First City National Bank In March 1987, David Craig, vice president of operations for First City National Bank of Philadelphia was considering a change in teller operations. Currently, the bank's tellers were arranged in pods to handle customer transactions. There were four pods containing three teller stations each. One pod was used primarily for savings accounts since some savings transactions took longer than other types of deposits or withdrawals. The major problem with the pod system was that one pod might be crowded while another was vacant. The distance between pods was such that the customers were unwilling to move from one to another. Mr. Craig was considering two alternatives to the pod system. The first was a single-line teller arrangement as shown in Exhibit 1. Using this plan, all customers would wait in a single line until a teller became available. The person at the head of the line would then move to the open teller. Mr. Craig thought that 10 tellers would be required to handle the bank's usual business. However, he could not be sure of the exact number without further study. page 1 Exhibit 1 also shows the second alternative teller arrangement. Using this more conventional plan, the customers would form separate lines in front of each of the teller windows. Thus for 10 tellers, a total of 10 different lines could be formed. In evaluating these alternatives, several issues were of utmost importance. First, Mr. Craig was concerned with both customer waiting time and teller efficiency. On the basis of past experience, Mr. Craig felt that more than 3 minutes of waiting time would be unacceptable to most customers. He also felt that teller utilization should be as high as possible, perhaps in the 80 to 90 percent range. Since demand varied during the day, the number of tellers provided would have to vary to meet the customer-service and tellerutilization goals. 70 60 50 Day 4/20/91: Friday Time Period: 11:45-12:45 Total Arrivals: 431 customers Average Time Between Arrivals: 8.4 Sec 40 30 20 10 0 Interarrival time . Exhibit 2 - Histogram of Interarrival times The statistical distribution of service time and arrival time is shown in Exhibits 2 and 3. The service time averages 45 seconds per customer and does not vary by time of day. On the other hand, the average time between arrivals does vary with the time of day. For example, between 11:45 and 12:45 on one particular day sampled, 431 customers arrived at the bank, with an average of 8.4 seconds between customers. page 2 70 60 50 Avg. = 45.5 seconds s= 37.8 seconds n = 308 observations 40 30 20 10 0 < 10 20 30 40 50 60 70 80 90 100 110 120 Service Time (Seconds) Exhibit 3 - Histogram of Service Times To estimate the average arrival rate during different times of the day, the data in Exhibit 4 were collected. Over the period between November 1, 1986, and February 28, 1987, arrivals were counted for each half-hour period. The days were then divided into normal days, peak days, and superpeak days, depending on the intensity of the flow. Although the average number of arrivals varied during each hour of the day, the statistical pattern of arrivals was stable during each particular hour. page 3 Normal Days Time of Day 8-8:30 8:30-9 9-9:30 9:30-10 10-10:30 10:30-11 11-11:30 11:30-12 12-12:30 12:30-1 1-1:30 1:30-2 2-2:30 2:30-3 3-3:30 3:30-4 4-4:30 4:30-5 5-5:30 Total Number of Arrivals 803 919 1207 2580 2599 2870 3384 4548 5804 5351 4355 3632 2321 1935 2151 2115 2291 2054 1598 Average Arrival Rate* 19 22 29 63 63 70 83 111 142 131 106 89 57 47 52 52 55 50 39 Peak Days Total Number of Arrivals 625 758 863 2033 2237 2283 2625 4060 5329 4923 3983 3150 2012 1960 2064 2238 2340 2191 1763 Average Arrival Rate* 22 27 31 72 80 82 94 145 190 176 142 113 72 70 74 80 84 78 63 Superpeak Days Total Number of Arrivals 331 418 571 1228 1382 1337 1577 2325 2908 2724 2271 1991 1282 1206 1250 1328 1346 1216 924 Average Arrival Rate* 25 32 44 94 106 103 121 179 224 210 175 153 99 93 96 102 104 93 71 Total normal days = 41, total peak days = 28, total superpeak days = 13. *The total number of arrivals is divided by the number of days to arrive at the average arrival rate. Exhibit 4 Chart Of Average Customer Arrival Rates In order to arrive at a decision, Mr. Craig requested an analysis