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A model to determine service facility requirements Redden, Charles Robert 1975

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A MODEL TO DETERMINE SERVICE FACILITY REQUIREMENTS by CHARLES ROBERT REDDEN B.S. Univers i ty of B r i t i s h Columbia, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IH 3UISS: ;]S3 ADMINISTRATION in the Faculty of Commerce We accept th i s thesis as conforming to the require^! standard UNIVERSlfY OF BRITISH COLUMBIA January, 1975 In presenting t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission for extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of M A ^ RT <^j..*sa.^' S> y\ ftt, The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date \ ( U v ^ I 1 i ABSTRACT Credit unions have d i f f i c u l t y in estimating the f a c i l i t y requirements which w i l l enable them to provide adequate member serv ice . Due to the recent growth in membership, most c red i t unions have had to enlarge t h e i r f a c i l i t y to one which would adequately accommodate the present membership as well as provide for expected future growth in operations. The high cost of expansion has made i t necessary to accurately determine t h e i r f a c i l -i t y requirements. B. C. Central Credit Union, a service centre which pro-vides professional assistance to credi t unions and co-operatives through-out B r i t i s h Columbia, was concerned with the so lut ion of t h i s problem and f i r s t looked to how other organizations t r i e d to solve t h e i r f a c i l i t y plan-ning problems. A number of organizations have developed, or attempted to develop, f a c i l i t y planning models. Some have resulted in complete f a i l u r e and have been ab-andoned. Others could not answer enough of the questions that c red i t union managers needed to know about their f a c i l i t y requirements. The management of B. C. Central Credit Union decided to acquire the services of the author f o r the purpose of designing and implementing a f a c i l i t y planning model. After a prel iminary invest igat ion of the problem and discussions with c red i t union managers, i t was decided that a simulation model would be the most appropriate management tool to use. i i The scope of the project was to develop and implement a simulation model to accurately determine present and future t e l l e r f a c i l i t y requirements (wick-ets and queuing area) which w i l l enable a c red i t union to provide adequate member service. The t e l l e r f a c i l i t y i s simulated under varying conditions to determine the required number of wickets and queuing area for a given c red i t union. It i s shown that the model is sens i t ive to the approximation of the t e l l e r service time d i s t r i bu t i on and the method of data co l l ec t i on on member ar -r i v a l s . A c red i t union's t e l l e r f a c i l i t y requirements as well as the level of service are shown to be very dependent on the operating pol icy to regulate the number of wickets which are ava i lab le . At present, two c red i t unions have benefited considerably from the simula-t ion model. In both cases, the management of the c red i t union had decided to bui ld a new enlarged f a c i l i t y because the ex i s t ing c red i t union could not adequately accommodate i t s members. The simulation model showed that only a change in the f a c i l i t y layout was required. The c red i t unions re -versed the i r decision to bui ld a new f a c i l i t y and simply changed the lay -out. Both are presently operating e f f e c t i ve l y with the new layout and have avoided the expense of a new bui ld ing. Table of Contents i i i Page 1. BACKGROUND 1 2. SCOPE 1 3. DATA COLLECTION 1 3.1 Queuing D i s c i p l i n e 3 3.2 Serv i c e Process 3 • 3.3 A r r i v a l Pattern 5 4. METHOD 1 4.1 Basic S t r u c t u r e 7 4.2 Model Input and Output 8 4.3 Method of P r o j e c t i o n 9 5. SERVICE POLICY 10 5.1 Introduction 10 5.2 Development 11 5.3 Formulation 11 6. COMPUTER PROGRAM 1.3 7. STEADY STATE AND SENSITIVITY ANALYSIS 13 7.1 Introdu c t i o n 13 7.2 Steady State A n a l y s i s 14 7.3 S e n s i t i v i t y A n a l y s i s 19 i v Table of Contents Page 8. VERIFICATION 27 9. CONCLUSIONS 28 10. RECOMMENDATIONS 29 10.1 Implementation 29 10.2 Extensions 29 11. SUMMARY 31 12. REFERENCES 32 13. APPENDIX 1 - A Sample Computer Printout 34 14. APPENDIX 2 - Model Documentation 50 15. APPENDIX 3 - Other F a c i l i t y Planning Models 59 16. APPENDIX 4 - A Case Study 62 1 1. BACKGROUND Due to recent growth in membership most c red i t unions have had a s i g -n i f i c a n t increase in the demand made on t h e i r f a c i l i t y . This increase in the demand has forced many credi t unions to enlarge t h e i r f a c i l i t y . The high cost of expansion has made i t necessary to accurately deter-mine the f a c i l i t y requirements and the continuing growth rate has made i t necessary to project the future requirements for the d i f f e r e n t op-erations in order to assure that the f a c i l i t y w i l l be adequate for at least 3 - 5 years. 2. SCOPE The purpose of th i s paper i s to develop a simulation model to accur-ately determine present and future t e l l e r f a c i l i t y requirements (wick-ets and queuing area) which w i l l enable the c red i t union to provide adequate member serv ice . 3. DATA COLLECTION Data from 3 major c redi t unions* was col lec ted so that the queuing d i s c i p l i n e , service process and a r r i v a l pattern could be adequately described within the model. * Richmond Savings Credit Union, Prince George & D i s t r i c t Credit Union, and Campbell River D i s t r i c t Credit Union. DATA COLLECTION (continued) The environment of the credi t union which the simulation model is con-cerned with i s described in Figure 1. Open Served Members Open Open Open Closed Closed Members Enter Served Members Figure 1 As shown in f igure 1, members enter the system and j o i n one of several queues. Service is performed in a FIFO, d i s c i p l i n e and then the member leaves the system. The number of avai lable wickets depends on the present demand being made on the t e l l e r f a c i l i t y . DATA COLLECTION (continued) 3.1 The queuing d i s c i p l i n e In the three c red i t unions* the mult iple queuing d i s c i p l i n e was used. However, from observing the real l i f e s i t u a t i o n , the FIFO operating charac ter i s t i c is ' e f f e c t i v e l y maintained with respect to the to ta l number of members waiting for serv ice , since members tend to jockey for pos i t ion between the several queues. Since the capacity of the queuing area i s determined by the number of members waiting for ser-v i c e , the FIFO queuing d i s c i p l i n e i s used in the simulation model. 3.2 The service process In those c r e d i t unions, t e l l e r s perform only simple services (e .g. cash withdrawals, deposi ts ) . When a member requires other types of service (e .g . general information about his account, t r a v e l l e r ' s cheques) he i s serviced by other personnel. The service time i s the time required by a t e l l e r to serve an i n d i v -idual member. Since the service-times vary s t o c h a s t i c a l l y , i t i s •Richmond Savings Credit Union, Prince George & D i s t r i c t Credit Union and Campbell River D i s t r i c t Credit Union. 4 3.2 The service process (continued) necessary to describe them by a probabil ty d i s t r i b u t i o n . Before es-timating th i s d i s t r i b u t i o n , i t was necessary to make two assumptions. The f i r s t assumption i s that a l l t e l l e r s have the same service rate d i s t r i b u t i o n with equal means. The second assumption i s that the d i s t r i b u t i o n of service rates and the mean is not s i g n i f i c a n t l y d i f -ferent within or between days. The data was col lec ted during a continuous period of the operating day when the system was moderately busy ( a l l wickets open and 1 - 3 members wait ing at each wicket ) . The main reason for c o l l e c t i n g data during th i s period was to insure that an accurate estimate of the set-up time was included ( i . e . that interval when the t e l l e r has f in i shed servic ing the member but not his t ransact ion) . An extensive amount of data was col lected from Richmond Savings Credi t Union. I t was thought that i f a large amount of data was c o l l e c t e d , not only would the d i s t r i b u t i o n function be accurately described, but tes ts . could be applied to determine whether a theoret ical d i s t r i b u t i o n could provide a s t a t i s t i c a l l y "good f i t " to the empirical d i s t r i b u t i o n . It was hypothesized that the d i s t r i b u t i o n of service times could be ap-proximated by a theoret ical d i s t r i b u t i o n (e .g . exponential d i s t r i b u - ' t i o n ) . A s t a t i s t i c a l test (1) of a sample of 375 service times was performed to determine the "goodness of f i t " of the test data to 7 5 3.2 The service process (continued) theoret ical d i s t r i b u t i o n s . (See Table 1.) Table 1 GOODNESS OF FIT TESTS FOR SERVICE RATES DISTRIBUTION X 2 X Prob Normal 164.0 0.00 Poisson Very large Very small Binomial Very large Very small Negative Binomial Very large Very small Gamma 41.0 0.00 Lognormal 15.5 0.21 Exponential 122.0 0.00 Table 1 shows that the service rates cannot be approximated by any of the above common theoret ica l d i s t r i b u t i o n s . For this reason, the ser-vice process is described in the simulation model by an estimate of the empirical d i s t r i b u t i o n . 3.3 The a r r i v a l pattern From d i rec t observation, i t was apparent that the member a r r i v a l rate varied within and between days. I t i s d i f f i c u l t to accurately des-6 3.3 The a r r i v a l pattern (continued) cr ibe the a r r i v a l pattern for any given day because the member a r r i v a l rate during the day i s dependent on many factors not contro l led by the c red i t union. For th is reason, in order to simulate the a r r i v a l pat-tern for a given day and to simulate the expected v a r i a t i o n i n the a r r i v a l rate for s i m i l a r such operating days the fol lowing data c o l l e c -t ion strategy was used. Data was col lected on member a r r i v a l s by recording the number of mem-bers that enter the c r e d i t union to use the t e l l e r f a c i l i t y during each 15 minute period of operating day. The time dependent a r r i v a l s input to the simulation would then be th i s recorded frequency d i s t r i b -ution which would represent the expected number of a r r i v a l s for the cor-responding time per iod. Since the c r e d i t union has no d i r e c t control over the i n t e r a r r i v a l rates wi th in such a short in terval they were as-sumed to be completely random. More formal ly , i t i s assumed that the time of the next a r r i v a l i s independent of the previous a r r i v a l , and the p r o b a b i l i t y of a r r i v a l i n an in terva l AV i s proportional to * Thus, to a given interval the i n t e r - a r r i v a l rates were assumed to be exponentially d i s t r i b u t e d about the recorded mean. 7 4. METHOD 4.1 Basic Structure A computer model, programmed in the simulation language GPSSV, was de-veloped to simulate the behavior of-the above system (for a complete documentation, see Appendix 2) . The queuing system which, the model had to describe i s seen below. Figure 1 - Queuing System Members > Enter Served Members M M M M M Served Members M • Open T e l l e r f a c i l i t y M Open (Number of ava i l -M Open ble wickets de-M Open pends on queue Closed length) . Closed As shown in Figure 1, members enter the system and j o i n a s ingle queue. Service i s performed in a FIFO d i s c i p l i n e and then the mem-ber leaves the system. The number of avai lable wickets depends on the present demand being made on the t e l l e r f a c i l i t y . In the above 8 4.1 Basic Structure (Continued) system, the maximum number of wickets which could be made avai lab le i s 6. 