r i v a l s  o ro  4*  o> oo ©  10  o M J» cn CD o w  1 1 3 0 : 0 0 :  1 0 : 3 0 : 0 0  1 1 3 2 : 0 0  # o f A r r i v a l s  # o f A r r i v a l s  # o f A r r i v a l s  O M  o ro  O) CO O M  9 : 3 0 : 0 0  8 : 3 0 : 0 0  1 0 : 3 2 : 0 0  9 : 3 2 : 0 0  8 : 3 2 : 0 0  1 1 3 4 : 0 0 |  1 0 : 3 4 : 0 0  9 : 3 4 : 0 0  8 : 3 4 : 0 0  1 1 3 6 : 0 0 ;  1 0 : 3 6 : 0 0  9 : 3 6 : 0 0  8 : 3 6 : 0 0  1 1 3 8 : 0 0  1 0 : 3 8 : 0 0  9 : 3 8 : 0 0  8 : 3 8 : 0 0  1 1 4 0 : 0 0 i  1 0 : 4 0 : 0 0  9 : 4 0 : 0 0  8 : 4 0 : 0 0  1 1 4 2 : 0 0 j  1 0 : 4 2 : 0 0  9 : 4 2 : 0 0  8 : 4 2 : 0 0  1 1 4 4 : 0 0 ;  1 0 : 4 4 : 0 0  9 : 4 4 : 0 0  8 : 4 4 : 0 0  1 1 4 6 : 0 0 j  1 0 : 4 6 : 0 0  9 : 4 6 : 0 0  8 : 4 6 : 0 0  1 1 4 8 : 0 0 I  1 0 : 4 8 : 0 0  9 : 4 8 : 0 0  8 : 4 8 : 0 0  1 1 5 0 : 0 0 )  1 0 : 5 0 : 0 0  9 : 5 0 : 0 0  8 : 5 0 : 0 0  1 1 5 2 : 0 0 j  1 0 : 5 2 : 0 0  9 : 5 2 : 0 0  8 : 5 2 : 0 0  1 1 5 4 : 0 0 !  1 0 : 5 4 : 0 0  9 : 5 4 : 0 0  8 : 5 4 : 0 0  1 1 5 6 : 0 0 :  1 0 : 5 6 : 0 0  9 : 5 6 : 0 0  8 : 5 6 : 0 0  1 1 5 8 : 0 0 :  1 0 : 5 8 : 0 0  3 '1 2 0 0 : 0 0 \  1 1 : 0 0 : 0 0  3  1 1 : 0 2 : 0 0  C D  :  C D1 2 0 2 : 0 0  |  9 : 5 8 : 0 0 1 0 : 0 0 : 0 0  8 : 5 8 : 0 0  H 3  9 : 0 0 : 0 0  C D  1 2 0 4 : 0 0  1 1 : 0 4 : 0 0  1 0 : 0 2 : 0 0  9 : 0 2 : 0 0  1 2 0 6 : 0 0  1 1 : 0 6 : 0 0  1 0 : 0 4 : 0 0  9 : 0 4 : 0 0  1 2 0 8 : 0 0  1 1 : 0 8 : 0 0  1 0 : 0 6 : 0 0  9 : 0 6 : 0 0  1 2 1 0 : 0 0  1 1 : 1 0 : 0 0  1 0 : 0 8 : 0 0  9 : 0 8 : 0 0  1 2 1 2 : 0 0  1 1 : 1 2 : 0 0  1 0 : 1 0 : 0 0  9 : 1 0 : 0 0  1 1 : 1 4 : 0 0  1 0 : 1 2 : 0 0  9 : 1 2 : 0 0  1 2 1 6 : 0 0 j  1 1 : 1 6 : 0 0  1 0 : 1 4 : 0 0  9 : 1 4 : 0 0  1 2 1 8 : 0 0 •  1 1 : 1 8 : 0 0  1 0 : 1 6 : 0 0  9 : 1 6 : 0 0  1 2 2 0 : 0 0  1 1 : 2 0 : 0 0  1 0 : 1 8 : 0 0  9 : 1 8 0 .0  1 2 2 2 : 0 0  1 1 : 2 2 : 0 0  1 0 : 2 0 : 0 0  9 : 2 0 : 0 0  1 2 2 4 : 0 0 i  1 1 : 2 4 : 0 0  1 0 : 2 2 : 0 0  9 : 2 2 : 0 0  1 2 2 6 : 0 0  1 1 : 2 6 : 0 0  1 0 : 2 4 : 0 0  9 : 2 4 : 0 0  1 2 2 8 : 0 0  1 1 : 2 8 : 0 0  1 0 : 2 6 : 0 0  9 : 2 6 : 0 0  1 0 : 2 8 : 0 0  9 : 2 8 : 0 0  1 2 1 4 : 0 0  !  cn co o ro  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 op o o p p p p p p p p p p p p p p O O O O o O O O OCSI pT f OCO pCO p CN o o CN Tf Tf CO OO CO 00 CO CO CN CN Tf CD CO Tf" Tt CO o o CO CO CO CO po p p p p o d CN CN CN 00 rd o co CO CO CO Tf Tf Tf Tf CN CN Tf *p. 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 o o p o p o p o p o p o p o o o o p o o p o o o o p CN Tf CN Tf OO OO CN cb cb Tf cb o o p p d > p CO CO CO CO T f T f T f T f T f CO CO CO CO CO co CO CO CO CO CO CO CO CO CO CO CO CO CO cb cb CO  o p o o o o o o o o o p o p o o p o o o p o p o p Tf CN CO CO CO Tf 00 o CN p T f cb p cb T— o CN p T— p p CN o TT CN CM CN p o o of Tf" o T f T f T f T f Tf" Tf" T f Tf" T f CM Tf Tf T Tf Tf Time  81  # of Users  o  11 :30:00  11 :36:00 11 :38:00 11 :40:00 11 :42:00  ro  # of Users  #of Users 8:30:00  9:32:00  8:32:00  10:34:00 |  9:34:00  8:34:00  10:36:00  9:36:00  8:36:00  10:38:00 |  9:38:00  8:38:00  10:40:00 |  9:40:00  8:40:00 8:42:00  10:32:00  11 :34:00  -fc  9:30:00  10:30:00 j  11 :32:00  # of Users j  10:42:00 I  9:42:00  11 :44:00  10:44:00  9:44:00  8:44:00  11 :46:00  10:46:00 I  9:46:00  8:46:00  10:48:00 (  9:48:00  8:48:00  10:50:00  9:50:00  8:50:00  10:52:00  9:52:00  8:52:00  10:54:00  9:54:00  8:54:00  10:56:00  9:56:00  8:56:00  10:58:00 11 :00:00 (  9:58:00  11 :48:00 11 :50:00 11 :52:00 11 :54:00 11 :56:00 11 .58:00  -I 3 12:00:00 , ro 12:02:00 I  11 :04:00 |  12:06:00  11 :06:00 j  10:04:00  12:08:00  11 :08:00  10:06:00  12:10:00  11 :10:00  10:08:00  12:12:00  11 :12:00 ;  10:10:00  12:14:00  11 :14:00  12:16:00  11 :16:00 )  12:18:00  11 :18:00  12:20:00  11 :20:00  12:22:00  11 :22:00 |  12:24:00  11 :24:00 \  12:26:00  11 26:00  12:28:00  11 •28:00  Time  12:04:00  -t 3 10:00:00 to 10:02:00  11 :02:00  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  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  •-pll • ITlfh m  f o  o S 0  O o  o o  o "  o  ^ 0  0  (  p  N 0 0 "  p (N 0  p  o  (N 0  o  o  o  CO CO 5 0 0 0  o T? 0  p 0  p  p IO o  0  p O ^  O ^ '  0  0  0  0  o  o  0 0 o  0 0  0  0 o  o  o  Service Time (minutes)  ^ b  ^ C ^ ^ r ^ S f N C N ( N f O