of the single- and multiple-line teller arrangements. For a given number of tellers, Mr. Craig wanted to know which arrangement provided the best customer service. He also specified that the analysis should include a calculation of the number of tellers required at various times of the day, so that a teller staffing plan could be devised. In addition to the statistical analysis, Mr. Craig wondered what the customer reaction to the single-line or double-line arrangement might be. Would the appearance of a long single line drive customers away, or would the customers perceive fast service from the rapidly moving line? Mr. Craig also wondered what the other advantages and disadvantages of the single line relative to the multiple lines might be. page 4 DISCUSSION QUESTIONS 1. Which alternative arrangement of teller lines should Mr. Craig select? Support your recommendation with appropriate analysis and consideration of customer reaction. 2. For the alternative you recommend, develop appropriate staffing levels for each hour of the day. 3. Should other alternatives, not described in the case, be considered? page 5 QUEUING TEMPLATES 1995 by David W. Ashley Revised September 3, 2001 This workbook computes queuing results for the following models: M/M/s M / M / s with finite queue length M / M / s with finite arrival population M/G/1 Click on the page tab to use the model of your choice. Enter the required parameters in the boxes. Parameters for all models are initially linked to those entered for M/M/s. Calculations are limited to 170 servers in the M/M/s model, a total of 170 servers plus queue capacity in M/M/s with finite queue length, and a population of 500 (170 servers) in M/M/s with finite arrival population. M/M/s P r o b a b i l i ty Arrival rate Service rate Number of servers Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits 1 2 1 Assumes Poisson process for arrivals and services. 50.00% 0.5000 0.5000 1.0000 0.5000 1.0000 0.5000 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 910 12 14 16 18 2 0 1 2 3 4 5 6 7 8 9 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 11 13 15 17 19 2 2 2 2 2 2 2 2 2 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98100 NUMB ER IN SYSTEM P r o b a b i l i ty M/M/s with Finite Queue Arrival rate Service rate Number of servers Maximum queue length Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits Probability that a customer balks 1 2 1 169 50.00% 0.5000 0.5000 1.0000 0.5000 1.0000 0.5000 0.0000 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 101112 13141516 1718192021222324252627282930 3132 33343536 37383940 4142 43444546 47484950 5152 53545556 575859606162636465666768697 0 1 2 3 4 5 6 77 8 98081828384858687 888990919293949596979899 7 7 7 7 7 77 7 100 NUMB ER IN SYSTEM P r o b a b i l i ty M/M/s with Finite Population Arrival rate Service rate Number of servers Population size Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits overall arrival rate 0.002 2 1 500 (per customer) (per server) 1 49.90% 0.5010 0.4931 0.9921 0.4941 0.9941 0.4990 0.6 0.4 0.2 0 0 2 4 6 8 101214 1618 20 22 24 26 28 303234 3638404244 4648505254 NUMB62 64IN 687 07 27 47 67 880 82 84 86 88 90 92 94 96 98100 104 108 112 116 120 124 128 13234 138 40 4244 148 50 5254 158 162 64 168 7 1717 17 17 8 80 184 188 192 196 200 204 208 212 216 220 224 228 232 236 240 244 248 252 256 260 264 268 07 27 4 67 8 282 286 290 294 298 302 306 310 1214 318 322 326 330 3234 338 40 4244 348 50 5254 358 362 366 37 3737 37 37 8 80 384 388 392 3 5658 60 ER 66 SYSTEM 102 106 110 114 118 122 126 130 1 136 1 1 1 146 1 1 1 156 160 1 166 1 0 2 4 6 1 182 186 190 194 198 202 206 210 214 218 222 226 230 234 238 242 246 250 254 258 262 266 27 2 27 2 280 284 288 292 296 300 304 308 3 3 316 320 324 328 3 3 336 3 3 3 346 3 3 3 356 360 364 368 0 2 4 6 3 382 386 390 394 0 384 388 392 396 400 404 408 41214 418 20 424 428 43234 438 40 4244 448 50 5254 458 462 64 468 7 4747 4 7 47 8 80 484 488 492 496 500 382 386 390 394 398 402 406 410 4 416 4 422 426 430 4 436 4 4 4 446 4 4 4 456 460 4 466 4 0 2 6 4 482 486 490 494 498 M/G/1 average service RATE Arrival rate Average service TIME Standard dev. of service time Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system 1 0.5 0.5 2 50.00% 0.5000 0.5000 1.0000 0.5000 1.0000