4.2 Model Input and Output 4.2.1 Model Input To determine the f a c i l i t y requirements for a given c r e d i t union, the fol lowing information is input to the simulation model: ( i ) The level of service the manager wishes to provide his members (which i s defined in terms of the d i s t r i b u t i o n of wait ing time and the d i s t r i b u t i o n of queue length) , ( i i ) The d i s t r i b u t i o n of t e l l e r service times and the member a r r i v a l d i s t r i b u t i o n from a maximum (peak) load condi t ion, ( i i i ) Any physical contraints (e .g . area or the maximum number of wickets ) . 4.2.2 Model Output The f a c i l i t y requirements can then be determined by evaluating the fol lowing output from the s imulat ion : ( i ) Frequency d i s t r i b u t i o n of the member wait ing time, ( i i ) Frequency d i s t r i b u t i o n of the number of members wait ing in the system. ( i i i ) Frequency d i s t r i b u t i o n of the queue length per avai lable wicket . 9 4.2.2 Model Output (Continued) ( i v ) Frequency d i s t r i b u t i o n of the number of wickets u t i l i z e d , (v) Histogram of the average number of t e l l e r s u t i l i z e d as a f u n c t i o n of the time of day. ( v i ) Histogram of the average number of members wai t i n g per a v a i l a b l e wicket as a fu n c t i o n of the time of day. ( v i i ) Histogram of the average number of member a r r i v a l s as a f u n c t i o n of the time of day. 4.3 Method of P r o j e c t i o n I t i s important to estimate the t e l l e r f a c i l i t y requirements f o r a s p e c i f i e d l e v e l of growth, so that the c r e d i t union can be prepared f o r the expected increase in demand. This model does not f o r e c a s t when a c e r t a i n l e v e l of growth w i l l occur, but ra t h e r i t shows the e f f e c t the increased demand w i l l have on the futu r e t e l l e r f a c i l i t y requi rements. Since there i s no information a v a i l a b l e on the previous a r r i v a l rates the f u t u r e peak loan conditions are approximated by i n c r e a s i n g the mean a r r i v a l rate f o r each i n t e r v a l . For example, in order to estim-ate the f a c i l i t y requirements f o r a 40% increase i n the demand, the mean a r r i v a l rate f o r each 15 minute i n t e r v a l would be increased by 40%. Since the a r r i v a l rates are assumed to be e x p o n e n t i a l l y d i s -t r i b u t e d , i n c r e a s i n g the mean a r r i v a l r a t e f o r each i n t e r v a l w i l l 4.3 Method of P r o j e c t i o n (Continued) 10 a u t o m a t i c a l l y increase the corresponding variance of each i n t e r v a l by the appropriate amount. 5. SERVICE POLICY 5.1 Introduction For a given queuing d i s c i p l i n e , s e r v i c e process and a r r i v a l p a t t e r n , the two major f a c t o r s which a f f e c t the member s e r v i c e and t e l l e r u t i l -i z a t i o n are: ( i ) The operating p o l i c y to regulate the number of wickets which are a v a i l a b l e f o r s e r v i c e ( i . e . opening/closing w i c k e t s ) . This p o l i c y would be used to maintain an adequate balance between the l e v e l of s e r v i c e to the members and the t e l l e r u t i l i z a t i o n ( since when a t e l l e r ' s wicket i s c l o s e d she i s f r e e to perform other d u t i e s ) , ( i i ) The p h y s i c a l l i m i t a t i o n s of the c r e d i t union i n terms of the num-ber of wickets and queuing area a v a i l a b l e . Since the main purpose of t h i s model i s to define the f a c i l i t y r e -quirements f o r a c r e d i t union t h a t intends to change i t s p h y s i c a l l i m -i t a t i o n ( i . e . expand), the e x i s t i n g p h y s i c a l c o n s t r a i n t s are not nor-mally part of the model input. The s e r v i c e p o l i c y i s then one of the major concerns which must be well defined and input to.the model be-fore the system behavior can be a c c u r a t e l y described. 11 5.2 Pol icy Development The operating pol icy of the model should e i ther r e f l e c t e x i s t i n g op-erat ing pol icy by management, or improve on t h e i r p o l i c y , yet be sim-ple enough for management to implement into t h e i r operations. In general , the desired balance between t e l l e r u t i l i z a t i o n and leve l of service to the member d i f f e r s between credi t managers. For this reason, the pol icy u t i l i z e d by the simulation must be f l e x i b l e enough to adapt to the d i f ferent credi t union manager's requirements, (e .g . one manager may tolerate only very small queues, while another may not concern himself with the s ize of the queue). The pol icy could e i ther be formulated in terms of wait ing time or queue length. Since the pol i cy should be easy for management to im-plement into t h e i r own credi t union, a po l i cy formulated in terms of queue length i s more p r a c t i c a l . 5.3 Pol icy Formulation Let the fol lowing notation be used to formulate the service p o l i c y : Parameters range of acceptable queue length/avai lable wicket minimum, maximum number of wickets which can be made a v a i l a b l e . (1) ( Y r Y 2 ) = (2) ( S r S 2 ) = 12 5.3 Pol icy Formulation (Continued) Variables (1) y = queue length/avai lable wicket (2) s = number of avai lable wickets Pol icy to increase/decrease the number of wickets . Wait U n t i l y < Y, or s = s - 1 s = s + 1 13 COMPUTER PROGRAM The simulation was formulated i n the programming language GPSSV. This language is espec ia l ly sui table for handling queuing problems and appeared most sui table for the problem in question. (For a complete documentation, see Appendix 2.) Model S t a t i s t i c s Statements - 680 l ines Core required - 15000 bytes GPSS Enti ty option - B Compile time. - .09 minutes CPU time - 6.37 minutes on IBM 360/67 Page printed - 40 STEADY STATE AND SENSITIVITY ANALYSIS 7.1 Introduction In order to test for the s e n s i t i v i t y to parameter changes and deter-mine when steady state conditions occur i t was necessary to input some test data. Since a large volume of data had been col lec ted on the a r r i v a l rate d i s t r i b u t i o n , i t was decided to select the test data which represented one f u l l operating day and contained a high degree 14 7.1 Introduction (Continued) of var ia t ion between adjacent time periods. The a r r i v a l rate d i s t r i b -ution was then transformed to represent the a r r i v a l rate from a large credi t union with a high degree of v a r i a b i l i t y in the member a r r i v a l ra te , This was accomplished by increasing the mean a r r i v a l rate for each interval by 50%. Thus the data on a r r i v a l rates would represent a large credi t union (where in p a r t i c u l a r the f a c i l i t y requirements are more d i f f i c u l t to determine), with a high degree of v a r i a t i o n in the a r r i v a l rate . I t was assumed that i f the model performed well under these conditions then i t would perform at least as well under more normal circumstances 7.2 Steady State Analysis 7.2.1 General Before steady state conditions could be determined, i t was necessary to gather information on the s t a t i s t i c s of in teres t . The fo l lowing sequence describes the method: 1) simulate the data for one complete cycle 2) record the values of the necessary s t a t i s t i c s 3a ) i f the predetermined number of cycles have been run - stop 3b)otherwise destroy a l l cumulative s t a t i s t i c s and return to step 1 with a d i f ferent point in the random number* stream. * The GPSSV compiler uses a m u l t i p l i c a t i v e congruential method for gener-a t i n g random numbers with a period equal to 2 - 1. 15 7.2.1 General (Continued) Since the simulation used a pol icy decision which attempted to main-ta in an adequate balance between t e l l e r u t i l i z a t i o n and level of ser-vice to the members i t was evident that the d i s t r i b u t i o n of wait ing time and queue length would be s i g n i f i c a n t l y skewed in an upward d i -rec t ion . However, i t was hypothesized that d i s t r i b u t i o n of the s ta -t i s t i c s of interest (average wait ing time, average queue length per avai lable wicket) could be c losely approximated by a normal d i s t r i b -ut ion . A s t a t i s t i c a l analysis (1) of a sample of 100 of each of the above s t a t i s t i c s was performed to determine the "goodness of f i t " of the test data to hypothetical d i s t r i b u t i o n s . (See Table 2.) Table 2 Goodness of F i t Tests for Sample S t a t i s t i c s STATISTIC DISTRIBUTION AVERAGE WAITING TIME AVERAGE QUEUE LENGTH MAXIMUM QUEUE LENGTH X 2 Prob X 2 Prob X 2 Prob Normal 3.76 0.81 3.98 0.86 4.87 0.77 Poisson 34.83 0.00 13.44 0.10 43.34 0.00 Binomial 87.02 0.00 62.34 0.00 89.29 0.00 Negative Binomial 5.81 0.67 1.36 0.99 10.24 0.18 Gamma 7.76 0.35 1.71 0.97 4.68 0.70 Lognormal 6.49 0.48 1.34 0.99 5.37 0.61 Exponential Very Very Very Very Very Very Large Smal 1 Large Smal 1 Large Smal 1 16 7.2.1 General (Continued) As shown in Table 2 , the three d i s t r i b u t i o n s of the sample s t a t i s t i c s provided a s t a t i s t i c a l l y good f i t to the normal d i s t r i b u t i o n . There-f o r e , s t a t i s t i c a l tes t ing on the three values involv ing normality as-sumptions could be used. 7.2.2 Run Length to Determine the Number of Wickets Required For the simulation to aid in def ining the t e l l e r f a c i l i t y requirements for a given p o l i c y , i t is essent ia l to have r e l i a b l e information on the area requirements as well as the level of service . Since the area requirements and the level of service are both dependent on the maxi-mum number of avai lable wickets S2, the area requirements and level of service for each case can be ob jec t ive ly compared. One of the major problems of simulation experiments i s determining the length of the simulation run. In th is simulation model, the computing cost of simulating a run of length one (equivalent to one operating day) i s $1.50. Therefore, i f th is model i s to be of prac t i ca l use, the run length must be kept to a minimum. I t i s of in teres t to observe the estimates of the queuing are required for d i f f e r e n t S2 as the length of the simulation increases. This w i l l serve to answer two major questions: 1) What i s the run length required to determine the number of wickets which w i l l minimize the area required? 2) What i s the run length required to determine the minimum area re-qui red? 17 7.2.2 Run Length to Determine the Number of Wickets Required (Cont.) Figure 2 shows the re la t ionship between the run length and the estim-ation of the area required for the t e l l e r f a c i l i t y as a function of the maximum number of avai lable wickets . The number of wickets de-termines the width of the t e l l e r f a c i l i t y and the maximum queue length per avai lable wicket determines the depth. Addit ional wickets reduce the maximum queue length, and thus the required depth of the t e l l e r f a c i l i t y . However, each addit ional wicket also increases the required width by a f ixed amount*. Therefore, i t i s possible to minimize the tota l area required for the t e l l e r f a c i l i t y . Figure 2 The Relationship Between the Area Required and Run Length ( s q . f t . 