o  b  0 d  0 o  0 0  0  m  0  2 o  0  11  0  0  0 b  0  0 o  0  0 S  b  0  o  to  ro ro  b  o  d  o  Service Time (minutes)  Floor 2 Tuesday March 13, 2001  Floor 2 Wednesday March 14, 2001  g  30 o cr £ 20  u.  10  R S  o  o  o  o  o  o  o  • R o  o  i  o  o  o  o *-  Service Time (minutes)  Floor 2 Thursday March 15, 2001  0  0  0  0  0  0  0  0  0  0  <7>  O  O  O  C M O  Service Time (minutes)  LO cn  T-  Floor 2 Friday March 16, 2001  O N  IO  Service Time (minutes)  O  O  O  O  O  Service Time (minutes)  85  Floor 2 Monday M a r c h 19, 2001  40 -  g 30 a  3 CT  © 20  U-  10 • ra  „  m  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 ;  „  3  GO  BP  • 0:13:00 g 0:15:00 ; |! 0:17:00 ; Z. 0:19:00 "  1  0:03:00 ; 0:05:00 ; 0:07:00 ; 0:09:00 ; 0:11:00 ;  i  Floor 4 Wednesday March 14, 2001  Floor 4 Friday March 16, 2001  p p p p o q q o o p o p p q q o o o j o o o o o oo O  Q  O  O  O  O  O  O  O  O  Service Time (minutes)  IV  PROBABILITY DENSITY FUNCTION FOR AN EXPONENTIONAL DISTRIBUTION 6>0 0<y <  J  oo  fiy) = •  otherwise  V  PROBABILITY DENSITY FUNCTION FOR A SHIFTED EXPONENTIONAL DISTRIBUTION  1  ~A  — e P  y  / h  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 collectedfroma random sample. The expected frequency for each category is five or more. ii.  T E S T STATISTIC  with degrees offreedom(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 : the data follows a specific distribution > H : the data do not follow a specific distribution 0  a  iv.  PASS/FAIL C R I T E R I A  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.  ASSUMPTIONS  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 STATISTIC  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 ~  iii.  N  HYPOTHESIS  H : the data followsa specific distribution H : the data do not follow a specific distribution 0  a  iv.  PASS/FAIL CRITERIA  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 Q , compute: c = np/100 Where n = total number of values p = percentile 3  Qi corresponds to the 25th percentile and Q3 corresponds to the 75 percentile. So, for Qi, the formula is c = 25n/100 and for Q the formula is c = 75n/100. th  3  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 c value. l  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 = Q - Q, 3  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 Q - Q|. S5- = Qj - S4 S5+ = Q + S4 3  3  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 13 30 40 49 57 65 76 88 96 110 120 130 141 155 165 173 188 197 223 244 277 290 307 321 343 359 382 400 429 452 472 507 538 610 658 720 763 845 920 1035 1200 1720  2 15 30 40 50 58 70 78 90 97 111 121 131 144 155 166 173 188 201 225 249 277 295 308 325 345 360 383 400 430 453 475 510 545 610 664 720 764 850 935 1052 1211 1750  4 15 30 41 50 58 70 78 90 97 112 126 134 145 156 168 177 188 206 228 250 278 296 308 332 347 360 385 402 433 454 480 510 545 610 665 720 764 853 935 1081 1285 1875  5 18 30 43 52 59 70 80 90 101 114 127 134 145 156 168 178 190 209 235 252 278 297 310 332 350 360 385 402 435 455 488 513 550 617 668 724 781 862 950 1103 1330 1960  5 20 31 43 53 60 71 80 90 102 118 128 135 146 157 170 180 190 210 235 257 279 300 310 335 353 368 386 405 440 457 488 518 575 618 671 726 790 866 985 1125 1383 2397  10 22 32 45 53 60 72 81 90 105 118 128 137 148 158 170 184 190 215 235 259 280 300 315 338 355 370 388 410 448 457 493 520 580 620 675 729 810 888 992 1147 1474 2401  10 22 37 45 54 60 75 82 92 105 120 130 138 152 160 170 185 195 218 235 260 280 302 315 339 355 370 390 410 448 460 495 523 581 625 688 735 814 892 997 1173 1477 2500  10 28 40 47 55 61 75 85 95 106 120 130 138 152 163 170 186 195 220 240 270 288 305 315 340 357 372 390 422 448 461 502 525 594 625 698 750 820 915 1006 1181 1500 4079  12 30 40 49 55 62 75 88 95 108 120 130 140 154 165 172 186 197 220 243 275 289 305 320 340 358 375 400 425 450 471 505 536 603 655 707 757 830 918 1020 1190 1517  /  90  X  BOX PLOT FOR TUESDAY  0  1000  2000  3000  4000  Tuesday  XI 10 55 75 110 138 163 197 260 300 325 370 402 455 520 617 735 853 1103 2397  SERVICE TIME DATA FOR FLOOR 1 TUESDAY 12 55 80 120 140 170 197 277 302 335 372 405 461 525 620 781 888 1125 2401  13 60 81 120 145 173 201 278 307 338 382 410 471 536 664 790 892 1200 