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Customers arrive to the checkout area of New Navy a rate of 80 customers per hour. T arrival times are exponentially distributed. At the checkout area, the service time customer is exponentially distributed, and on average each cashier can serve 10 custo hour. The store has hired 12 cashiers to checkout customers and the checkout area co a single pool of cashiers with one common line in front of them. What is the probability that a customer's service time is less than 12 minutes?

Customers arrive to the checkout area of New Navy a rate of 80 customers per hour. T arrival times are exponentially distributed. At the checkout area, the service time customer is exponentially distributed, and on average each cashier can serve 10 custo hour. The store has hired 12 cashiers to checkout customers and the checkout area co a single pool of cashiers with one common line in front of them. What is the probability that a customer's service time is less than 12 minutes?

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter12: Queueing Models
Section12.5: Analytic Steady-state Queueing Models
Problem 9P
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1. Customers arrive to the checkout area of New Navy a rate of 80 customers per hour. The inter- arrival times are exponentially distributed. At the checkout area, the service time of each customer is exponentially distributed, and on average each cashier can serve 10 customers per hour. The store has hired 12 cashiers to checkout customers and the checkout area consists of a single pool of cashiers with one common line in front of them. a. What is the probability that a customer's service time is less than 12 minutes? b. How long do customers have to wait in the queue on average, in seconds? Please use the La table (not the M/M/s spreadsheet), and show your work. c. What is the average total number of customers in the checkout area? This includes the customers who are waiting to be checked out and the customers being checked out. Please use the La table (not the M/M/s spreadsheet), and show your work. d. New Navy is considering having 13 cashiers instead of 12. An extra cashier will cost an extra $10/hour. New Navy believes that for each second that the average waiting time in the queue improves, their revenue will increase by $1/hour. (Note that this improvement is in total, not per customer.) Should they hire an extra cashier to have 13 instead of 12? (You can use the MMS spreadsheet for this question, but please explain your answer.) e. Suppose now that New Navy decides to replace the single common line for its customers with 12 separate lines, one line in front of each of the 12 cashiers. What is now the average number of customers waiting to be checked out? This only includes the customers who are waiting to be checked out (not the customers being served).
Transcribed Image Text:1. Customers arrive to the checkout area of New Navy a rate of 80 customers per hour. The inter- arrival times are exponentially distributed. At the checkout area, the service time of each customer is exponentially distributed, and on average each cashier can serve 10 customers per hour. The store has hired 12 cashiers to checkout customers and the checkout area consists of a single pool of cashiers with one common line in front of them. a. What is the probability that a customer's service time is less than 12 minutes? b. How long do customers have to wait in the queue on average, in seconds? Please use the La table (not the M/M/s spreadsheet), and show your work. c. What is the average total number of customers in the checkout area? This includes the customers who are waiting to be checked out and the customers being checked out. Please use the La table (not the M/M/s spreadsheet), and show your work. d. New Navy is considering having 13 cashiers instead of 12. An extra cashier will cost an extra $10/hour. New Navy believes that for each second that the average waiting time in the queue improves, their revenue will increase by $1/hour. (Note that this improvement is in total, not per customer.) Should they hire an extra cashier to have 13 instead of 12? (You can use the MMS spreadsheet for this question, but please explain your answer.) e. Suppose now that New Navy decides to replace the single common line for its customers with 12 separate lines, one line in front of each of the 12 cashiers. What is now the average number of customers waiting to be checked out? This only includes the customers who are waiting to be checked out (not the customers being served).