1100 1050 1000 950 900 850 . 800 750 . 700 \ \ 5 wk. : ] 7 wk. -] 6 wk. 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 (Run Length) * The dimensions of a t e l l e r ' s wicket are bh f t . x 7h f t . 18 7.2.2 Run Length to Determine the Number of Wickets Required (Cont.) I t would appear from Figure 2, that a minimum run length of 20 days sim-l a t i o n is required i n order to determine the number of wickets which w i l l minimize the area required and run length of 30 is required to de-termine the minimum area required. 7.2.3 Run Length Required to Estimate the Level of Service I t was of interes t to determine the e f fect of run length on the estimate of the level of service . A confidence in terva l for the mean wait ing time in the queue was determined for the d i f f e r e n t S2. Since the norm-a l i t y assumptions were s a t i s f i e d , a confidence in terva l for the d i f f e r -ent means could be instructed (See f i g u r e . 3 ) , from the assumption that : Prob C p i - ^ . C S , - « 5?; r ' j J i t ^ . O * >-«-where 5? • = > ->r • ^ > = f- 21> - sample mean p ; & pi 2C> - population mean population variance 51 •*• I Git,- « _Gj_ = sample variance of the mean V i = n > - l - degrees of freedom A 95% confidence in terva l was constructed for the average wait ing time as a function of the run length of the s imulat ion. Figure 3 shows con-fidence intervals when S2 = 5 , 6, and 7 wickets . 19 7.2.2 Run Length required to Estimate the Level of Service (Cont.) I t would appear from Figure 3 , that a simulation of run length equal to 20 is s u f f i c i e n t for estimating the level of serv ice . 7.3 S e n s i t i v i t y Analysis 7.3.1 General It was essential to measure the e f fect of parameter changes to the ser-vice p o l i c y , the method of estimating the d i s t r i b u t i o n of service times 20 7.3.1 General (Continued) and the method of estimating the a r r i v a l rates on the system behavior. Af ter the effects were evaluated for a f u l l range of reasonable condi-t i o n s , the decision maker could then select that set of parameter v a l -ues which best suited his d e f i n i t i o n of an adequate level of s e r v i c e , as well as determining the method of estimating the above d i s t r i b u t i o n The evaluations were performed with the ident i ca l random number stream in order to achieve the above. 7.3.2 Parameter Changes to the Service Pol icy ( i ) S = The number of avai lable wickets . There i s some argument as to the minimum number of wickets (S I ) , which should be avai lable at a l l times. C l e a r l y , there must be at least one wicket open but apparently not more than 2 wickets . Consequently, the values of SI = 1 and SI = 2 were included in the ana lys i s . The maxi-mum number of avai lable wickets (S2) i s a physical constraint rather than a parameter for the service p o l i c y . From other analysis (See Appendix 1) , the value of S2 = 7 i s ce r ta in ly the maximum number of wickets that th is p a r t i c u l a r c redi t union requires . Table 3 shows that the d i s t r i b u t i o n of wait ing is generally insens i t ive to the above values of SI . I f the upper portion of the d i s t r i b u t i o n (the 95 - 99 percenti les) are compared for any f ixed Y2, then the differences are very s l i g h t . Table 4 shows that the d i s t r i b u t i o n of queue length cor-responding to the 99 percenti le is not sensi t ive to the value of SI . 21 7 . 3 -2 Parameter Changes to the Service Pol icy (Continued) Table 3 The re la t ionship between the d i s t r i b u t i o n of wait ing time in the queue and decision p o l i c y . X = Waiting SI = Minimum Number of Wickets Time In 1 2 Queue Y2 = Pol icy to Increase Tel lers Y2 = Pol icy to Increase Te l lers (Minutes) 2 3 4 5 2 3 4 5 $ 2 44.0 23.5 16.9 13.0 46.7 28.0 21.1 18.3 « 4 87.0 59.5 39.2 29.2 86.9 61.3 41.2 34.0 $ 6 88.9 69.8 50.9 > $ 5 . 7 " 89.5 69.9 54.2 $ 8 ^^98.2^ -"96.2^ 90.5 76.4 -"98.0 ,96AS 90.5 77.1 $10 . 98 .5^ ^96 .5^ 92.7 ^99.3 . . 9 8 . 4 ^ ^96. 92.6 a z 99.9 99.5 . 9 8 . 6 ^ "W.lV 99.8 ^99.5 ^ -98.6 ' $14 100 99.9 99.6 ^98.8' 100 99.9 99.6 -99.1 / $16 100 100 99.9 99.7 100 100 99.9 99.8 $18 100 100 100 100 100 100 100 100 $20 100 100 100 100 100 100 100 100 A l l values inside the array are cumulative percentages for the cor-responding column. Table 3 shows the re la t ionship between the d i s t r i b u t i o n of wait ing time in the queue and the decision p o l i c y . The shaded area represents that region of major concern to credi t union management. That i s , t h e i r i n -terpretat ion of the maximum wait ing time in the queue. (95 - 99 per-cent i le ) . 22 7.3.2 Parameter changes to the Service Pol icy (Continued) Table 4 The re la t ionship between the d i s t r i b u t i o n of queue length per a v a i l -able wicket and decision p o l i c y . Y = Queue SI =• Minimum Number of Te l lers Length 1 2 Per Wicket Y2 = Pol icy to Increase Tel lers Y2 = Pol icy to Increase Te l lers (Members) 2 3 4 5 2 3 4 5 * 1 4.6 3.3 2.6 2.2 9.6 7.3 6.0 5.3 « 2 38.9 21.7 17.0 13.7 44.5 28.8 22.2 19.6 4 3 85.5 55.0 37.2 29.1 86.7 58.9 42.0 36.7 44 96.0 89.4 64.7 48.5 96.3 90.4 66.9 53.7 45 98.7 86.7 91.1 72.7 98.9 9.7.1 92.0 74.5 46 99.8 99.1 97.4 92.9 99.8 99.2 97.7 93.7 * 7 99.9 99.8 99.4 98.2 100 99.9 99.5 98.8 * 8 99.9 99.9 99.9 99.7 100 100 99.9 99.8 4 9 100 100 99.9 99.9 100 100 100 100 & 10 100 100 100 100 100 100 100 100 411 100 100 100 100 100 100 100 100 412 100 100 100 100 100 100 100 100 A l l values inside the array are cumulative percentages for the corres-ponding column. Table 4 above shows the re la t ionship between the d i s t r i b u t i o n of queue length per avai lable wicket and decision p o l i c y . The underlined values represent that region of major concern to c redi t union management ( i . e . 23 7.3.2 Parameter Changes to the Service Po l i cy (Continued) the 99 percent i l e ) . When analysing the d i s t r i b u t i o n of queue length, the s e n s i t i v i t y anal -y s i s of the decision pol icy by varying the pol icy parameters serves a d i f f e r e n t purpose from the analysis of the d i s t r i b u t i o n of wait ing time. Whereas the d i s t r i b u t i o n of wait ing time is considered a measure of the level of service given to members, the d i s t r i b u t i o n of queue length i s also a measure of the system requirements in terms of the queuing area required. The queuing area required i s generally defined to be that area which w i l l adequately accommodate 99% of the members. Since the multi-queue d i s c i p l i n e i s practised and in general the queuing area i s rectangular in shape, a system designed to accommodate a maximum queue/available wicket of 10 members w i l l require twice the queuing area that a system designed to serve a maximum queue/available wicket of 5 members. ( i i ) Y = The acceptable queue length/avai lable wicket The minimum acceptable queue length (Yl) of one appears, in general , to be agreed upon. That i s , a wicket i s closed i f the t e l l e r i s i d l e . The maximum acceptable queue length (Y2) appears to d i f f e r consider-ably among management. In order to accommodate most styles of manage-ment the values of Y2 = 2 , 3 , 4 , and 5 were included in the ana lys i s . 24 7.3.2 Parameter Changes to the Service Pol icy (Continued) Tables 3 and 4 show the s e n s i t i v i t y of the d i s t r i b u t i o n of wait ing time and queue length to the above values of Y2. Since the mean service time is approximately 2.2 minutes, i t i s not surpr i s ing that for i n -creasing values of Y2 the d i s t r i b u t i o n s would s h i f t upwards in th i s manner. 7.3.3 The D i s t r i b u t i o n of Service Times It was of interes t to examine the difference between the s t a t i s t i c s on the level of service and the estimated requirements when the service rate d i s t r i b u t i o n was approximated by the expotential d i s t r i b u t i o n vs the empirical d i s t r i b u t i o n . (See Table 5.) Table 5 The s e n s i t i v i t y of service rate d i s t r i b u t i o n on the s t a t i s t i c a l out-put. ! Dis t r ibut ion S t a t i s t i c Area Requirements ( s q . f t . ) Level of Service Average Waiting Time (Min.) Average Queue Length (Members) Maximum Queue Length (Members) Empirical 5.86 4.02 9.6 1125 Exponential 5.04 3.79 9.6 1125 25 7.3.3 The D i s t r i b u t i o n of Service Times (Continued) Table 5 shows that the exponent ia l^dis tr ibut ion tends to underestimate the average wait ing time in the queue and to a lesser degree the aver-age queue length per avai lable wicket . This i s caused because the v a r i -ance for the exponential d i s t r i b u t i o n is less than the variance of the empirical d i s t r i b u t i o n . However, there i s no dif ference in the maximum queue length per avai lable wicket (and hence the estimated area require-ments). This indicates that these s t a t i s t i c s are pr imar i ly a function of operational po l i cy and not on the d i s t r i b u t i o n of the service rates . I t should be noted in some environments where the t e l l e r s perform other time consuming duties (e .g . providing general information about the mem-bers account, issuing t r a v e l l e r s cheques, e tc . ) the variance of the t e l -l e r service rates would be much greater. The exponential d i s t r i b u t i o n would then underestimate the average wait ing time to a much greater de-gree. , 7.3.4 The D i s t r i b u t i o n of A r r i v a l Rates The system behavior with respect to wait ing time in the queue and queue length during the operating day i s c l e a r l y a function of the v a r i a b i l i t y * of the a r r i v a l rate d i s t r i b u t i o n . * For example see H i l l i e r and Lieberman, p. 301. 26 7.3.4 The D i s t r i b u t i o n of A r r i v a l Rates (Continued) This model assumes that for short enough interva ls the a r r i v a l rate can be considered exponentially d is t r ibuted about the mean. In this manner the a r r i v a l rate d i s t r i b u t i o n can be considered as exponen-t i a l l y d i s t r ibuted with the mean varying over time. In order to es-timate the ef fect of the in terva l length on the output of the simu-l a t i o n , i t was decided to simulate the data under i d e n t i c a l condi-t i o n s , (e .g . the same random number stream) with one exception. That i s the estimated average wait ing time, queue length and area re-quirements •for the t e l l e r f a c i l i t y were compared for d i f f e r e n t i n t e r -val lengths. (See Table 6.) Table 6 The s e n s i t i v i t y of the estimates of area requirements and the level of service to the interval length between successive data co l l ec t ions on member a r r i v a l s . Interval Length JArea Requirements (Minutes) 15 30 45 60 75 ( s q . f t . ) 880 790 710 700 620 Level of Service Average Waiting Average Queue Time (Minutes) 5.86 5.40 4.80 4.90 4.55. Length (Members) 4.02 3.74 3.49 3.54 3.38 Maximum Queue Length |( Members) 9.6 8.2 7.0 6.8 5.6 > 27 7 . 3 . 