4079  20 61 90 120 145 180 220 279 308 345 385 410 480 545 675 810 918 1285  30 62 90 126 146 185 223 280 310 355 386 430 493 550 720 814 920 1383  30 65 92 127 152 186 235 288 315 355 390 435 495 575 720 820 935 1517  40 70 97 130 154 188 249 290 315 357 390 448 502 580 720 830 950 1750  40 72 105 131 156 195 250 295 315 358 400 450 507 581 726 845 985 1875  53 75 105 135 160 195 257 296 320 360 400 454 513 594 729 850 997 1960  91 I  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  Floor 2 Wednesday March 14, 2001  120 100  tn  Arri  r>o 80  60 o 40 20 0  < O  Time  Floor 2 Thursday March 15, 2001  40  Time  Floor 2 Friday March 16, 2001  Floor 2 Monday March 19, 2001  tn ta  120 100  80 < 60 o 40 * 20 0 >  O ,_  -•-  C M  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 o f 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 o f 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  Floor 2 Friday March 16, 2001  O  Inter-arrival Time {minutes)  -r-  T-  T-  T-  Inter-arrival Time (minutes)  94  Floor 2 Monday March 19, 2001  u c  3 CT Lg L) C D  o p O  b  o o O  o o O  b  O  o o O  b  o o O  O  b  o o T  b  o o -  "  b  o o ~  b  o o I  -  b  o o I  b  o o -  b  o o T  -  b  o o T  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 ppOy^^c^rjCNrocO'^^'^-Lninoooi-'-  Inter-arrival Time (minutes)  Floor 4 Friday March 16, 2001  o p  o  o o o o o o o o o o o o o o o 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-^--^-TfTrio b b b b b b b b b b b b b b b b b  Inter-arrival Time (minutes)  95  X V  QUEUEING ANALYSIS "WORST CASE"  CHARTS  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 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 2 3 4 5 6 7 8 9 10  4903.33 2599.11 1789.45 1316.82 955.15 696.04 556.47 495.50 470.84 461.07  4448.33 2144.11 1334.45 861.82 500.15 241.04 101.47 40.50 15.84 6.07  9.78 9.42 8.80 7.53 5.28 2.79 1.21 0.49 0.19 0.07  1.00 1.00 1.00 0.99 0.96 0.88 0.78 0.68 0.61 0.55  #Of Copiers  Mean Time In System  Mean Time In Queue  Mean Queue Length  Utilization  1 2 3 4 5 6 7  3124.22 1698.84 1196.79 913.23 715.20 575.43 488.26  2723.22 1297.84 795.79 512.23 314.20 174.43 87.26  6.79 6.47 5.94 5.05 3.74 2.32 1.22  #Of Copiers  Mean Time In System  Mean Time In Queue  Mean Queue Length .  1 2 3 4  579.37 419.56 369.46 347.25  252.37 92.56 42.46 20.25  0.72 0.49 0.31 0.17  Floor 2:  1.00 1.00 . 1.00 0.99 0.96 0.89 0.80  Floor 4:  Utilization  0.94' 0.86 0.78 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:  Servi evel  CD U  100% 95% 90% 85% 80% 75% 70% 65% 60%  0  30  60  Waiting Time (in seconds) 90 120 150 180 210  17 11 10 9 9 8 8 8 8  16 10 10 9 9 8 8 8 8  16 10 9 9 8 8 8 8 7  15 10 9 8 8 8 8 7 7  15 10 9 8 8 8 7 7 7  15 9 9 8 8 8 7 7 7  14 9 8 8 8  14 9 8 8 8 7 7 7\ 7 7 \ 7 7 \7 s  240  270  300  13 9 8 8 7 7 7 7 7  13 9 8 8 7 7 7 7 7  13 8 8 8 7 7 7 7 7  240  270  300  9 4 4 3 3 3 3 2 2  8 4 4 3 3 3 3 2 2  8 4 3 3 3 3 2 2 2  240  270  300  13 9 8 8 8 7 7 7 7  13 9 8 8 7 7 7 7 7  Example  Floor 4:  Servi evel  100% 95% 90% 85% o 80% _i 75% 70% 65% 60%  0  30  60  Waiting Time (in seconds) 90 120 150 180 210  13 8 7 6 6 5 5 5 4  12 7 6 6 5 5 5 •4 4  12 7 6 5 5 4 4 4 4  11 6 5 5 4 4 4 4 3  0  30  60  Waiting Time (in seconds) 90 120 150 180 210  18 11 10 10 9 9 9 8 8  17 11 10 9 9 9 8 8 8  16 10 10 9 9 8 8 8 8  16 10 9 9 8 8 8 8 7  11 6 5 4 4 4 4 3 3  10 j 5 .5 4 4 4 3 3 3  10 5 4 4 4 3 3 3 3  9 5 4 4 3 3 3 3 2  Floor 2:  Servi evel  o  100% 95% 90% 85% 80% 75% 70% 65% 60%  15 10 9 9 8 8 8 8 7  15 9 9 8 8 8 8 7 7  14 9 9 8 8 8 7 7 7  14 9 8 8 8 8 7 7 7  13 9' 8 8 8 7 7 7 7  97  XVII QUEUEING ANALYSIS " M E A N CASE"  CHARTS  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 i n seconds. Floor 1: #Of Copiers 1  2 3 4  5 6 7 8 9 10  Mean Time In System  Mean Time In Queue  Mean Queue Length  Utilization  4642.38 1826.99 741.79 516.95 470.06 458.69 455.87 455.19 455.04 455.01  4187.38 1371.99 286.79 61.95 15.06 3.69 0.87 0.19 0.04 0.01  9.20 5.81 1.41 0.31 0.07 0.02 0.00 0.00 0.00 0.00  1.00 0.96 0.74 0.56 0.45 0.38 0.32 0.28 0.25 0.23  Mean Time In System  Mean Time In Queue  Mean Queue Length  Utilization  3038.22 1604.46 1058.36 732.94 543.75 453.86 416.97  2643.22 1209.46 663.36 337.94 148.75 58.86 21.97  6.69 6.12 