Expected Number of People Waiting in Line (L) for Various Values of S and X/μ 1 0.10 0.0111 0.15 0.0264 0.0006 0.20 0.0500 0.0020 1.0 1.2 14 1.6 1.8 20 0.25 0.0833 0.0039 0.30 0.1285 0.0069 0.35 0.1884 0.0110 0.40 0.2666 0.0166 0.45 0.3681 0.0239 0.0019 0.50 0.5000 0.0333 0.0030 0.55 0.6722 0.045 0.0043 0.60 0.9090 0.0593 0.0061 0.65 12071 0.0767 0.0084 0.70 1.6333 0.0976 0.0112 0.75 2.2500 0.1227 0.0147 0.80 3.2000 0.1523 0.0189 0.85 4.8165 0.1873 0.0239 0.0031 0.90 8.1000 0.2285 0.0300 0.0041 0.95 18.0500 0.2767 0.0371 0.0053 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 40 42 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.6 6.8 7.0 7.2 74 7.6 7.8 8.0 8.2 84 86 8.8 9.0 92 94 9.6 2 9.8 10 3 4 5 0.3333 0.0454 0.0067 0.6748 0.0940 0.0158 13449 0.1778 0.0324 0.0059 2.8441 0.3128 0.0604 0.0121 7.6731 NUMBER OF SERVICE CHANNELS, S 7 6 0.5320 0.1051 0.0227 0.0047 0.0090 0.8888 0.1730 0.0390 0.0158 1.4907 0.2770 0.066 2.1261 4.9322 12.2724 0.4205 0.1047 0.0266 0.0065 0.6581 0.1609 0.0425 0.0110 1.0000 0.2411 0.0659 0.0180 8 1.5282 0.3541 0.0991 0.0282 0.0077 2.3855 0.5128 0.1452 0.0427 0.0122 3.9060 0.7365 0.2085 0.0631 0.0189 7.0893 1.0550 0.2947 0.0912 0.0283 16.9366 1.5181 0.4114 0.1292 0.0412 0.5694 0.1801 0.0590 2.2164 3.3269 5.2675 0.7837 0.2475 0.0827 1.0777 0.3364 0.1142 9.2885 1.4857 0.4532 0.1555 21.6384 2.0708 0.6071 0.2092 9 0.0084 0.0127 0.0189 0.0273 0.0389 0.0541 0.0742 4.4471 1.3471 6.3133 1.7288 9.5102 2.2324 16.0379 2.9113 35.8956 3.8558 10 0.0087 0.0128 0.0184 0.0260 5.2264 7.3441 11 1.6364 2.0736 10.9592 2.6470 18.3223 3.4160 40.6824 4.4805 0.1006 0.0361 0.0125 0.1345 0.0492 0.0175 2.9375 0.8102 0.2785 43004 1.0804 0.3680 6.6609 1.4441 0.5871 0.1779 0.0663 0.0243 0.0085 11.5178 1.9436 0.6313 0.2330 0.0683 0.0330 0.0119 26.3726 2.6481 0.8225 0.3032 0.1164 0.0443 0.0164 3.6878 1.0707 0.3918 0.1518 0.0590 0.0224 5.2979 1.3967 0.5037 0.1964 8.0768 1.8040 0.6454 0.2524 0.1008 0.0398 0.0153 13.7992 2.4198 0.8247 0.3222 0.1302 0.0523 0.0205 31.1270 3.2441 1.0533 0.0775 0.0300 0.0113 12 exhibit 7A.9 13 0.4090 0.1666 0.0679 0.0271 0.0105 0.5172 0.2119 0.0876 0.6521 0.2677 0.1119 0.8202 0.3364 01420 1.0310 0.4211 0.1789 1.2972 0.5250 0.2243 45.4769 5.1156 6.8210 14 0.0357 0.0141 0.0463 0.0187 15 0.0595 0.0245 0.0097 0.0761 0.0318 0.0129 0.0966 0.0410 0.0168 0.6530 0.2796 0.1214 0.0522 0.0220 0.8109 03469 0.1520 0.0663 0.0283 10060 0.4288 0.1891 0.0834 0.0361 1.2484 0.5236 0.2341 0.1043 0.0459 1.5524 0.6501 0.2885 0.1208 0.0577 6.0183 1.9366 0.7980 0.3543 0.1603 0.0723 8.3869 24293 0.9788 0.4333 0.1974 0.0899 12.4183 3.0732 1.2010 0.5267 0.2419 01111 20.6160 3.9318 1.4752 0.5437 0.2952 01367 1.8165 0.7827 0.3699 0.16731 2.2465 0.9506 0.4352 0.2040 253