4 The D i s t r i b u t i o n of A r r i v a l Rates (Continued) Table 6 shows that both the required area estimates and the level of service estimates are sens i t ive to the method of estimating the d i s t r i -bution of member a r r i v a l s . The longer in terva ls (and thus fewer of them) reduce the natural var ia t ion of the time dependent a r r i v a l s . This causes an averaging ef fect (smoothing the peaks and troughs) of the queue length which would resul t in understating the area require-ments, as well as overestimating the level of service . 8. VERIFICATION In order to ver i fy the simulation model in a prac t i ca l s i t u a t i o n , the present and future t e l l e r f a c i l i t y requirements were determined for Campbell River and D i s t r i c t Credit Union, (See Appendix 1.) The management of Campbell River and D i s t r i c t Credit Union had o r i g i n -a l l y decided to b u i l d a new enlarged f a c i l i t y because the e x i s t i n g credi t union could not adequately accommodate i t s members. The simu-l a t i o n model showed that only a change in the f a c i l i t y layout was re-quired. Management reversed i t s decision to b u i l d a new f a c i l i t y and simply changed the layout. The credi t union i s presently operating ef-f e c t i v e l y with the new layout and has avoided the expense of a new b u i l -ding. 28 9. CONCLUSIONS 9.1 Model Usage 9.1.1 Model Input The s e n s i t i v i t y analysis shows that the model i s sens i t ive to both the d i s t r i b u t i o n of service times and the d i s t r i b u t i o n of a r r i v a l rates . Since the service process could not be approximated by a theoret ica l d i s t r i b u t i o n an estimate of the empirical d i s t r i b u t i o n should be used. A l s o , since the a r r i v a l rates were found to be time dependent, the interval length between successive data c o l l e c t i o n should be approx-imately 15 minutes. 9.2 Steady State The steady state analysis shows that a simulation run of 30 cycles i s required to ensure steady state condit ions. 9.3 Model Service Pol icy The s e n s i t i v i t y analysis shows that the model i s sens i t ive to the parameter values of the service p o l i c y . The manager of a given cred-i t union should f i r s t be consulted in order to determine the service pol icy which he u t i l i z e s . These parameter values (range of accept-able queue length/avai lable wicket and the minimum/maximum number of 29 9. CONCLUSIONS (Continued) 9.3 Model Service Pol icy (Continued) wickets which can be made avai lable) can then be input to the model in order to r e f l e c t the real world behavior. 10. RECOMMENDATIONS 10.1 Imp!ementation Credit unions or banks which have a large enough operation (minimum t e l l e r f a c i l i t y of 4 wickets) should make use of the simulation model whenever they are considering expanding t h e i r present f a c i l i t y . The model, should be used to accurately determine present and future t e l -• l e r f a c i l i t y requirements, the level of member service to the provided and the necessary input required to determine a t e l l e r schedule. 10.2 Extensions This simulation model could be extended to provide further ins ight into the queuing process and ass i s t the c red i t union or bank managers in the i r evaluation of a l ternat ive p o l i c i e s . Improvements could be made reducing the number of assumptions made by the model and test ing t h e i r v a l i d i t y . 30 10.2 Extensions (Continued) Further development could include: 10.2.1 Estimate the s i g n i f i c a n t differences between the fo l lowing queue d i s c i p l i n e s : ( i ) Single Line ( i i ) Quick Line By u t i l i z i n g a quick l i n e (a queue designated for those members with simple transactions such as deposit or withdrawal) the c red i t union manager is u t i l i z i n g a form of p r i o r i t y where the members requir ing minimal service are processed sooner than other members. The simu-l a t i o n could evaluate the effectiveness of th is queue d i s c i p l i n e and determine the optimal number of quick l ines for any given c red i t union. 10.2.2 Determine the ef fect of changing the time required by a t e l l e r to serve a member on the number of t e l l e r s and wickets required. The d i s t r i b u t i o n of t e l l e r service rates may be s i g n i f i c a n t l y effected by: - improving f a c i l i t y layout - r e d e f i n i t i o n of t e l l e r a c t i v i t i e s . 31 10.2.3 Adapt the model to other contexts For credi t unions and banks, the model could also be used to determine the number of loan o f f i ce rs and loan of f i ces required. The number of checkout stations and the queuing area required in supermarkets, l iquor stores and a i r l i n e booking o f f i ces could also be determined by the model with very minor changes. 11. SUMMARY The simulation model as developed in this paper has been used i n three credi t unions and has provided the information necessary to accurately determine present and future t e l l e r f a c i l i t y requirements which w i l l enable the credi t ' union to provide adequate member service . Although the internal complexities of the simulation model are not well under-stood by credi t union managers, i t s c a p a b i l i t y to accurately deter-mine t h e i r present and future ,service f a c i l i t y requirements i s known to many. It i s expected during 1975 that at least f i v e addit ional credi t unions w i l l make use of and benefit from the simulation model. 32 REFERENCES 1. FREQ Universi ty of B r i t i s h Columbia Computing Centre, 1973. 2. Gordon, Geoffry: System Simulation (New Jersey, Prent ice -Hal l I n c . , 1969). 3. GPSS/360 Introductory User's Manual (IBM Corp . , Form No. H20-0304). 4. GPSS/360 User's Manual (IBM Corp . , Form No. H20-0326). 5. H i l l i e r , F . S . , Lieberman, G . J . : Introduction to Operations Research (San Francisco, Holden Day, I n c . , 1972). 7. Naylor , Thomas H . , Joseph L. B a l i n t f y , Donald S. Burdich and Kong Chu: Computer Simulation Techniques (New York, John Wiley and Sons, I n c . , 1966). APPENDICES Appendix 1 Appendix 2 Appendix 3 Appendix 4 A Sample Computer Printout Model Documentation Other F a c i l i t y Planning Models A Case Study APPENDIX 1 A SAMPLE COMPUTER PRINTOUT — •- • * F A C I L I T I E S * o * » * * » » * » » » » < • * « * » » » * * » » • * * » * * » * * * * » * j ® ' -AVERAGE UTILIZATION OURING- & FACILITY 1 NU,J.3tR ENTRIES 017 AVERAGE, TIMF/TRAN 1381. P35 TOTAL TIME AvAIL. UNAvAIl. CUFfcENT TIME TIME STATUS Rt i- O E M A VAT L A P ILITY ino.n ( R fl^tiAi. i SEIZING PREEMPTING e 2 3 4 — oil 1257 1521 1364. 333 1312.171 1344. 725 .240 .357 .473 100.0 100.0 100.0 ] © 5 6 7 2339 2 33 8 2572 1*38.685 1 373.C45 1313. 712. .5»Z .696 .761 100.0 1C0.0 100.0 1 I a 2 74 3 1338.001" .796" i u O . 0 © * * * * * * * * * * * * * « . » * M . * O * * * » S TORAGF S • ! j - * » * * » * * * * * * * * * * * * * * * * * * * * * * * < * * * * * * * * * * * * * i ! -AVERAGE UTILIZATION DURING-n ir c CT.T— — V A X I K J M " ; CONTENTS i 6 '• 0 STORAGE 1 TA~PAC"ITV <» AVE9AGE ENTRIES C L N T E M S 7.719 14581 AVERAGE TOTAL AVAIL. U N A V A I L . U l t -K tT .M TIME/UNIT T IKE TIKE TI«F STATUS 2439.731 .964 H fc r L t M AVA1LAHILITY 1 00. 0 C U * *•. t I CONT E NTS 6 e Appendix 1 - A Sample Computer Printout 36 U) Ul UJ. 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I Ul M o» w UJ ul Hwai r /• oj a H i>) m i- i i ro i—• ro -J r M o -r • I CT- O O CT> I " m r-1 7J n i Ul Ul UJ UJ • i u» f> UJ r._ *• -4 sO' ro JN iA kl Ul M M M O '"l Ul w j - *> *o ro cJ l\) ;v) - K . *-* O O J") TO -ui *J o .V y • | -j -j i> JN i o « o r a iiJ <k) ro !-• M n- o r O- - 1 T "\J v/| M ^ - (S) M M cr m CD o o ^ o o c D t n t 356 363 372 378 384 391 454 460 466_ 472 478 tail 490 496 502 508 " 514 520_ '526 532 53B 544 604 615 ? . 7 5 11 If) 1_ 4 4 _ 4 4" 4 3 3 8 3 ' 2 24 8 •"5 7 36* 373 3 79 3 = 5 3)2 455 461 467 ' 4 73i " 479 4_P_5_ "451 4 07 5C3 509 515 5 21_ "52 7 533 5 39 545 606 616 8 8 7 2 599 16 359 366 374 380 366 393 456 462 468 474 460 486 492 498 504 51 0 516 _52 2_ 52 8 534 540 546' 609 619 7 12 9 1_ 5 4 _ 4 V" 4 3 a 8 6 2 4603000 2 360 363 375 381 387 _451_ 457 4.63 469 4 75" 481 487_ ~4 93 499 505 511" 517 523 529 535 541 601 510 625 2 7 11 36] 369 376 362 3S9 45? 459 464 _470_ 476 482 4f a 4 9 4 500 506 "512" 518 5?4 T30" 536 542 602' 611 371 37 7 i r , i « 0 453 9 465 '71 "'4 7 7 4 R.3 4 a q ""4 = 5"" 501' 507 ' 513" c19 C -5 S 537 603 614 10 11. 8 3 ... j 360C0 ******* ********************************* ' * _ " * ~ « . FULL WO3D MATRICES * ********* ******************************* ! FULLWORD MATRIX i ! ROW/COLUMN 1 2 3 . 4 5 6 7 8 9 10 i 1 i • ' . • 24 ROWS 2-5, 10 COLUMNS 15 1-10 APE ZERO 12 10 15 11 • 13 11 9 : | j ! ROW/COLUMN 11 12 13 14 15 16 17 IB 19 20 j 6 14 12 7 9 10 6 5 26 ! 0 i i 1 RTWS 2-5T COLUMNS T1-2CT ARE 7cRil i 0 | RCW/COLUMN 21 22. 23 24 25 26 27 2P 29 30 1 e ! 1 17 ROWS 2-5, 1 5 COLUMNS 20 21-30 ARE ZERO ' 30 38 22 25 25 14 15"; ! « ROW/COLUMN 31 32 33 34 35 3 6 37 38 39 40 1 • 1 ROWS 2-5, 1 . COLUMNS 24 31-40 ARE ZERO 0 0 0 0 0 0 0 ! t 001 ooj *0«(S 1-5, COLUMNS 41-50 ARE ZERO TABLES 3 O +-> c s-o_ s-cu +J Q . E -o o QJ col I T3 OJ Q Q . TABLE 2 ENTRIES IN T A 3 L E MEAN ARGUMENT 14296 U"P;R L IM[T ""' 0 5 59.574 STANDARD OEVIATION 41.750 SUM OF ARGUMENTS OBSERVED F RC QUE NCY 154 362 1015 20 2 5" 3 0 35 PER CFNT OF TOTAL 6.67 2.67 468 494 469 540" 541 515 "TT2"7 3.45 3. 28 "3.77" 3.73 3.60 CUMUI ATI VE PERCENTAGE 6 . 5 9 . 3 CUMULATIVE "' REM1INDCR' 9 3 . 3 9 C . 6 40 45 53 5 5 60 .65 583 631 724 753" 720 9?2 12.6 16.0 19. 3 23.1" 26.9 30. 5 7 J 75 8 0 8 5" 90 95 fT4~~ 700 6] 1 "472" 335 337 100 105 110 ""11 5~ 12 0 125 T T T " 135 143 "2TT" 221 22 5 19 3" 157 100 145" 150 155 "T5T~ 120 127 !02"" 11 5 103 4.07 4.41 5.06 "5.30" 5.0 3 S . a i 4.39 4.27 "3.3 0" 2.34 '2.35 1.90 1.54 1.57 "1.34 " 1.09 .69 34.5 39.0 44.0 "49.3 54.4 60.2 "f 7. 3 83.9 80. 6 "76."?" 73.0 69.4 P5l'"7"9"7. 000 MULTIPLF """ Cf' » !4» ' " -.000 . Of.3 N0N-WF IGHT'FO 65.9 70.8 75.1 78.4 " 80. 9 83.1 65. 4 60.9 55.9 50.6" 45.5 39.7 T T S T .251 . 335 ~4'l'9"~ .503 .587 C£V I AT I ON "FROM ME A M" -1.426 -1.307 . 67] .755 . 639 ""."923" 1.007 1. 091 -1.067 -.947 -.703 -.585 16T" 165 170 "HS"" 180 185 11 1 115 "9 7" 49 35 1.07 .83 .39 " .71" .80 .72 '-.*5 .77 .80 ".67 .34 .24 85.0 •36.6 88.1 89.5 90.0 91.3 ~3~4T0~ 29. 1 24. 8 "21 :~s~ 19.1 16.8 1.174 1.258 1. 342 "1.426" 1. 510 1. 594 14.9 13.3 11.3 "10.4" 9. 3 8.6 1.675 1. 76 2 1 . "4 6 1.950" 2.014 2. C98 .468 .3 49 .229 .109" .010 .12? • >4v .369 .45? .603" .728 . 848 '52.4 93.2 94. 1 94.8 95.6 96.3 7. 5 6.7 5.8 5.1 4.3 3.6 963 1. . C 8 8 1.2 0.7 1.327" 1.447 1.567 97. 0 97.7 93.6 99.2 99.6 99. 8 2.5 82 2.266 2.349 2.433" 2. 517 2.601 l.oh6 1. f06 1.926 2.046" 2.165 2.265 2.2 1.3 " .7" .3 . 1 2.685 2.769 2.653 "2.537" 3.021 3.) 05 Z74TJ5-. 2.525 2.644 "2.764 " 2.8 94 3.004 : 1 ! e 190 15 19 5 •'. RE-fAINING FREQUENCIES AH E ALL 7.1 .10 .02 :RO 99.4 100.0 . 0 .0 3. 139 3.2H TABLE ENTRIES IN TABLE 1380 MEAN' ARGUMENT 2ft.380 UPPER L IMIT 0 O^SSRvEn FREQ'J'NCY 364 10 20 30 Vo" 51 60 IF. 8 171 125 9 7" 120 122 70 1 J ioo" no 120 103 58 21 8 1 PER OF CENT TOTAL 26.37 REMAINING FREQUENCIES ARE ALL ZERO 13. 62 12.39 _ 9.05 7.02"' 8.69 8.J)4_ 7. 4 6 4.20 1.5.? .57" .07 .14 STANDARD DEVIATION 25.937 SUM OF ARGUMENTS 36405.COO CUMULATIVE PERCENTAGE 26.3 CUMULATIVE REMAINDER 72. 6 39.9 52.3 61.4 68.4 77.1 86.0 60.0 47. 6 38.5 "31. 5 ' 22.8 13.9 93.4 "7.6 99.2 99.7 99. ,3 100.0 6.5 2. 3 . 7 .2 . 1 .0 NON-WF. IGHTEO MULTIPLE CF MF AN - . COO DEVIATION FROM M'jfl -1.017 . 379 . 7*8 1.137 1.516 ' 1. 395 2. ?74 2.65 3 3.022 3 .4)1 "3.790" 4.165 4. 