4.97 3.17 1.54 0.63 0.24  1.00 1.00 0.99 0.93 0.82 0.70 0.60  Mean Time In System  Mean Time In Queue  Mean Queue Length  Utilization  289.75 258.18 257.04 257.00  32.75 1.18 0.04 0.00  0.02 0.00 0.00 0.00  0.14 0.07 0.05 0.04  Floor 2: #Of Copiers 1  2 3 4  5 6 7  Floor 4: #Of Copiers 1  2 3 4  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:  Servi evel  CO  u  100% 95% 90% 85% 80% 75% 70% 65% 60%  0  30  60  Waiting Time (in seconds) 90 120 150 180 210  11 6 5 5 5 4 4 4 4  10 6 5 5 5 4 4 4 4  10 6 5 5 4 4 4 4 4  10 6 5 5 4 4 4 4 4  9 5 5 5 4 4 4 4 4  9 5 5 4 4 4 4 4 4  9 5 5 4 4  9 5 5 4 4 4 4 \ 4 \ 4 -4 \ 4 4 \4  240  270  300  9 5 4 4 4 4 4 4 3  8 5 4 4 4 4 4 4 3  8 5 4 4 4 4 4 3 3  Example  Floor 2:  Servi evel  100% 95% 90% cu 85% o 80% _i 75% 70% 65% 60%  0  30  60  Waiting Time (in seconds) 90 120 150 180 210  240  270  300  15 9 8 8 7 7 7 6 6  14 9 8 7 7 7 . 7 6 6  14 8 8 7 7 7 6 6 6  13 8 7 7 7 6 6 6 6  12 8 7 7 6 6 6 6 6  12 7 7 6 6 6 6 6 5  11 7 7 6 6 6 6 6 5  11 7 7 6 6 6 6 5 5  11 7 6 6 6 6 6 5 5  0  30  60  Waiting Time (in seconds) 90 120 150 180 210  240  270  300  4 2 2  4 2 2  4 2 2  4 2 2  4 2 1  3 2 1  3 2 1  3 2 1  ]  }  ]  1  1  1  1  i  Floor 4:  i rvice  100% 95% 90% 85% o > 80% ID w 75% to 70% 65% 60%  13 8 7 7 7 6 6 6 6  12 7 7 7 6 6 6 6 6  3 2 1 1  3 2 1  3 2 1 1  \ 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 2 + exp(325) ' exp(327) 10 + exp(391) exp(401) 40 + exp(415) exp(455)  Service Time Distribution 49 + exp(208) exp(257) 40 + exp(415) exp(455)  Inter-Arrival Time Distribution 196 + exp(1570) exp(1760)  Inter-Arrival Mean . , Number of Time .. Users Distribution exp(96.7) exp(96.7) exp(69.3) exp(69.3) exp(83.1) exp(83.1)  299.26 298.96 416.79 414.14 344.71 343.09  Inter-Arrival Mean Number of Time ,. Users Distribution exp(1760) exp(1760) exp(202) exp(202)  16.4 16.27 142.27 142.27  Mean Service Time Number of Distribution .. Users exp(257) exp(257)  17.15 16.27  Mean ^ Queue . Length 0.72147 0.71985 1.163 1.1506 0.0616 .0.05145  Mean Queue . Length 0.01352 0.0129 0.000017535 0.000036178  Mean Queue . Length 0.01263 0.0129  . Mean Time Mean Time . ~ . . in Queue in System ' T  0  251.93 251.05 81.979 81.623 5.0661 4.294  581.75 579.63 477.5 479.91 456.72 451.62  .. Mean Time Mean Time . ~ . . in Queue in System 0  3  22.452 22.351 0.00353 0.00729  272.71 273.57 449.01 447.65  Mean Time Mean Time . ^ . . in Queue in System ' 0  21.723 22.351  287.58 273.57  100  STATUS Q U O CfflxJ  } Afra - F | 1 10 - Run Mode] 3 File £drt ifew loot* AtiangefibjectRun Wrd itow y«slp  o Common o Advanced Transfer o Advanced  Floor 1  PTOCPS-.  Basic Process Reports Ti Navg iate -: ^ Top-Uvel Model  ©  TT I Tq i  r ri ruestn  T Fl lues Derasli |2] T M Tues Overveiw (3)  I  ft  I  i-  1/1 O O 1462 8804 Usern i'terrupted (20051 .711)  For Hep l; press Fi ' Page 38 Sec 2 434 /5  0*  '4 Arena - F | 2 7 - Hun Model I &a £# M ew loots Arrange Qbeict Bun Vmdow Hep l •  6-'  *• M  & IA  O Common O Advanced Transfer  i  M •  rV?  Floor 2  ^ Top-level Model r Ft Tues 11) T Fl TuesDelals(2) If* Fl Tues Overveiw 3 |)  rot neip, pienri lobreo'tlslseteoed  LU  1/100 46436 .831 User n iterrupted (28171 ,569) My Compuetr !  526KB  M Arena - [F4 1 - Run Made) 3 Ffe Edi View loots Arrange Object Run Window Help » H » II H • i>?  Common Advanced Transfer o Advanced Process o Basic Process O Reports Is Navigats - • Topievel Model T Fl Tues(1) T f Tues Details (2) y F1 Tues Oveivrew (3] «>  <>  J  Fot Help, press Fl  1/100 2553.6122 Uninterrupted 5  2T  12773 13121  j j My Computet  RECOMMENDATION  \M Arena  - [Fl 7 - Run Mode] EH File Edit. View loots Arrange Object Bur. Window Help  BEsIBI  -|t?j x|  •  H >•  14 • T | ;?  Common o 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 Page 39 Sec 2  1/1IM 1495(1863 Usei interrupted 44/44  At 5.6"  Ln 6 Coll  (1128.37)  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.  KOERNER LIBRARY  September 2001 1200 •  October 2001 1200  November 2001  December 2001 1200  January 2002  o I*  300 -)-  103  March  2002  April  2002  1200  1200  900  600  300  B.  WOODWARD BIOMEDICAL LIBRARY  September 2001 500  October 2001 500  November 2001  December 2001 500  500  co  Day  Day  104  January  2002  February  2002  500  jfl re >  400 300  k_  < ». o  200 100 0  Day  March  2002  Day  Day  April  2002  Day  May 2002  C.  