Transcribed Image Text:Expected Number of People Waiting in Line (L) for Various Values of S and X/μ 1 0.10 0.0111 0.15 0.0264 0.0006 0.20 0.0500 0.0020 1.0 1.2 14 1.6 1.8 20 0.25 0.0833 0.0039 0.30 0.1285 0.0069 0.35 0.1884 0.0110 0.40 0.2666 0.0166 0.45 0.3681 0.0239 0.0019 0.50 0.5000 0.0333 0.0030 0.55 0.6722 0.045 0.0043 0.60 0.9090 0.0593 0.0061 0.65 12071 0.0767 0.0084 0.70 1.6333 0.0976 0.0112 0.75 2.2500 0.1227 0.0147 0.80 3.2000 0.1523 0.0189 0.85 4.8165 0.1873 0.0239 0.0031 0.90 8.1000 0.2285 0.0300 0.0041 0.95 18.0500 0.2767 0.0371 0.0053 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 40 42 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.6 6.8 7.0 7.2 74 7.6 7.8 8.0 8.2 84 86 8.8 9.0 92 94 9.6 2 9.8 10 3 4 5 0.3333 0.0454 0.0067 0.6748 0.0940 0.0158 13449 0.1778 0.0324 0.0059 2.8441 0.3128 0.0604 0.0121 7.6731 NUMBER OF SERVICE CHANNELS, S 7 6 0.5320 0.1051 0.0227 0.0047 0.0090 0.8888 0.1730 0.0390 0.0158 1.4907 0.2770 0.066 2.1261 4.9322 12.2724 0.4205 0.1047 0.0266 0.0065 0.6581 0.1609 0.0425 0.0110 1.0000 0.2411 0.0659 0.0180 8 1.5282 0.3541 0.0991 0.0282 0.0077 2.3855 0.5128 0.1452 0.0427 0.0122 3.9060 0.7365 0.2085 0.0631 0.0189 7.0893 1.0550 0.2947 0.0912 0.0283 16.9366 1.5181 0.4114 0.1292 0.0412 0.5694 0.1801 0.0590 2.2164 3.3269 5.2675 0.7837 0.2475 0.0827 1.0777 0.3364 0.1142 9.2885 1.4857 0.4532 0.1555 21.6384 2.0708 0.6071 0.2092 9 0.0084 0.0127 0.0189 0.0273 0.0389 0.0541 0.0742 4.4471 1.3471 6.3133 1.7288 9.5102 2.2324 16.0379 2.9113 35.8956 3.8558 10 0.0087 0.0128 0.0184 0.0260 5.2264 7.3441 11 1.6364 2.0736 10.9592 2.6470 18.3223 3.4160 40.6824 4.4805 0.1006 0.0361 0.0125 0.1345 0.0492 0.0175 2.9375 0.8102 0.2785 43004 1.0804 0.3680 6.6609 1.4441 0.5871 0.1779 0.0663 0.0243 0.0085 11.5178 1.9436 0.6313 0.2330 0.0683 0.0330 0.0119 26.3726 2.6481 0.8225 0.3032 0.1164 0.0443 0.0164 3.6878 1.0707 0.3918 0.1518 0.0590 0.0224 5.2979 1.3967 0.5037 0.1964 8.0768 1.8040 0.6454 0.2524 0.1008 0.0398 0.0153 13.7992 2.4198 0.8247 0.3222 0.1302 0.0523 0.0205 31.1270 3.2441 1.0533 0.0775 0.0300 0.0113 12 exhibit 7A.9 13 0.4090 0.1666 0.0679 0.0271 0.0105 0.5172 0.2119 0.0876 0.6521 0.2677 0.1119 0.8202 0.3364 01420 1.0310 0.4211 0.1789 1.2972 0.5250 0.2243 45.4769 5.1156 6.8210 14 0.0357 0.0141 0.0463 0.0187 15 0.0595 0.0245 0.0097 0.0761 0.0318 0.0129 0.0966 0.0410 0.0168 0.6530 0.2796 0.1214 0.0522 0.0220 0.8109 03469 0.1520 0.0663 0.0283 10060 0.4288 0.1891 0.0834 0.0361 1.2484 0.5236 0.2341 0.1043 0.0459 1.5524 0.6501 0.2885 0.1208 0.0577 6.0183 1.9366 0.7980 0.3543 0.1603 0.0723 8.3869 24293 0.9788 0.4333 0.1974 0.0899 12.4183 3.0732 1.2010 0.5267 0.2419 01111 20.6160 3.9318 1.4752 0.5437 0.2952 01367 1.8165 0.7827 0.3699 0.16731 2.2465 0.9506 0.4352 0.2040 253
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