548 -.245 .1 39 ""'.5 25" .910 1.296 1.681 2.067 2.4 52 2.638 3.22 3 3.60= TABLE 4 ENTRIES IN TASLE 2423 MEAN ARGUMENT 43.028 STANDARD DEVIATION 25.375 SUM OF ARGUMENTS 104257.000 f!Cl-WE IGHTEO UPOFR OBSERVED PER CENT CUMUt AT IVE CUKULATIVE L IMIT FREQ'. IENCY OF TOTAL PERCENTAGE . REMAINDER 0 152 6.27 6.2 93.7 10 149 6.10 12.3 87.6 2 0 219 9.03 21.4 76. 5 30 284 11.72 33.1 t 6. 8 40 291 12.00 45.1 54.8 50 34 4 14.19 59.3 40.6 60 339 13.99 73.3 26.6 70 310 12.79 86.1 13.8 80 175 7.22 93.3 6. 6 90 67 3.59 96.9 3.0 100 54 2.22 99.1 . 8 110 14 .57 99.7 .2 120 4 " ;i"6 " 99.9 .0 130 2 .08 100.0 .0 REMAINING FREQUENCIES l ' ARE ALL ZERO MULT I PL E OF MEAN ' "-. 000" .232 . 4 6 4 .697 .929 1.162 1.394" 1.626 1. 859 DEVIATION FROM MEAN -1".6?5"' -1.301 -.9 07 -.513 -.119 .274 1.062 1.457 2.324 2. 556 "2.7B8" 3.021 1. "51 2.245 2.639 "2i~033~ 3.427 TABLE 5 ""£"NTR'TE5—IN -T4BTT" 3711 "flEAVTKGMMENT" 42.211 "STANDAWDEVIATION"" 29.500 "SUM-TIF A R r,Ou?KT S~ 156647.000 NON— WEIGHTEO UPPER LIMIT 0 10 2 0 3 0 0'3sSavFt) FREQUENCY 318 57 r 343 2 89 PEP. CENT OF TOTAL S.56 — 9.9<r 9.24 10.48 C U M U L A T I V E ' PERCENTAGE 8.5 : ~18.5" 27.8 30.2 REMAINDER 91.4 B1T4-72. 1 61.7 MULTIPLE CF ^E AN -.000 ;23"6~-.473 .710 DEVI AT I ON FRTAI MEAN -1.430 -.752 -.413 1 v 4) 50 60 7 0 8 0 90 100 11 3 12 3_ "13 0 143 15 0 163 170 34 7 404 476 48 8 2C7 133 43 14 1 7 "" 3 18 8 14 7 9.35 10.83 12. 82 13.15 3.27 3.5 8 1.29 .37 .45 ~" .24 .48 .21 47.5 51.5 71.3 64.5 92.7 _9 6.^3_ 97.6 93.0 98.4 98.7" 99.2 99. 4 .37 .18 99.3 100.0 52.3 41.4 28.6 15.4 7. 2 3. 6 2. 3 1.9 1_. 5 1 .2"" .7 . 5 .=47 1. I »4 1.421 1.658 1.895 2. 132 2. 369 2.605 2.842_ ~'ir. 079 3.216 3.553 2.790 4.027 -.0 74 . 21 4 .602 .041 1.280 1 . 6 113 -.5 8 297 636 975 " 314 C53 3.992 4.331 R E M A I N I N G F R E Q U E N C I E S ARE_ ALL. J_E*_0_ TABLE ENTRIES IN TABLE 565 ^ FAN ARGUMENT 61.495 U°PE R L IMIT 0 OBSERVED FREQUENCY 3 PER OF CENT TOTAL .00 10 20 30 ' 4 0' 53 6 0 5 9 31_ 75 64 60 .33 1.59 5.4 8 "13.2 7" 11.32 10.61 STANDARD DEVIATION 20.937 "CUMULATIVE" PERCENTAGE .0 .8 2.4 7.9 21.2 32. 5 43.1 70 30 90 "100" 92 93 105 26" 16.28 17.34 18.58 4.60" R E M A I N I N G F R E Q U E N C I E S ARE ALL ZERO 59.4 76. 3 95.3 "100.0" CUMULATIVE REMAINDER 100.0 SUM OF ARGUMENTS 34745.COO "MULTIPLE CF MEAN -.000 NON-WEIGHTED 99. 97, 92. '76. 67. 56.8. 40.5 23. 1 4.6 — . 0 " " . 162^  .32f .487 ".650" .813 .975 ] .13" 1. 300 1.463 "U'6'2'6" "DEVI AT I ON" FROM MEAN -2.937 2^74"E"9--1.781 -1.504 "-1.C26" -.549 -.071 TABLE 7 ENTRIES IN TABLE_ "640" UP"ER MEAN ARGUMENT .... ... — - 3 I < 2 8 2 - -LIMIT 0 10 20" 30 4 0 OBSERVED FREi'JENCY 55 77 12 3 117 6 3 50 60 70 "80" 90 100 68 27 37 56 8 PER CENT TT5TAT7 8. 59 12.03 19.21 18.23 13.62 T0T6"2~ 4.21 5.78 " e.74 1. 24 .62 STANDARD D E V I A T I O N 24.000" SUM OF AFCUMFNTS 20021.000 CUMULATIVE 8.5 20.6 " ' 39.8 5 8.1 68. 7 -79T3-83.5 89.3 98.1 99.3 100.0 CUMULATI VE EMAINDER 91.4 79. 3 60. 1 4! .8 31.2 20. 6 16.4 10. 6 "1.8 .6 .0 NGN-WEIGHTED" MULT I P L F —CF t'EAN -.000 .3)9 " .639 .958 1 .278 1. 5PK 1.917 2. 237 2. 557 2.876 3. 196 CEVIATION FROM "EAN -1.303 • "• -.4 70 -.053 .363 ~T7T9~ 1.196 1.613 2.029 2.446 2. 863 REMAINING FRE0'JhNCIS5 ARE ALL ZbRU "TASTE 8" ENTRIES IN TABLE 5579 "EAN ARGUMENT 89.571 STANDARD D E V I A T I O N 41.875 SUM OF ARGUMENTS 499722.000 NOM-WEIGHTEO • » i I U PRE R LIMIT 0 10 20 30 OISERVEO "" PER-CENT FRcO'JF'ICY OF TOTAL 65 1.16 61 1.09 93 1.75 135 2.41 CUMULATIVE PERCENTAGE 1.1 2.2 4.0 6.4 '"' CUMULATI VE' REMAINDER 9 e . 8 97.7 95.9 93. 5 MU T I PI f or MEAN -.000 • 111 .223 .334 CrVI AT.irj:: F^n.y **t;A\' -2.129 -l.VOO -1.661 -1.422 , 40 50 50 223 3.94 355 6.36 454 8.13 10. 3 16.7 24. 8 89.6 83.2 75. 1 .446 .558 .669 -1.183 s -.c45 -.706 o 70 RO 9 3 626 11.22 617 11.05 453 8.11 36.0 47.1 55.2 63.9 52.8 44.7 .781 .893 1. 004 -.467 -.228 .010 ft » 100 110 120 47 0 8.42 43 7 7. 47 327 5. 86 63.7 71.1 77.0 36.2 28. 8 22.9 1.116 1 .22f. 1.339 .249 .487 .726 • a 130 140 15 0 242 4.33 229 4. 10 209 3.74 81.3 85.4 89. 2 18.6 14.5 10.7 1.45) 1.562 1.67^  .965 1.2 04 1 . 4 42 ft » : 160 170 130 !F2 3.26 219 3.92 146 2.61 92.4 96.4 99.0 7.5 3.5 .9 1.766 1. 597 2.009 1.6 SI 1.920 2. 1 59 * 0 REMAINING no 200 FREOUENCI 50 .89 4 .07 ES ARE ML ZERO "  99.9 100.0 " " ' .0" .0 2.121 " 2. 232 "" 2.395 "' " " " 2.637 ft tf TABLE 10 0 r ENTRIES IN TABLE 14293 "MEAN ARGUMENT 16.232 STANDARD DEVIATION 15.503 SUM OF ARGUMENTS 232098.COO NCN-WEIGHTEO c UPPER LIMIT 0 ORSERVEO PER CENT FREQUCN'CY OF TOTAL 1404 9.31 CUMULATIVE PERCENTAGE 9.6 CUMULATIVE REMAINDER 9C.1 MULT IPLE OF MEAN -.000 • DEVI ATICN FROM MEAN T-1.047 c 1 2 3 521 3.64 5)0 . 3.56 562 3.93 13.4 17.0 20.9 "' 86.5" 82.9 79.0 .061 .123 .16*. -.582 -.918 -.153 » 4 5 6 519 3.62 567 3.96 621 4.34 24.5 28.5 32.3 75.4 71.4 67.1 .246 . 308 .369 -. '39 -.724 -.660 V - 7 8 9 639 4.46 559 3.90 5C7 3.54 37.3 41.2 44.8 62.6 58.7 5 5. 1 .43! .492 .554 -.5 95 -.531 -.466 * c 10 11 12 489 3.42 461 3.22 431 ' 3.01 48.2 51.4 54.4 51. 7 48.5 45.5 . 616 .677 .739 -.-02 -.337 -.273 13 14 15 404 2.82 '89 2. 71 143 2.43 57.3 60.0 62.4 4 2.-6" 39.9 37. 5 . HUU .962 .92'--.208 -.144 -.079 fa. 1 6 17 18 310 2.16 275 1.92 231 1.61 64.6 66.5 68.1 .J5. i 23.4 31.8 . 9 85 1.047 1. 108 -. 01 5 .049 .) 13 fa C -19 20 21 205 1.43 182 1.27 213 1.48 69. t 70.8 72.3 3L.4 29.1 27. 6 1. I '0 1.232 1.293 • I'M .242 .3 07 -c ! 22 23 24 216 1.51 154 1.07 123 .36 '3.8 74.9 75.8 26.1 25.0 24.1 1.3^ 5 1.416 1.478 .Ml .'36 .500 « 25 26 27 123 .S6 123 .86 145 1.01 76.6 77.5 78.5 23.3 22.4 21.4 1.540 1.60) 1.663 .5a5 .6 94 G 3^  4-> Z ! O 4-> C • i — S-Q . S-o 4-> Z3 C L E O c_> 1 ! ro <9 c • f'l 29 31) 31 . 32 33 l'i'T 129 n o 113 114 Of. l.o? .90 .76 .79 .79 .67 7'i. 5 80.4 ei.2 8 2.0 82.8 83.5 20. 4 19. 5 18.7 1 7.9 1 7. 1 16.4 1. ',""• '""""' 1. 736 1. C48 1.909 1. 971 2.032 .823 .6B7 .952 -i.oi6 ; 1.031 j - 1 * » i 34 3 5 36 IIS 11 7 146 .32 .81 1.02 84. 3 85.1 86.1 15.6 14.3 13.8 2. 0°4 2.156 2.217 1.145 1.2'0 1.274 y • 37 33 39 14 1" 131 .1.0 4 .98 .91 .72 87.1 88.0 PB.8 12.8 11.9 11.1 2. 279 2. 340 2.4 02 • 1 . 3 3 9 1.4 03 • i 1.468 40 41 42 83 76 80 .56 .53 .55 69.3 89.9 90.4 1 0 . 6 10.0 9 . 5 2 . 4 6 4 2.525 2.587 1 . 3 32 . 1.5 97 ; 1 61 43 44 4 5 84" 81 R3 .58 .56 .58 91.0 °1.6 92.2 8.9 8.3 7.7 2.648 2.710 . 2.772 " "" 1.726 - • - ; 1.790 1.855 ' O 46 47 43 81 96 95 .56 .67 .66 "2.7 9 3 . 4 94.1 7.2 6.5 5.8 2. H33 2.895 ?. 956 1.919 ; 1.984 2.04? o 49 50 . 51 97" 96 79 "" ""' .67" . 67 .55 94.7 95.4 96.0 "5.2 4.5 3.9 3.018 3.030 3.141 2.113 " — : 2.1/7 I 2.242 52 53 54 73 82 72 . 54 .57 .50 96.5 97. 1 S^.6 3.4 2. 8 2.3 3.203 3.264 3. 326 2.i06 2.371 2 . ' 3 5 • 55 56 57 61 37 32 .42 .25 .22 98.0 98. 3 98.5 1.9 1.6 1.4 3.388 3.449 3.511 2.500 " 4 2.564 2.629 ; • . 53 59 60 31 26 31 .21 .18 .21 96.7 °8.9 99.1 1.2 1.0 .8 3.572 3.634 3.696 . 2.6^3 i 2.758 j 2.822 ! o 61 62 63 34 29 2 7 .23 .20 .18 99.3 99.6 99.7 .6 .3 .2 3.757 3.319 3.331 2.3S7 2.951 ! 3.016 • 64 65 65 1 3 6 2 .12 .04 .01 99.9 99.9 99.9 .0 .0 .0 3.942 . 4. 004 4.065 3.080 3.145 ' 3.209 '; t ... REMAINING 67 69 3 i *.LL Z " .02 .00 EP.n 99.9 100.0 ."0" .0 4.127 4. 189 3.338 j «-TABLE 11 •If. AN r SUM OF ARGU'lFNTS ' 3774.000 i c \ -ENTRIES IN TA3LE 1330 "ARGUMENT ' ~ " 2.734 STANDARD DEVIATION 3.351 NPN-WEl3HTEO ! 0 -UPPE*. LIMIT 0 G tlS = FPFC'J * v F 0 rNCY 546 PER CENT OF TOTAL . 39.56 CUMULTm-Vi: PERCENTAGE 39.5 CUMULATIVE REMAINDER 60.4 Mill 11 PLh nr MEAN -.000 UbVlAI I'J.'J : FROM >'::4M 1 --:,1 5 . • • _ • ! t> 2 4 6 ?R9" 202 123 20.94 14.63 8.91 '•" 60.5 75.1 84.0 39.4 24.8 15.9 .731 " •-' 1.462 2.19? _> ? 1 9 i .377 . " 7 4 m - ; a 10 12 115 65 21 8.33 4.71 1.52 92.3 97.1 98.6 '.6 2.8 1.3 2. 425 3.656 4.3B7 15'J i 2.167 j 2.764 i m -416 11 "IT" 5 2 .79 .36 .14 ~" 99.4" 99.7 99.9 .5 .2 .0 . ?> 119 5.850 6.581 3.3 61 " " " 1 3.957 | 4.554 ; 4» 7 1 1 REMAINING FREQUENCIES AK E ALL ZERO .0 .151 CO CI- TABLE 12 ENTRIES IN TAULE A'-liH APGUMENT -(-> 3 o +-> •r-i . Q_ J-<u o-o OJ 1^ <=c: i T 3 C OJ =a:: 2423 UPPER 6.114 STANDARD DEVIATION 4.179 SUM OF AFGUMfcNTS L IMIT 0 2 _ OBSERVED FREQUENCY" ' 256 279 PER "'"OF"" 4 6 8 "ib" 12 14 360 457 4?0 26 7" 139 101 16 18 20 22 24 49 24 3 2 1 REMAINING FREQUENCIES ARE ALL ZEF 0 CENT TOTAL" 10. 56 11.51 14. 8 5 18.86 17.74 11.01" 7.50 _4._16_ 2.02 .99 .33 .08 " .04 CUMULATIVE PERCENTAGE 10.5 22.0 CUMULATIVE "REMAINDER" 89.4 . 77.9 36.9 55. 7 73.5 84.5" 92.3 96.5 I 4816.000 _MULTIPLE_ OF'MEAN -.000 . 327 M ' - N - W E ;r,HTi o DEVIATION^  FROM M F AN -1.462 -.cut. 67.0 44. 2 2 6.4 "15.4" 7.6 3.4 C. 5 9V.5 99. 8 99. 9 ' 100.0 1 . .4 .1 .0 .0 .6f 4 .981 ) .308 1.6 3 5" 1. 962 2.289 ? . 6T"6 2.943 3.270 3.597" 3.924 -.505 -.027 .451 ".920 1.406 1.8 86 ~2Y3"6v" 2. P43 3.322" 3.600" 4.2 79 TABLE 13 _ ENTRIES IN TABLE 3711 MEAN ARGUMENT 7.922 STANDARD DEVIATION 5.Y57 SUM""0F ARGUMFNTS 29400.COO NON-WEIGHTEO. ftv i 1 UPPER IIMIT 0 OBSERVFC FREQUENCY 454 PER CFNT OF TOTAL 12.23 CUMULAT IVE PERCENTAGE 12.2 CUMULATIVE REMAINDER f 7.7 MULTI PLF OF MEAN -.000 DEVIATION FROM MFAN -1.375 % — -2 4 6 374 369 196 • 10.07 9. 94 10.67 22.3 32.2 42.9 77.6 67.7 57. 0 .252 .504 .757 -1.026 -.68) -.333 ft 3 10 12 453 436 37 9 12. 34 11.74 10.18 55.2 67.0 77.2 44.7 32.9 22.7 1.009 1.262 1.514 .013 .360 .70S ft 14 16 19 339 249 151 9.13 6.70 '4.06 . 86.3 93.0 97.1 13.6 6.9 2. 8 1. 767 2.019 2. 272 l.L'55 1.A02 1 .750 a 20 22 24 55 17 6 1.4ft .45 ' .16 93.5 99.0 99.2 1.4 .9 .7 2.524 2.776 3. 029 2.09/ 2.444 2. 792 • : a 26 23 30 9 7 6 .21 .18 .16 99.4 99.6 99.7 .5 .3 .2 ?. 281 3. 534 3. 796 ~?;i 3-? 34 873.834 1 i REMAINING 32 34 FREQUENCIES i 5 ARE ALL ZERO .08 .13 99.8 100.0 .1 . 0 4.UJU 4.291 41B14.529 i TABLE 14 «. i tNIRltb IN JA3LC 5S5 MEAN ARGUMENT 13.980 STANDARD DEVIATION SUM 5.675 UF iP.GUML'TS 7899.000 N3N-WEIGHTEO •1 % I U F P t K LIMIT 0 •JSERVEC FREQUENCY 2 PER CENT OF TOTAL .35 CUMUL AT J VE PERCENTAGE .3 CUMULATIVE REMAINDER 99.6 MULTIPLE OF MEAN -.000 "D't'VlAI IL'N FROM «E(N -2.463 l t i ft Xi © f— — ' 2 6 1 . 0 6 1 .4 9 8 . 5 . I ' ? - 2 . 1 1 J N 4 1 " 3 . 36 4 . 7 9 5 . 2 . 2 8 6 - 1 . 7 5 8 6 33 5 . 3 4 • 1 0 . 6 8 9 . 3 . 4 2 9 - 1 . 4 0 6 (ST. 9 ? 3 5 . 9 4 1 6 . 4 8 3 . 5 . 572 - 1 . 0 5 3 1 0 51 9 . 0 2 2 5 . 4 7 4 . 5 . 7 1 5 - . 7 0 1 ,3 12 " 3 1 6 . 4 6 4 1 . 9 5 8 . 0 . B5e - . " 1 £S ) 14 91 1 6 . 1 0 5 3 . 0 4 1 . 9 1 . 0 0 1 . 0 0 3 16 6 2 1 0 . 9 7 6 9 . 0 3 0 . 9 1 . 1 4 4 o 18 4 0 . 7 . 0 7 7 6 . 1 2 3 . 8 1 . 2 8 7 ,7 0 6 2 0 5 4 9 . 5 5 8 5 . 6 1 4 . 3 1 . 4 3 0 . 1 . 0 6 0 22 34 6 . 0 1 9 1 . 6 8 . 3 1 . 5 7 3 1 . 4 1 ? o 2 4 29 5 . 1 3 9 6 . 8 3 . 1 1 . 716 1 . 7 6 5 o 2 6 9 1 .41 9 8 . 2 1 . 7 1 . 8 5 9 2 . . U 7 23 . 10 1 . 7 6 1 0 0 . 0 .0 2 . 0 0 2 2 . 4 7 0 Q RE M A I N I N G F R E Q U E N C 1 ES ARE A L L Z E R O O T A B L E 15 o E N T R I E S P T A H L G MEAN ARGUMENT STANDARD D E V I A T I O N SUM OF ARGUMENTS 6 4 0 9 . 0 0 1 7 . 5 5 4 5 7 6 1 . 0 0 0 NO N - WE I 0 H T t 0 0 U P P E R ~ 0 3 S E R V E D PER C E N T C U M U L A T I V E C U M U L A T I V E MULT I PL E D E V I A T I O N L I M I T F R E Q U E N C Y OF T O T A L P E R C E N T A G E R E M A I N D E R C F " E A N FROM MEAN 0 7 4 1 1 . 5 6 1 1 . 5 8 8 . 4 - . 0 0 0 - 1 . 1 9 1 2 59 9 . 2 1 2 0 . 7 7 9 . 2 . 2 2 2 - • ^ 25 4 7 3 1 1 . 4 0 3 2 . 1 6 7 . 8 . 4 4 4 - . 6 6 2 •at 6 8 7 1 3 . 5 9 4 5 . 7 5 4 . 2 . 6 6 6 - . 3 9 7 ° 8 8 3 1 2 . 9 6 5 8 . 7 4 1 . 2 . 8 F 8 - . 1 32 10 4 0 6 . 2 5 6 4 . 9 3 5 . 0 1 . 1 1 0 . 1 3 2 o 12 35 5 . 4 6 7 0 . 4 • 2 9 . 5 1 . 3 3 3 . 3 9 6 Cf 1 4 4 1 5 . 4 0 7 6 . 8 2 3 . 1 1 . 5 5 5 . 6 6 1 1 6 4 5 7 . 0 3 8 2 . 9 1 6 . 0 1 . 7 7 7 . 926 13 36 5 . 6 2 8 9 . 5 1 0 . 4 1 . 9 9 9 ' 1 . 1 9 1 o 2 0 15 2 . 3 4 " 9 1 . 8 8 . 1 2 . 2 2 1 1 . 4 5 5 2 2 1 0 . 1 . 5 6 9 3 . 4 6 . 5 2 . 4 4 4 1 . 7 2 0 O r 2 4 14 2 . 1 8 9 5 . 6 4 . 3 2 . 6 6 6 1 . 9 8 5 CJ 2 6 6 . 9 3 9 o . 5 3 . 4 2 . 8 8 8 2 . 2 5 0 2 8 4 . 6 2 9 7 . 1 2 . 8 3 . 1 1 0 2 . 5 1 4 <3 3 0 5 . 7 8 9 7 . 9 2 .0 3 . 