DAVID L A M LIBRARY  September 2001  October 2001  50 • 40 •  re > 30 ^ 20 -  o  10  0 Day  Day  105  March 2002  T  r  ^  o  c '  o -  c  T  D  -  T  O -  T  c -  April 2002  C  s  M  j  C  i  N  n J  c  C  o N  T  I  C  -  O  Day  T  -  ^  h  -  o  c T  -  o 1  c -  o T  c -  T  r -  j  C  c  S  I  N  C  i  M  n  C  a  N  o  J  Day  May 2002 50  w  40  TO  ._  >  >3U  -  o  20  *  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.  KOERNER LIBRARY  107  B.  WOODWARD BIOMEDICAL LIBRARY  108  C.  DAVID L A M LIBRARY  29 September 2001  26 September 2001  10  10  2o 4-  >6 ! 4 o  CM  Hour  CM  CM  CM  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.  NOTATION  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.  FORMULAS  \=—  P  vii.  XI  —  n\  m  PP-  X1 + X2  n\ + m  q =1- p P  P  ASSUMPTIONS  a. b. c. d.  data is collected from a random sample. the two proportions are independent. Tt) and 7t are not close to 0 or 1. np > 5 and nq > 5. 2  viii.  T E S T STATISTIC  ix.  HYPOTHESIS  H: H:  x.  X2  2=  P  0  7i] - 712  a  Ti] - 7 i  2  = 0 < 0 or 7 i i  - 7t  2  > 0  PASS/FAIL CRITERIA  The hypothesis is rejected when the p-value is of a small value. 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  = the probability of a successful charge transaction when machine i is used and the bar code is placed on area j . Ttjj  ii.  f0  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  MODEL  l ° g i t ( ^ ) = log  = B +/?,x, + B x 0  2  2  + /? x,x + e 3  2  •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.  Ill  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.  KOERNER LIBRARY September 26  o o  00  o o  Oi  o o © o o o o  CN  o o  CO  o o Tt T -  o o u-S T o o  CD  o o 1^ T —  o o  00  o o  ai  T-  o o o CM o o CM  o o csi CM o o  CO C M  0.80 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180  2001 2 1 1 2 2 1 2 2 2 3 3 3 2 2 2 2 2 2 3 2 2 2 2 2 2 * 2 2 2 2 2 4 3 3 2 2 1 2 2 1 2 1 1 2 1 1 1 1 1  September 29  March 20  March 23  May 14  May 17  2001 1 1 1 1 1 1  2002 2 1 1 2 2 2 2 2 2 3 3 2 3 3 3 3 3 3 3 3 2 3 3 3 3 3 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 1 1 1 1  2002 1 1 1 1 1 1  2002 1 1 1 2 1 1 2 2 1 2 2 1 2 2 1 2 2 2 2 2 2 2 2 1 2 2 2 2 2 1 2 2 1 2 1 1 2 1 1 2 2 1 2 1 1 1 1 1  2002 1 1 1 2 1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 3 2 2 2  1 1 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 1 1  1 1 1 1 1 1 1 1 1 •i  1 1 1 1 1 1 1 1  112  B.  o o  00 o o ai o o o o o *o o  CM o o  CO o o Tj"  o o  in  T-  o o  CO T—  o o  o o  CO o o  ay O T-  o o  CM o o  CN O O CN CN O o  CO CN  WOODWARD BIOMEDICAL LIBRARY  0.80 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180  September 26  September 29  March 20  March 23  2001  2001  2002  2002  14 2002  1  1  2 2 2 2  1  1 2 2 1 2 2 1  2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 1 1 2 2 1 2 1  1 2 1 1 2 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1 1 1  2 1 1 2 1 1 -'\ 2 2 1 2 2 1  2 2 1 2 2 2 2 1 1 2 2 1 2 2 1 2 2 2 2 2 1  2 1 1 2 2 1 1 1 1 2 1  1 1  1 1  1 1 1 1  17 2002  May  May  2  2  1  i  1  1 1  1  1  1  1 1  1  1  1  1  1 -  1  1 1  1  1 1  1 1  1 1  1 1  1 1  1 1  1  1  1 2 1 1  2 2 1 2 1 2 1 2 1 1  2 2 1 2 2 1 2 2 1 1 1 1 1 1 1  1 1 1 1 1 1 1 1 1  1 1 1  1  1  1 1  1 1 1 1 1  1  1 1 1  1  1  1  1 1 1 1 1 1 1 1 1  1 1 1  1  1  1  1 1 1  1  1  1 1  113  23:00 22:00 21:00  20:00  19:00  18:00  17:00  16:00  15:00  14:00  13:00  12:00  11:00 10:00  9:00  8:00  C.  DAVID L A M LIBRARY  0.80 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180 60 120 180  September 26  September 29  March 20  March 23  May 14  May 17  2001  2001  2002  2002  2002  2002  1  ]  3  3 3  ]  j j  3  1  ] ]  J  \  \  \ \ J 1 1  | | 3 3 j 3 3  \  \  \  j j J  3  ]  \  3  ]  3  3 3 j  3 j  3  3 3  3  3 j  ]  ]  3 j  j j  ]  ]  ]  3 j 3  j  j j  3 3 3  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.  