3 3 2 2 . 7 7 9 32 1 0 — 1.56" 9 9 . 5 . 4 i. 554 i. 0 44 3 4 3 . 4 6 1 0 0 . 0 .0 3 . 7 7 7 3 . 2 0 8 O RE MA I N I N G F R E O U E N C I t S ARE ALL ZERO TABLE 16 ~E N THTE S~ I N~"T A BT"E ~ 55 79 "MEAN" ARGUMENT ' 3 0 . 5 5 1 " S T A N D A R D D E V I A T I ON 1 5 . 0 6 6 SUM OF' "ARGUMENTS ~ 1 7 0 4 4 8 . 0 0 0 N 0 N - W E I G H T E D U P P E R C 8 S E R V E 0 P E R C E N T C U M U L A T I V E C U M U L A T I V E M U L T I P L E I H V I A T I G H j L I M I T F R E Q U E N C Y O F T O T A L P E R C E N T A G E R E M A I N D E R O F " F A N F R O M M E A N 1 0 . 7 ? 1 . 2 9 1 . 2 • 9 8 . 7 - . 0 0 0 - 2 . 0 2 7 i 2 2 4 . 4 3 ... 1 ^ 7 - 9 8 . 2 • 06 5 - 1 . 5 9 5 i 4 58 1 . 0 3 2 . 7 9 7 . 2 . 1 3 0 - 1 . 7 6 2 i 6 9 2 1 . 5 4 4 . 4 9 5 . 5 . 1 9 6 - 1 . 6 2 9 i 8 79 1 .41 5 . H 9 4 . 1 . ' 6 1 j 1 0 1 3 7 2 . 4 5 8 . 2 9 1 . 7 . 3 2 7 - 1 . 3 5 4 12 1 7 6 3 . 1 5 1 1 . 4 8 8 . 5 . 392 - 1 . 2 3 1 1 4 2 0 9 3 . 7 4 " 1 5 . 1 " " e 4 . 8 " . 4 5 8 - 1 . 0 9 5 ] 1 6 2 4 3 4 . 4 4 1 9 . 6 8 0 . 3 . 5 2 3 - . 9 6 5 1 3 2 5 3 4 . 5 3 2 4 . 1 7 5 . 8 . 5 8 9 - . 8 3 ? r 20 254 4.55 28.7 71. 2 .654 -.700 22 36 6 6.56 35.2 64. 7 .720 - .5 67 ' 24 227 4.06 39.3 60.6 .785 -.434 26 224 4.01 4 3. 3 56. 6 .351 - . - 0 ? 28 272 4.87 •48.2 51.7 .916 -.169 • 30 229 4.08 52.3 47.6 .931 -.036 -} 32 214 3.83 56.1 43. a 1.047 . .0 9'; \ 34 206 3. 69 59.8 40.1 1.112 .223 36 263 4.71 64. 5 35.4 1.178 .361 38 272 4.87 69.4 3 0. 5 1.243 . 4 94 40 137 3.35 72.7 27.2 1.'09 .627 42 156 2.79 75.5 24.4 1 .374 .759 44 165 2.95 73.5 21.4 1.440 .892 46 154 2.93 81.4 . 18.5 1.505 1.025 43 191 3.42 84.9 15.0 1. 571 1.153 50 193 3.45 "38.3" 11.6 1.636 1.2 90 52 157 2. 81 91.1 8.8 1.702 1.423 54 154 2.76 93.9 6.0 1.767 1.556 56 93 1.75 95.6 4.3 1.832 1.680 59 63 1.12 96. H 3.1 1.993 1.921 60 57 1.02 97.8 2. 1 1.963 1.954 62 63 . "1.12 98.9 1.0 2.029 2.087 64 45 .90 99. 7 .2 2.094 2.220 65 8 .14 99.9 .0 2.160 2.352 68 4 .07 100.0 .0 2.225 2.485 REMAINING FREQUENCIES ARE \LL ZERO TABLE 18 ...... ENTRIES IN TABLE MEAN ARGUMENT STANOARO DEVIATION SUM OF ARGUMENTS 1536 15.319 15.210 23530.000 NON-WtIGHTEO UPPER OTSE^VEO PER CENT CUMUL AT IVE CUMULATIVE MULTIPLE DEVIATION LIMIT FSEOU F NC Y """OF TOTAL "PERCENTAGE REMAINDER OF MEAN FROM MEAN 3 103 6.70 6.7 93.2 - . COO -1.007 1 63 4.10 , 10.9 89.1 . 065 -.941 2 69 4.49 15.2 d4. 7 .130 -.675 3 72 4.63 19.9 80.0 .195 - . eoo 4 64 4.16 24.1 75.8 .261 -.744 5 ""'67 4.36 28.5 71.4 .326 - .6 /0 6 43 3.1 2 31.6 68. 3 .391 -.612 7 66 4.2 9 35.9 64.0 .456 -.546 8 63 4.42 40. 3 59.6 .522 -.481 9 61 3.97 44.3 55.6 .587 -.415 10 72 4. 68 49.0 50.9 .652 . -.349 11 70 4.55 . 53.5 46. 4 . / IB -.Hi 12 57 3.71 57.2 42 .7 . 783 -.218 13 53 3.45 60. 7 3 ° . 2 .94 8 -.152 14 39 2.4 ! 63.2 i a . ( .913 15 43 Z.T) 66. 0 33.9 .979 -.020 16 34 2.21 . 68.2 31.7 1.044 .044 17 " 35 2.27 70.5 24.4 1. 104 .110 18 30 1.95 72.4 27.5 1.175 .176 -19 30 1.95 74.4 25. 5 1.240 .241 20 23 1.92 76.2 23. / 1. 305 .30 ' 21 26 1.69 77 .9 22.0 1.370 .373 22 15 1.04 78.9 21.0 1.436 .439 23 22 1.43 80.4 19.5 1. 501 .004 24 22 1.43 81.8 18.1 1.566 .570 25 10 .65 82.4 17. 5 1.631 .636 f 26 11 . 71 93.2 16.7 1.697 .702 -~ 27 15 .97 84.1 15.8 1.762 .767 28 9 .58 84.7 15.2 1. 327 .833 29 16 1.04 •35.3 14.1 l. A93 .309 30 12 .73 96. 5 13.4 1. 958 .965 \ 31 7 ,45 87.0 ' 12.9 2.023 1.030 ( 3 2 3 .19 87.2 12.7 2.0E3 1.396 33 3 .19 8 7.4 12.5 2. 1 54 1.162 34 5 .32 87.7 12.2 2.219 1.228 35 9 .58 8«. 3 11.6 2.234 1.293 36 12 .73 89.1 10.8 2.350 1.359 37 10 .65 89.7 ' 10.2 2.415 1.4 25 39 3 . 19 89. 9 10.0 2.490 1.491 3 9 3 . 19 90.1 9. 8 2. 545 1. 5 56 40 5 . 39 90.5 9.4 2.611 1.622 41 7 .45 91,0 8. 9 2. 676 1.6 68 .42 6 .39 91.4 8.5 2. 741 1.754 43 3 . 1 9 91.6 8.3 2.306 1.31? 44 14 .91 92.5 7.4 2. 372 1..".° = ! 45 7 .45 92.9 7.0 2.937 1.951 ! 46 4 .26 93.2 6. 7 3.002 2.317 47 .9 .58 93.8 6. 1 "3. 053 ' 2 . 0 8 2 " - "' " 43 5 .32 94.1 5.8 3.133 2. 14? 49 1 . 06 94. 2 5.7 2.193 2.214 50 3 .19 94.4 5.5 2. 263 2.J80 51 6 .39 94.7 5.2 3.329 2.345 52 6 . 39 95.1 4. 8 2.394 2.411 53 3 .19 95.3 4.6 3.459 2.477 54 8 .52 95.8 4.1 3.525 2.542 55 6 .39 96.2 3. 7 3. 590 2.509 56 9 .58 .96.9 3.1 3.65 5 2.674 57 3 .19 97.0 2.9 3.720 2. 740 -58 4 .26 97.3 2.6 3. 786 ' 2. 605 59 5 '.IV 97.6 2.3 3.851 2. e 71 60 4 .26 97.9 2.0 3.916 2.937 61 8 .52 98.4 1. 5 3.9ei 3.003 62 4 .26 98.6 1.3 4.047 3.066 63 2 .13 93.8 1. 1 4.112 3.134 64 3 .19 99.0 .9 4.177 3.200 65 ' "" 4 .26 99.2 .7 4.243 3.2 66 66 0 .00 99. 2 .7 4. 308 3.331 67 1 .06 99.3 .6 4. 373 3.397 63 1 . 06 94.4 .5 4.42B 3.463 69 1 .06 99.4 .5 4. 504 3.529 70 4 .26 99.7 . 2 4.569 2.594 7T " 2 ' "i 1 3 " 99.8 " " " " " . 1 •~~ 4.634 " 72 1 .06 99.9 .0 • 4.700 3.726 73 1 . 06 100.0 .0 4.765 3.792 REMAINING FREQUENCE E5 ARE ALL ZESO. f TA6LE 20 " " - :- . . . . . . ... . _ . .. '--•• - ; . .._ : ENTRIES IN TABLE MEAN ARGUMENT STANDARD OEVIATION SUM OF APGUMFNTS 1536 4.444 2. 144 6626.000 NON-WFIGHTED UPPER O3SERVE0 . PER CENT CUMULATIVE CUMULATIVE MULTIPLE DEVIATION LIMIT FREQUENCY OF TOTAL PERCENTAGE REMAINDER OF MEAN FROM MEAN 0 ' 0' '".00 ' .0 "' " " 100.0 - -.000 " • -2.072 " 1 0 .00 .0 100.0 .225 -1.605 2' 378 24.60 24.6 75.3 .450 -1.139 r 3 26 7 17.33 41.9 58. 0 . 675 -.573 4 26 0 16.92 58.9 41.0 .900 -.107 5 16 2 10.54 69. 4 30. 5 1.125 .25? 6 137 e.91 78.3 21.6 l . ' f O .725 7 51 3.3 4 82.2 17.7 1.57 5 1.191 \ 9 273 17.77 100.0 . 0 1.800 . 1.653 ? REMAINING FRETUENC IES ARE ALL ZERO TABLE 25 - . ENTRIES IN TA3LE MEAN ARGU MENT STANDARD DEVIATION SUM OF ARGUMF NTS 14258 39 .724 19.937 567985.000 NTN-WEIGHTEO UPPER 03SERVE0 PFR CENT" CUMUL AT IVE CUMULATIVE MULTIPLE DEVIATION ' LIMIT FREQJENCY OF TOTAL PERCENTAGE REMAINDER OF MEAN FROM H=AN 0 0 .00 .0 100.0 -.000 -1.99? 2 12 .08 .0 59. 9 . 050 -1.892 4 41 .28 .3 99.6 .100 -1.791 6 276 1.93 2. 3 97.6 .151 -1.691 . 9 137 .95 3.2 96.7 . 201 -1.59! 10 42 2 2.95 6.2 93.7 .251 -1.490 ' 12 276 1.93 8.1 91 .8 .302 -1.390 14 244 1.70 9.8 90. 1 . 352 -1.290 15 622 4.3 5 14.1 85.8 .402 -1.159 18 326 2.26 16. 4 82. 5 .45 3 -1.059 20 601 4.20 20.6 79. 3 . 503 opt) 22 370 2.58 23.2 76.7 .553 -.689 24 ' " 3 1 4 " " ~" 2.19' 2 5.4 74. 5 .604 -.788 26 621 4.34 29.8 70. 1 .654 -.666 28 32 9 2.30 32. 1 67.8 .704 -.566 3 0 59 2 4.14 36. 2 63.7 . 75 5 -.487 32 . 391 2.73 38.9 61.0 . 805 - . 367 • 34 306 2.14 41.1 58.9 .855 -.287 36 791 " 5.53 46.6 "53.3" . 906 - . ! f6 39 407 2.94 49.5 50.4 .956 - .0 36 40 65 0 4. 54 54.0 45.9 1.006 .013 42 i 9 2 . 2.74 56.7 4S.2 1.05/ .1 JA 44 363 2.53 59.3 40.6 1.107 .214 46 794 5.55 6 4 . e 35.1 1.157 .314 48 438 3.06 67.9 32.0 1.208 .415 50 534 3.73 71.6 28.3 1.258 .515 52 3? 5 2.76 74.4 25. 5 1.309 .615 54 275 1.93 76. 3 2 3.6 1.35V . /15 56 588 4. 11 80.4 19.5 1.409 .916 58 353 2.46 92.9 17.0 1.460 .416 60 29 7 2.07 85.0 14.9 1.510 l . U U 62 199 1.39 86.4 12.5 1 .560 1.117 64 128 .89 87.3 12.6 1.61 1 1.217 66 292 2.04 89.3 1U.6 l . to j l.il 1 58 18 5 1.29 90. 6 9. 3 1.71 1 1.416 % 70 175 1.22 91.8 8. 1 1.762 1.516 72 202 "1.41 93.2 6.7" J.B12 "1.61B 74 117 .31 94. 1 5. 9 1. 862 1.719 % : 76 219 1.53 95.5 4.3 1.913 1.519 73 156 1.09 ^6. 7 3.2 1.V63 80 94 .65 97.3 2.6 2. 01? 2.020 % j 92 70 .48 97.9 2. 1 2. 064 2.1 20 84 55 .24 s>a. 1 I.H 2. 114 2 . t c U 86 79 .55 98.6 1.3 2.164 2.321 % 88 64 .44 99.1 .8 2.215 2.421 n r, O o c © 94 96 c 3 100 102 104 _106_ 109 110 112 4 7 24 4 13 0 11 .32 .16 .02 .12 . 0 0 •_07_ .00 .00 .04 .OO" .04 .00 99.4 99.6 99.6 99.7 99.7 99. B 99.3 99. 8 99.9 99.9" 99.9 99. 9 2. 265 2.315 2.366 2.416 2.466 2.517 ~275"6T" 2.613 2.66 3 "2.719" 2.769 2.319 2.52! 2. 121 2.122 2.522 2. =22 3.02 3 3.127T 3.2 2 3 . 3.324 "3. 42 4" 3.524 3. 625 114 116 11 3 120 REMAINING FREQUENCIES .00 .02 • 00_ .02 ' 99.9 99.9 99. 9_ "100.0 2.859 2.920 2.S7C "2"."02"C~ ARE ALL ZEFO 3. 725 3.325 3.926 4"."02 6~ cm CD Q Q a o o O ! C V c 49 STATISTICS OF INTEREST Table 2 Frequency d i s t r i b u t i o n of the member wait ing time. The in terva l length for each class i s 0.5 minutes. For example, the fol lowing output shows that for the 14298 members which were serviced a t o t a l of 832 waited between 6.5 and 7.0 minutes before re-ceiving service . Table 10 Frequency d i s t r i b u t i o n of the number of members wait ing i n the system. For example, 62.4% of the time there were at most 15 members wait ing for serv ice . Table 20 Frequency d i s t r i b u t i o n of the number of wickets u t i l i z e d . For example, 60% of the time there were at least 4 t e l l e r s avai lable for serv ice . Table 25 " Frequency d i s t r i b u t i o n of the queue length per avai lable wicket . The in terva l length for each class i s 0.2 members. For example the fol lowing output shows that 91.8% of the time, the average queue length was at most 7. APPENDIX 2 MODEL DOCUMENTATION Segment 1 This segment simulates the a r r i v a l , queuing, and service to members. Generate V$Membr, FN$XP0IS [Gbye ) No. Assign 3,FN$PAR 3 Assign 4,FN$PAR4 Queue 2 GATE SNF 1 Members enter. A r r i v a l s are poisson with (Time varying) . Can c redi t union accommo-date addit ional members? Table which' waiting time w i l l be entered Table which length w i l l be entered Enter s ingle l i n e queue Wait for avai lable t e l l e r III P r i o r i 7 ty Assign 5,X626 Increase p r i o r i t y I n i t i a l i z e loop, store in parameter 5. Decrement Parameter Is there a wicket which has not been examined? Is there a wicket in ; ,F the range P5-X626 which is not in use? Is the wicket open? 2 Segment 1 (Continued) Appendix 2 CM LT) 2 Logic S P2 Close the wicket Enter 1 Reduce number of avai lable t e l l e r s . Sei ze P2 Seize t e l l e r Depart 2 Leave queue Tabulate P3 Record waiting time in table according to a r r i v a l rate . Tabulate P4 Record queue length in table according to a r r i v a l rate . - Model Documentation Tabulate 2 Tabulate 10 Save value 610+,1 Save value 611+, Ml Save value 620+,Q2 Advance VSIMPER Record wait ing time in general table . Record queue length in general tab le . Update number of members serviced. Add members waitinq time to X611. Add queue length to X620 Wait for service . C O L O Segment 1 (Continued) Appendix 2 - Model Documentation 3 (GbyeVf Logic R P2 Release P2 Leave 1 Reopen Wicket Release t e l l e r Increase number of avai lable t e l l e r s Terminate I Leave Credit Union Segment 2 Append This segment calculates arid records average queue length, average waiting time, member a r r i v a l ra te , and n o . - o f t e l l e r s ava i lab le during the period;-I t also determines whether t e l l e r s are overstaffed or under staffed and takes the appropriate ac t ion . Generate 3000, , , ,2 Savevalue V$Tellrt,X601 Savevalue V$Time, V$RATEl Savevalue V$Waitit,V$wai| Timer set for every 5 minutes, Record no. of t e l l e r s avai lable during period. Record a r r i v a l rate during period. Savevalue V$Lngth,V$Line Record waiting time during t period Record Queue length during period. - Model Documentation 1 Savevalue 619,1 Savevalue 610,1 Savevalue 611,1 Savevalue 620,0 Assign 1 ,X626 Increment counter for regis ter ing save values. I n i t i a l i z e accumulated member s t a t i s t i c s . I n i t i a l i z e accumulated waiting time s t a t i s t i c s . I n i t i a l i z e accumulated queue s t a t i s t i c s . Record no. of ava i lab le wickets , plus 1 in parameter 1. IStrt Are there too many t e l l e r s avai lable? Segment 2 (Continued) Yes Appendix 2 Decrement counter T e s t G \ PI i s greater than zero? Yes Is there more than 1 t e l l e r available? Is the t e l l e r free? Is the wicket open? Record the decrease in avai lable t e l l e r s Model Documentation Logic S P2 Enter 1 Close the wicket Reduce the no. of avai lable t e l l e r s Wait 1 time unit Return to^ s tar t Record ,no. ,of avai lable wickets plus 1 in parameter 1. Segment 2 (Continued) Appendix 2 [Here V - ^ f ^ 6 ^ ^ Is there a closed wicket? Are there too few t e l l e r s available? Decrement counter H e r e U l l - - ^ p i t Q U > > Is PI greater than zero? Is the t e l l e r free? Yes Logic R PI Is the wicket closed? Open a wicket. Model Documentation 3 Savevalue 601 + ,1 Leave 1 Record the increase in avai lab le t e l l e r s . Increase no of avai lable t e l l e r s Wait 1 time uni t . insfifer Return to there. ier / Advance 1 Wait 1 time uni t . Terminate.: J Leave model Segment 3 This segment changes the mean a r r i v a l rate . Generate 9000 Timer set to 15 minute intervals Save value 603+.1 Record a r r i v a l rate during interval Savevalue 604,NXI(X602,Xd03) Change the a r r i v a l rate Terminate i Leave model. Segment 4 Appendix 2 - Model Documentation This segment ends business for the day. Generate V$Day, , , ,5 Savevalue 603,1 Savevalue 604,MXI(1,1) Savevalue 609+,V$Day Savevalue 619,1 Timer set to length of day I n i t i a l i z e index of materix^for next day I n i t i a l i z e a r r i v a l rate for next day. Increase tota l time to date I n i t i a l i z e savevalue index Terminate C O Segment 5 This segment terminates the simulation Appendix 2 - Model Documentation Advance 1 Generate V$Stdy Assign 1,449 Savevalue 606,150 Timer set for end of simulation No. of averages to be computed. I n i t i a l i z e s tar t ing point for computation Wait 1 time unit Savevalue 606+.1 Savevalue . 607,X=X606 Next Savevalue 1 Savevalue X606,V$Avg Enter accumulated sum in X607 Enter average in Savevalue X606 A l l averages computed? APPENDIX 3 OTHER FACILITY PLANNING MODELS 60 OTHER MODELS 1. COMPANY: F i r s t National Bank, S t a t i s t i c s and Standards Sect ion. PURPOSE: To determine t e l l e r requirements for branch o f f i ces. MODEL DESCRIPTION: Analyt ic model based on t e l l e r a c t i v i t y and the corresponding work standards as defined from Methods-Time-Measurement (MTM) s tudies . CURRENT STATUS: Never been used, project cancel led. 2. COMPANY: Aer Lingus, I r i s h International A i r l i n e , Dublin A i r p o r t , I re land. PURPOSE: To determine manpower requirements for the a i r -l i n e booking o f f i c e s . MODEL DESCRIPTION: ( i ) Analy t i ca l model ( i i ) Assumptions - exponential service times - poisson a r r i v a l rate (mean a r r i v a l rate i s f ixed for 1 hour i n t e r v a l s ) . 61 OTHER MODELS (Continued) 2. COMMENTS: The model does not answer enough of the questions c r e d i t union managers wish to know about t h e i r f a c i l i t y requirements. 3. COMPANY: Woods, Gordon & C o . , Management Consultants, Tor-onto, Canada PURPOSE: To determine the number of t e l l e r s and wickets required in bank branches. MODEL DESCRIPTION: ( i ) Analy t i ca l model ( i i ) Requirements are determined by c o n t r o l l i n g the average waiting time in the queue. ( i i i ) Assumptions - exponential .service times - poisson a r r i v a l rates COMMENTS: The model does not answer enough of the questions c r e d i t union managers wish to know about t h e i r f a c i l i t y requirements. APPENDIX 4 A CASE STUDY A DETERMINATION OF THE REQUIRED TELLER FACILITY FOR CAMPBELL RIVER DISTRICT CREDIT UNION 63 Table of Contents SCOPE OBJECTIVES METHOD DATA COLLECTION - Present t e l l e r time schedule - Member service time - Other t e l l e r a c t i v i t i e s - Presently avai lable wickets - Member a r r i va l s - General assumptions EVALUATIONS - Te l l e r and wicket requirements - Present level of service - Effect of growth on wicket and t e l l e r requirements - Te l l e r schedule - Area requirements for wickets and member wait ing for service CONCLUSIONS RECOMMENDATIONS APPENDICES 4A. Space Required to Accommodate Members and Wickets 4B. Level of Service for Peak Day of the week 4C. Te l l e r Schedule 64 1. SCOPE Determine for Campbell River D i s t r i c t Credit Union t h e i r f a c i l i t y re-quirement i n order to provide adequate member service for present credi t union s ize and for a 50% and 100% growth in membership. 2. OBJECTIVES For Campbell River D i s t r i c t Credit Union, determine (at present c red i t union s ize and for a 50% and 100% growth): 2.1 The required number of wickets 2.2 The required number of t e l l e r s . 2.3 T e l l e r schedule 2.4 Member service level in terms of wait ing time and queue length 2.5 The required member service area 2.6 The required member wait ing area 65 3. METHOD Simulation Model In order to achieve the ob jec t ives , the member service process (length of time before service and the number of wait ing for service) was evaluated by a simulation model. A computer program was developed which simulates the t e l l e r services to members. The simulation was used for four main purposes: - to evaluate the present level of member service at Campbell River D i s t r i c t Credit Union. - to evaluate the e f fect on member service on increasing/decreasing the number of t e l l e r s at present and with a 50% and 100% growth. - to provide basic information required to determine the t e l l e r schedule. - to provide basic information required to determine the required member wait ing area and t e l l e r service area at present and with a 50% and 100% growth. 66 4. DATA COLLECTION In order to evaluate the queuing behavior and determine the level of member serv ice , the fol lowing information was required: 4.1 Present T e l l e r Time Schedule The present t e l l e r schedule according to the ex i s t ing pol icy i s des-cribed in Table 1. Table 1 T e l l e r Schedule Day of the Off ice Hours 1st S h i f t No. of 2nd S h i f t No. of Total No. Week i Staff S taf f Monday 9:00 - 1:00 8:30 - 1:30 3 - 3 Tuesday 1 0 : 0 0 - 5:30 9:00 - 5:30 7 - - 7 Wednesday 10:00 - 5:30 9:00 - 5:30, 7 - - 7 Thursday 10:00 - 5:30 9:00 - 5:30 7 - 7 Friday 10:00 - 6:00 9:00 - 6:00 3 9:00 - 5:30 3 6 Saturday 9:00 -12:30 8:30 - 1:30. 3 — 3 Total No. of Man days 33 NOTE: 1 part-time t e l l e r also avai lable on Friday between 9:00 - 2:00. 67 4.2 Member Service Time The average member service time is 2.1 minutes/member. 4.3 T e l l e r a c t i v i t i e s requires other than serving members A major part of each t e l l e r s work day is used for various other a c t i v -i t i e s than member s e r v i c e , as described in Table 2, f o l l o w i n g . Table 2 TELLER ACTIVITIES OTHER THAN MEMBER SERVICE DURING OFFICE HOURS Requi red Total Time Time per Per T e l l e r Acti v i t i e s Constraint Frequency Act i v i t y Per A c t i v i t y Ta l ly ing sums At reasonable intervals 3 10 min. 30 mini Balancing Before 3:00 p.m. 1 20 min. 20 min. Coffee Break At reasonable time 1 15 min. 15 ,min. Lunch Break At reasonable time 1 60 min. 60 min. Total T e l l e r Time: 125 min. per t e l l e r Note: 125 minutes or 28% of most t e l l e r work days are not avai lable for member service during regular o f f i c e hours. 68 4.4 The maximum number of avai lable wickets at Campbell River Dis-t r i c t Credit Union = 6. 4.5 Member a r r i v a l s at t e l l e r wicket during July 20 - August 4, 1973 The number of members entering Campbell River D i s t r i c t Credit Union to use the t e l l e r f a c i l i t y was measured for each 15 minute period during the day. Summary see Table 3, f o l l o w i n g . Table 3 MEMBER ARRIVAL SUMMARY Member A r r i v a l Rates Total Member Average Minimum Maximum Day of the Week A r r i v a l per Member A r r i v a l s A r r i v a l s Day A r r i val Per Hour Per Hour Per Hour F r i . , July 20 527 66 50 91 S a t . , July 21 224 64 33 87 Tues. , July 24 376 50 42 72 F r i . , July 27 434 54 33 84 Sat. , July 28 247 70 44 103 Mon., July 30 131 33 19 40 Tues. , July. 3:1 435 58 45 72 Wed., Aug. 1 339 45 28 57 Thurs . , Aug. 2 515 69 30 115 F r i . , Aug. 3 591 74 58 90 . S a t . , Aug. 4 226 65 38 87 For sample: 4045 59 19 115 69 4.6 General Assumptions - The length of time required by a t e l l e r to service members i s not dependent on the hour of the day. - The wait ing area required for a member = wickets length x 2 feet . 5. EVALUATIONS The evaluations i l l u s t r a t e how the s i tua t ion at Campbell River Dis -t r i c t Credit Union is best described within the model, s p e c i f i c a l l y : 5.1 T e l l e r a n d wicket requirements (within and between days). 5.2 Level of service to members. 5.3 Effect of growth on wicket and t e l l e r requirements. 5.4 T e l l e r schedule. 5.5 Area requirements for wickets and members wait ing for serv ice . 70 5.1 T e l l e r and wicket requirements 5.1.1 Present wicket requirements The maximum number of wickets which are required to provide s a t i s f a c -tory member service varies between days. See Table 4. Table 4 THE MAXIMUM NUMBER OF REQUIRED WICKETS Day of the Maximum number of week wi ckets requi red Thursday 5 Friday 4 Saturday 4 At present, 6 wickets are a v a i l a b l e , whereas, only 5 wickets are required to provide adequate member service . 5.1.2 Present t e l l e r requirements The maximum number of t e l l e r s required for serving members in each hour of the day of the week var ies . See Table 5. Table 5 TELLER REQUIREMENTS Day of the week 9-10 10-11 Hour of 11-12 12-1 the Day 1-2 2-3 3-4 4-5 5-6 Thursday Fri day Saturday 3 3 3 4 3 3 4 3 3 4 2 2 2 3 3 3 5 4 5 4 During the hour of 4:00 - 5:30, on Thursday, 5 t e l l e r s are required to provide adequate member service . On a l l other days, the maximum is 4 or l ess . 5.2 Level of service to members during the week Assuming that layout changes suggested for the ex i s t ing f a c i l i t y w i l l be implemented, there w i l l be no s i g n i f i c a n t changes in member serv-ice level during the week. The member service level w i l l be as described in Table 6. 72 Table 6 MEMBER SERVICE LEVEL Number of Members Member wait ing wai t ing/ava i lab le time in minutes Day t e l l e r Average Maximum Average Maximum Thursday 3 7 3 9 Friday 2 6 4 9 Saturday 3 7 4 9 Member service does not d i f f e r s i g n i f i c a n t l y during the busy, days of the week. NOTE:. Maximum implies that 99% of a l l values are equal to or less than the indicated value. 5.3 Effect of growth on wicket and t e l l e r requirements For a 50% and 100% increase in the demand made on the t e l l e r f a c i l -i t y , and assuming that members receive the same level of s e r v i c e , the fol lowing evaluations were made: 5.3.1 Wickets required to meek peak periods - present membership - 50% increase in membership 73 5. EVALUATIONS (Continued) - 100% increase in membership Increasing the number of wickets and changing the t e l l e r schedule so that the required number of t e l l e r s are avai lable w i l l s i g n i f i c a n t -ly improve the level of service to members. The fol lowing analysis gives a broad conceptual approach to the problem of determining the required number of wickets and member service . Level of service for present membership In this s e c t i o n , as i n previous sec t ions , the level of service i s determined by the member wait ing time and the number of members wait ing for serv ice . By increasing the number of wickets from 4 to 5 the wait ing time during the peak day i s s i g n i f i c a n t l y decreased as shown in Figure 1. The maximum wait ing time is decreased form 19 minutes to 11 minutes while the average wait ing time is decreased from 7 minutes to 4 min-utes. I t should be noted that any addit ional wickets would not s i g n i f i c a n t l y decrease the wait ing time. 74 5. EVALUATIONS (Continued) Member wait ing time (minutes) 30 28 26 24 22 20 18 16 \ 14 12 10 8 6 4 2 \ Max. (See footnote) Average -w -6 No. of wickets Figure 1 - Relationship between member wait ing time during the peak day and maximum avai lable wickets . Note: Maximum implies that 99% of a l l values are equal to or less than the plot ted value for the corresponding number of wickets . With an increase in the number of wickets from 4 to 5 , the number of members wait ing at each wicket during the peak day is s i g n i f i c a n t l y decreased. Figure 2 shows that the maximum number wait ing at each wicket i s decreased from 9 to 7, while the average number wait ing at each wicket i s decreased from 4 to 3. 