o o CO  o o O)  o o o  o o V"  o o CM  o o  CO  o o  KOERNERLIBRARY  0.80 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7  September 26  September 29  March 20  March 23  May 14  May 17  2001 12.24 0.43 0.02 0.00 0.00 0.00 0.00 43.10 2.74 0.22 0.02 0.00 0.00 0.00 840.00 16.70 2.23 0.32 0.04 0.01 0.00  2001 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.44 0.04 0.00 0.00 0.00 0.00 0.00 149.71 8.76 1.02 0.12 0.01 0.00 0.00 61.69 4.12 0.38 0.03 0.00 0.00 0.00 237.93 11.38 1.41 0.18 0.02 0.00 0.00 30.19 1.73 0.11 0.01 0.00 0.00 0.00  2002 8.25 0.22 0.01 0.00 0.00 0.00 0.00 180.00 9.82 1.18 0.14 0.02 0.00 0.00  2002 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 5.06 0.09 0.00 0.00 0.00 0.00 0.00 28.34 1.58 0.10 0.01 0.00 0.00 0.00 660.00 15.96 2.11 0.30 0.04 0.00 0.00 174.78 9.65 1.15 0.14 0.01 0.00 0.00  2002 0.00 0.00 0.00 0.00 0.00 0.00 0.00 4.19 0.06 0.00 0.00 0.00 0.00 0.00 22.60 1.14 0.06 0.00 0.00 0.00 0.00 48.54 3.16 0.26 0.02 0.00 0.00 0.00 45.37 2.92 0.24 0.02 0.00 0.00 0.00 100.00 • 6.49 0.69 0.07 0.01 0.00 0.00 103.02 6.65 0.72 0.07 0.01 0.00 0.00  2002 0.00 0.00 0.00 0.00 0.00 0.00 0.00 13.10 0.49 0.02 0.00 0.00 0.00 0.00 41.17 2.59 0.20 0.01 0.00 0.00 0.00 98.24 6.40 0.68 0.07 0.01 0.00 0.00 51.92 3.41 0.29 0.02 0.00 0.00 0.00 73.33 4'.91 0.48 0.04 0.00 0.00 0.00 119.25 7.46 0.83 0.09 0.01 0.00 0.00  00  225.88 16.95 3.35 0.72 0.15 0.03 00  20.45 2.80 0.42 0.06 0.01 0.00 00  25.00 3.47 0.55 0.08 0.01 0.00 00  35.76 4.98 0.85 0.14 0.02 0.00  00  25.28 3.51 0.56 0.08 0.01 0.00 00  73.05 9.11 1.71 0.33 0.06 0.01 00  387.54 20.09 3.97 0.88 0.19 0.04 00  786.05 23.00 4.54 1.02 0.22 0.05 00  00  100.27 11.29 2.17 0.43 0.08 0.01  26.59 3.70 0.59 0.09 0.01 0.00  115  o o  in  September 26  September 29  March 20  March 23  May 14  May 17  0.80  2001  2002  1  00  2001 184.07 9.95 1.20 0.14 0.02 0.00 0.00 47.20 3.06 0.25 0.02 0.00 0.00 0.00 161.54 9.20 1.08 0.13 0.01 0.00 0.00 2.79 ' 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  2002 879.13 16.83 2.25 0.32 0.04 0.01 0.00  2002 51.05 3.35 0.29 0.02 0.00 0.00 0.00 740.00 16.33 2.17 0.31 0.04 0.00 0.00 48.54 3.16 0.26 0.02 0.00 0.00 0.00 43.35 2.76 0.22 0.02 0.00 0.00 0.00 5.06 0.09 0.00 0.00 0.00 0.00 0.00 10.47 0.33 0.01 0.00 0.00 0.00 0.00 37.30 2.29 0.17 0.01 0.00 0.00 0.00 0.34 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  2002 32.70 1.93 0.13 0.01 0.00 0.00 0.00  2 3 4  7  24.11 3.34 0.52 0.08 0.01 0.00 351.43 13.39 1.72 0.23 0.03 0.00 0.00 315.65 12.86 1.64 0.22 0.03 0.00 0.00  1  00  5  6 7  1  cb  2 3 4 5 6 7  o o t^!  2 3 4 5  o o  1  6  o o CO  2 3 4 5  6 7  1  o o  d>  2 3 4  5 6 7  1  o o © CN  2 . 3 4 5 6 7  1  o o  CM  2 3 4 5  6 7  1  O O  CM CM  2 3 4 5 6 7  1  O O CO CM  2 3 4 5 6 7  00  36.28 6.87 1.62 0.38 0.09 16.19 0.69 0.03 0.00 0.00 0.00 0.00 23.40 1.20 0.07 0.00 0.00 0.00 0.00 12.36 0.44 0.02 0.00 0.00 0.00 0.00 ' 9.68 0.29 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  00  1035.21 23.80 4.69 1.06 0.23 • 0.05 00  82.26 9.91 1.88 0.37 0.07 0.01 00  34.37 4.79 0.81 0.13 0.02 0.00 557.14 15.35 2.02 0.28 0.04 0.00 0.00 557.14 15.35 2.02 0.28 0.04 0.00 0.00 140.00 8.38 0.96 0.11 0.01 0.00 0.00 16.46 0.70 0.03 0.00 0.00 0.00 0.00 49.92 3.26 0.28 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  00  32.20 4.49 0.75 0.12 0.02 0.00 157.09 9.04 1.06 0.12 0.01 0.00 0.00 75.00 5.02 0.49 0.05 0.00 0.00 0.00 49.09 3.20 0.27 0.02 0.00 0.00 0.00 75.42 5.04 0.50 0.05 0.00 0.00 0.00 82.57 5.49 0.56 0.05 0.00 0.00 0.00 84.00 5.58 . 0.57 0.06 0.00 0.00 0.00 1.36 0.01 0.00 0.00 0.00 0.00 0.00  00  34.87 4.86 0.82 0.14 0.02 0.00 0.93 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  116  B.  o o CO  . o o O) •  o o o  o o T_  o o c\i  o o  CO  o 'o  o o u>  WOODWARD BIOMEDICAL LIBRARY  0.80 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7  September 26  September 29  March 20  March 23  May 14  May 17  2001 340.00 13.23 1.69 0.22 •0.03 0.00 0.00 9.23 0.27 0.01 0.00 0.00 0.00 0.00 23.48 1.21 0.07 0.00 0.00 0.00 0.00 31.14 1.80 0.12 0.01 0.00 0.00 0.00 41.77 2.64 0.21 0.01 0.00 0.00 0.00 49.51 3:23 0.27 0.02 0.00 0.00 0.00 51.20 3.36 0.29 0.02 0.00 0.00 0.00 38.97 2.42 0.18 0.01 0.00 0.00 0.00  2001 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 5.16 0.09 0.00 0.00 0.00 0.00 0.00 5.90 0.12 0.00 0.00 0.00 0.00 0.00 7.61 0.19 0.00 0.00 0.00 0.00 0.00 16.60 0.71 0.03 0.00 0.00 0.00 0.00 5.01 0.09 0.00 0.00 0.00 0.00 0.00 15.00 0.61 0.03 0.00 0.00 0.00 0.00  2002 12.18 0.43 0.02 0.00 0.00 0.00 0.00 9.23 0.27 0.01 