75 5. EVALUATIONS (Continued) Queue Length (members) Max. queue (see footnote) Average queue No. of wickets Figure 2 - Relationship between members wait ing during the peak day at each wicket and maximum avai lable wickets . Note: Maximum implies that 99% of a l l values are equal to or less than the plot ted value for the corresponding number of wick-ets . Level of service for 50% growth in membership By increasing the number of wickets form 5 to 7, the wait ing time during the peak day decreases s i g n i f i c a n t l y , as shown i n Figure 3. The maximum waiting time is decreased from 25 minutes to 11 minutes while the average wait ing time is decreased from 9 to 5 minutes. 76 5. EVALUATIONS (Continued) Member wait ing time (minutes) 30 28. 26 7 No. of wickets Figure 3 - Relationship between.member wait ing time during the peak day and avai lable wickets for a 50% increase i n membership. 24 22 20 18 16 14 12 10 8 6 4 2 \ \ Max. Average Level of service for 50% growth in membership By increasing the number of wickets from 5 to 7, the number of members wait ing during the peak day at each wicket is s i g n i f i c a n t l y decreased. Figure 4 shows that the maximum number of wait ing at each wicket i s decreased from 12 to 7, while the average number wait ing at each wicket i s decreased from 5 to 3. 77 5. EVALUATIONS (Continued) 12 10 6 4 2 Queue length (members) \ -A/V-Maximum Average 5 6 7 No. of wickets Figure 4 - Relationship between members wait ing at each wicket during the peak day and maximum avai lable wickets . for a 50% i n -crease in membership. Level of service for 100% growth in membership By increasing the number of wickets from 8 to 9, the wait ing time decreases s i g n i f i c a n t l y , as shown in Figure 5. The maximum wait ing time is decreased from 17 minutes to 13 minutes while the average wait ing time is decreased from 6 minutes to 5 minutes. 78 5. EVALUATION (Continued) Member wait ing time (minutes) 30 28 26 24 22 20 18 Maximum 12 10 8 Average 6 —— 2 W - — — 8 9 10 No. of wickets Figure 5 - Relationship between wait ing time during the day and avai lable wickets for 100% increase in membership. Level of service for 100% growth in membership By increasing the number of wickets from 7 to 9, the number of mem-bers wait ing during the peak day at each wicket i s s i g n i f i c a n t l y de-creased. Figure 6 shows that the maximum number wait ing at each wicket i s decreased from 12 to 7, while the average number wait ing is decreased from 5 to 3. 79 EVALUATIONS (Continued) 12 10 8 6 4 Queue Length (members) • M r Maximum Average 8 9 Number of wickets 10 Figure 6 - Relationship between members wait ing at each wicket during the peak day and the maximum avai lable wickets for a 100% growth in membership. 5.4 T e l l e r schedule for busy day (Thursday, Fr iday, Saturday). See Appendix C. 5.5 Area requirements for wickets and members wait ing for service The amount of space required to accommodate members wait ing is depen-dent on the number of members wait ing for service at any given time. 80 5. EVALUATIONS (Continued) If too few wickets are provided so that the Credit Union cannot meet the demand during peak periods, then large queues w i l l develop and the wait ing time w i l l be very large. I f too many wickets are provided so that the Credit Union never uses a l l wickets even during peak per iods , then again space is poorly u t i l i z e d . The fol lowing (see Figure 7) shows the space required to accommodate wickets and members for a given number of wickets . As seen below, the minimum space required to accommodate wickets and members i s 550 sq f t . , at present, and 840 sq. f t . when the membership has i n -creased by 50%. sq. f t 1300 1200 1100 1000 900 800 700 600 500 400 \ 100% Growth* 50% Growth* \ Present Membership-L-A/V C U . presently operating with 6 wickets 7 8 Wickets 10 11 * Growth = membership 81 5. EVALUATIONS (Continued) Figure 7 - Relationship between space required for the t e l l e r f a c i l -i t y , ( i . e . Wickets and member wait ing area and number of wickets) The e x i s t i n g f a c i l i t y with 6 wickets, and member wait ing area i s suf-f i c i e n t for a growth of 30%. 6. CONCLUSIONS 6.1 The present f a c i l i t y with 6 wickets and member wait ing area i s s u f f i c i e n t for a 30% growth in membership. 6.2 In order to provide adequate member serv ice , the Campbell River D i s t r i c t Credit Union w i l l require the fol lowing number of wickets as the membership increases: Membership Wickets Required Present - 8200 5 8300 - 10000 6 10100 - 12000 7 12100 - 13900 8 14000 - 15900 9 6. CONCLUSIONS (Contined) 82 Note: I f the Manager feels that 9 wickets in one branch i s too many then he should consider a branch o f f i c e . 6.3 The fol lowing t e l l e r schedule would benefit the Credit Union by increasing t e l l e r u t i l i z a t i o n and reduce member wait ing time s i g n i f -i c a n t l y . 6.3.1 T e l l e r schedule for Thursday NUMBER OF TELLERS PER SHIFT Schedule S h i f t 1 S h i f t 2 9:00-5:30 1:30-5:30 Present 5 50% growth in members 7 100% growth i n members 6 4 6.3.2 T e l l e r schedule for Friday NUMBER OF TELLERS PER SHIFT Schedule S h i f t 1 9:00-6:00 Present 6 83 6. CONCLUSIONS (Continued) 6.3.3 T e l l e r schedule for Saturday NUMBER OF TELLERS PER SHIFT Schedule S h i f t 1 8:30-12:30 Present 4 100% growth 9 7. RECOMMENDATIONS 7.1 I t is not necessary to expand the present t e l l e r f a c i l i t y (wick-ets and wait ing area) u n t i l a 30% growth in membership is r e a l i z e d . 7.2 Increase the number of presently avai lable t e l l e r s on Saturday to 4. 7.3 Decrease the number of presently avai lable t e l l e r s on Thursday to 5. 7.4 Decrease the number of presently avai lable t e l l e r s on Friday to ' 5. 7.5 Re-evaluate the number of t e l l e r s required on Monday, Tuesday, RECOMMENDATIONS (Continued) and Wednesday, since the demand made on the t e l l e r f a c i l i t y during these days i s less than the demand made on any of the above days. SUB-APPENDICES TO APPENDIX 4 Appendix 4.A Space required to accommodate members and wickets Figure 1 - Space requirements for present member-s h i p , 50% growth and 100% growth. Appendix 4.B Level of service for peak day of the week Relationship between maximum.ayaiTable wickets and member wait ing time. Figure 2 - Present membership Figure 3 - 50% increase in membership Figure 4 - 100% increase in membership Relationship between maximum avai lable wickets and the number of members wait ing at each avai lable wicket . Figure 5 - Present membership Figure 6 - 50% increase in membership Figure 7 - 100% increase in membership 86 SUB-APPENDICES TO APPENDIX 4 (Continued) Relationship between maximum avai lable wickets and the number of members w a i t i n g . Figure 8 - Present membership Figure 9 - 50% increase in membership Figure 10 - 100% increase in membership Appendix 4.C T e l l e r Schedule T e l l e r schedule for Thursday Figure 11 - Present membership Figure 12 - 50% growth in membership Figure 13 - 100% growth i n membership T e l l e r schedule for Friday Figure 14 - Present Membership T e l l e r schedule for Saturday Figure 15 - Present Membership Figure 16 - 100% increase i n membership Figure 1 - Relationship between space required for the t e l l e r f a c i l i t y ( i . e . wickets and member waiting area) and the number of wickets. 100% Growth' \ \ \ 50% Growth* \ Present MembershiD - C U . presently operating here. 400 Wickets *Growth = Membership Appendix 4.A Minutes Figure 2 - Present Membership - Relationship between Maximum avai lab le wickets and member waiting time. 40 minutes \ \ \ \ \ Maximum \ \ Average 3 4 5 Wickets 6 7 Appendix 4.B Figure 3 - 50% increase in membership - Relationship between maximum avai lable wickets and member waiting time. \ \ \ ^ Maximum .Average 5 6 7 Wickets 8 Appendix 4.B tes Figure 4 - 100% increase in membership - Relationship between maximum avai lable wickets and member waiting time. Maximum Average 9 10 11 Wickets Appendix 4.B cr> Members 12 11 10 9 8 7 6 5 4 3 2 Figure 5 - Present Membership - Relationship between maximum avai lable wickets and the number of members waiting at each avai lable wicket Queue length at each wicket Present \ \ \ \ \ Maximum \ \ Average 4 5 Wickets Appendix 4.B Members. 12. 11 10 9 8 7 6 5 4 Figure 6 - 50% increase in membership - Relationship between maximum avai lab le wickets and the number of members waiting at each avai lable wicket \ \ \ \ Maximum Average Wickets Appendix 4.B Members 12 11 10 9 8 Figure 7 100% increase in membership - Relationship between maximum avai lab le wickets and number of members waiting at each avai lab le wicket . \ \ \ \ Maximum N Average 9 10 Wickets 11 Appendix 4.B Figure 8 - Present Membership of members wait ing. Relationship between maximum avai lab le wickets and the number \ \ \ \ Maximum Average 3 4 5 Wickets 6 7 Appendix 4 .B Members 55 50 45 40 35 30 25 20 15 10 5 Figure 9 - 50% increase in membership - Relationship between maximum avai lable wickets the number of members waiting \ \ \ Maximum Average 5 6 7 Wickets 8 Appendix 4.B VO Members 60 55 50 45 40 35 30 25 20 15 10 "5 Figure 10 - 100% increase in membership - Relationship between maximum avai lable wickets and the number of members waiting \ Maximum Average 9 Wickets 1 0 11 Appendix 4-B Tel lers Figure 11 - Te l ler Schedule for Thursday - Schedule - 5 t e l l e r s 9:00 - 5:30 Other t e l l e r act ivi t i -es Opening Preparation] Member Service 10 ~ 11 12 " 1 2 (Hour: of the day) Appendix 4.C Figure 12 - 50% growth i n membership - T e l l e r schedule f o r Friday Schedule 7 t e l l e r s 9:00 - 5:30 T e l l e r s 7 6 5 Opening Preparation 10 11 Other T e l l e r A c t i v i t i e s Member Service 12 (Hour of day) Appendix 4.C Figure 13 - 100% growth 5 in membership - T e l l e r Schedule for Thursday Schedule - 6 t e l l e r s 9:00 - 5:30, 3 t e l l e r s 1:30 - 5:30 ers 9 8 7 6 5 4 3 2 1 Opening -• Preparation Other Te l l e r A c t i v i t i e s Member Service 10 11 12 1 (Hour of Day) Appendix 4.C o o Figure 14 - Present Membership - Te l ler Schedule for Friday Schedule - 5 t e l l e r s 9:00 - 6:00 T e l l e r s , 5 4 3 2 1 Opening Preparation! 10 11 12 Other Te l l e r A c t i v i t i e s Member Service T — 2 (Hour of the Day) Appendix 4.C Tel le rs 5 4 3 2 1 Figure 15 - Present Membership - T e l l e r Schedule for Saturday Schedule - 4 t e l l e r s 8:00 - 12:30 Other Te l ler A c t i v i t i e s Member Service 10 11 12 (Hour of the day) "3 Appendix 4.C CM O Tel 1ers 9 8 7 6 5 4 3 2 1 Figure 16 - 100% increase in membership - Tell.er' ;Schedule for Saturday Schedule - 9 t e l l e r s 8:30 - 12:30 E o + J ro S-(O CD-CD S-Q_ o> E •r-•E CD Q . O Other t e l l e r a c t i v i t i e s Member Service 10 11 (Hour of the Day) 12 Appendix 4.C 

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