0.00 0.00 0.00 0.00 32.01 1.87 0.13 0.01 0.00 0.00 0.00 15.99 0.67 0.03 0.00 0.00 0.00 0.00 48.27 3.14 0.26 0.02 0.00 0.00 0.00 161.54 9.20 1.08 0.13 0.01 0.00 0.00 14.23 0.56 0.02 0.00 0.00 0.00 0.00 18.90 0.87 0.04 0.00 0.00 0.00 0.00  2002 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 8.74 0.24 0.01 0.00 0.00 0.00 0.00 42.13 2.67 0.21 0.01 0.00 0.00 0.00 10.24 0.32 0.01 0.00 0.00 0.00 0.00 2.07 0.02 0.00 0.00 0.00 0.00 0.00 10.59 0.34 0.01 0.00 0.00 0.00 0.00 14.61 0.58 0.02 0.00 0.00 0.00 0.00  2002 1.54 0.01 0.00 0.00 0.00 0.00 0.00 12.91 0.47 0.02 0.00 0.00 0.00 0.00 5.31 0.10 0.00 0.00 0.00 0.00 0.00 12.73 0.46 0.02 0.00 0.00 0.00 0.00 3.58 0.05 0.00 0.00 0.00 0.00 0.00 12.18 0.43 0.02 0.00 0.00 0.00 0.00 7.92 0.20 0.01 0.00 0.00 0.00 0.00 6.21 0.13 0.00 0.00 0.00 0.00 0.00  2002 5.01 0.09 0.00 0.00 0.00 0.00 0.00 3.02 0.03 0.00 0.00 0.00 0.00 0.00 13.66 0.52 0.02 0.00 0.00 0.00 0.00 7.92 0.20 0.01 0.00 0.00 0.00 0.00 11.29 0.38 0.01 0.00 0.00 0.00 0.00 15.59 0.64 0.03 0.00 0.00 0.00 0.00 11.64 0.40 0.01 0.00 0.00 0.00 0.00 3.72 0.05 0.00 0.00 0.00 0.00 0.00  117  o o  CO T-  o o  1^  o o  00  o o 6)  o o © CN  o o CN  o o  CN  O  o  C O CN  0.80 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7  September 26  September 29  March 20  March 23  May 14  May 17  2001 24.71 1.30 0.08 0.00 0.00 0.00 0.00 17.21 0.75 0.03 0.00 0.00 0.00 0.00 13.47 0.51 0.02 0.00 0.00 0.00 0.00 18.47 0.84 0.04 0.00 0.00 0.00 0.00 7.92 0.20 0.01 0.00 0.00 0.00 0.00 3.72 0.05 0.00 0.00 0.00 0.00 0.00 1.54 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  2001 9.90 0.30 0.01 0.00 0.00 0.00 0.00 9.73 0.29 0.01 0.00 0.00 0.00 0.00 2.61 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  2002 30.57 1.76 0.12 0.01 0.00 0.00 0.00 56.13 3.72 0.33 0.03 0.00 0.00 0.00 29.44 1.67 0.11 0.01 0.00 0.00 0.00 11.11 0.37 0.01 0.00 0.00 0.00 0.00 25.71 1.38 0.08 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 13.10 0.49 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  2002 16.39 0.70 0.03 0.00 0.00 0.00 0.00 33.20 1.97 0.14 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  2002 11.46 0.39 0.01 0.00 0.00 0.00 0.00 9.90 0.30 0.01 0.00 0.00 0.00 0.00 11.29 0.38 0.01 0.00 0.00 0.00 0.00 1.54 0.01 0.00 0.00 0.00 0.00 0.00 1.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  2002 5.45 0.10 0.00 0.00 0.00 0.00 0.00 7.13 0.17 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  118  C.  o o 00  o o o>  o o o  o o T  ~  o o c\i  o o  CO  o o Tt T—  o o  IO  T —  DAVID L A M LIBRARY  0.80 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7  September 26  September 29  March 20  March 23  May 14  May 17  2001 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.28 0.01 0.00 0.00 0.00 0.00 0.00 1.80 0.01 0.00 0.00 0.00 0.00 0.00 1.54 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.02 0.00 0.00 0.00 0.00 0.00 0.00 0.34 0.00 0.00 0.00 0.00 0.00 0.00 1.02 0.00 0.00 0.00 0.00 0.00 0.00  2001 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.93 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  2002 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.34 0.00 0.00 0.00 0.00 0.00 0.00 0.67 0.00 0.00 0.00 0.00 0.00 0.00 0.67 0.00 0.00 0.00 0.00 0.00 0.00 1.02 0.00 0.00 0.00 0.00 0.00 • 0.00 0.76 0.00 0.00 0.00 0.00 0.00 0.00 4.29 0.07 0.00 0.00 0.00 0.00 0.00 1.36 0.01 0.00 0.00 0.00 0.00 0.00  2002 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.97 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  2002 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.71 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.36 0.01 0.00 0.00 0.00 0.00 0.00 2.07 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.67 0.00 0.00 0.00 0.00 0.00 0.00  2002 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.67 0.00 0.00 0.00 0.00 0.00 0.00  119  September 26  o o CD  o o  1^  o o oo  o o  cn  o o o CM  o o  CM  o o C CM M  O O CO CM  0.80 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7  2001 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00. 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  September 29 2001 0.67 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  March 20 2002 5.06 0.09 0.00 0.00 • 0.00 0.00 0.00 0.34 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.34 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.67 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  March 23  May 14  May 17  2002 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.54 0.01 0.00 0.00 0.00 0.00 0.00 0.34 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  2002 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.34 0.00 0.00 0.00 0.00 0.00 0.00 0.34 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.34 0.00 0.00 0.00 0.00 0.00 0.00 0.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  2002 2.07 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 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 timeperiods. A.  KOERNER LIBRARY  Month Low Low Low Low Low Low Moderate Moderate Moderate Moderate Moderate Moderate Peak Peak Peak Peak Peak Peak  B.  Day Low Low Low Peak Peak Peak Low Low Low Peak Peak Peak Low Low Low Peak Peak Peak  Hour Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak  60 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3  Waiting Time 120 1 1 2 1 2 2 1 2 2 2 2 2 2 2 2 2 2 3  180 1 1 1 1 1 2 1 1 1 1 2 2 1 1 2 1 2 2  WOODWARD BIOMEDICAL LIBRARY  Month Low Low Low Low Low Low Moderate Moderate Moderate Moderate Moderate Moderate Peak Peak Peak Peak Peak Peak  Day Low Low Low Peak Peak Peak Low Low Low Peak Peak Peak Low Low Low Peak Peak Peak  Hour Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak Low Moderate Peak  60 1 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2  Waiting Time 120 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2  180 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  121  C.  DAVID L A M LIBRARY Waiting Time Month  Day  Hour  60  Low  Low  Low  2  120 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  Moderate Peak  2  1  1  Moderate  Low Low  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  180 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. KOERNER LIBRARY  A.  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  7.61  0.85  0.09  Moderate  Peak  Peak  122.28 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  17.94  2.41  0.35  Peak  Peak  Peak  1422.82 CO  92.77  10.75  2.06  122  B.  W O O D W A R D BIOMEDICAL LIBRARY  Month  Day  Hour  1  Low  Low  Low  Low  Low  Moderate  Low  Low  Peak  Low  Peak  Low  Low  Peak  Moderate Peak  0.00 1.07 4.34 1.77 7.14 12.37 0.00 3.29 9.15 3.20 14.12 26.93 0.00 5.75 20.38 5.34 24.05 45.06  Low  Peak  Moderate  Low  Low  Moderate  Low  Moderate  Moderate  Low  Peak  Moderate  Peak  Low  Moderate  Peak  Moderate  Moderate  Peak  Peak  Peak  Low  Low  Peak  Low  Moderate  Peak  Low  Peak  Peak  Peak  Low  Peak  Peak  Moderate  Peak  Peak  Peak  C.  Number Of Servers At Circulation Desk 2 3 0.00 0.00 0.00 0.00 0.07 0.00 0.01 0.00 0.17 0.00 0.44 0.02 0.00 0.00 0.04 0.00 0.26 0.01 0.04 0.00 0.55 0.02 1.47 0.09 0.00 0.00 0.11 0.00 0.98 0.05 0.10 0.00 1.25 0.07 2.89 0.23  4 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02  Number Of Servers At Circulation Desk 2 3 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 o.oo • 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.01 0.00  4 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  D A V I D L A M LIBRARY  Month  Day  Hour  Low  Low  Low  Low  Low  Moderate  Low  Low  Peak  Low  Peak  Low  Low  Peak  Moderate Peak  Low  Peak  Moderate  Low  Low  Moderate  Low  Moderate  Moderate  Low  Peak  Moderate  Peak  Low  Moderate  Peak  Moderate  Moderate  Peak  Peak  Peak  Low  Low  Peak  Low  Moderate  Peak  Low  Peak  Peak  Peak  Low  Peak  Peak  Moderate  Peak  Peak  Peak  1 0.01 0.07 0.57 0.16 0.48 0.69 0.02 0.35 0.85 0.27 0.93 1.05 0.03 0.63 1.04 0.33 1.13 1.50  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.  K O E R N E R LIBRARY  124  B.  W O O D W A R D BIOMEDICAL L I B R A R Y  September 26 2001  September 29 2001  (A <D  t  <•> „ 2 -  «  o  >_ a> a>  z<  E  Time  Time  C.  DAVID L A M LIBRARY  September 26 2001  September 29 2001 at  t  a  O i-  CO  0)  z  E  O  i-  CM  CO  CO  Time  OJ  o CM  *— CM  CM  CM  Time  May 14 2002  May 17 2002 CO  > ' -a  2-  «  T,  2  05 u O co 1— co CO Z 1  E  co  cn  o  i—  CM  co  CM  Time  CM  CM  CM  CO  O)  O  f-  Time  126  

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