Activity 8.7 - Group Case Study: Winter Park Hotel Donna Shader, manager of the Winter Park Hotel, is considering how to restructure the front desk to reach an optimum level of staff efficiency and guest service. At present, the hotel has five clerks on duty, each with a separate waiting line, during the peak check-in time of 3:00 pm. To 5:00 pm. Observation of arrivals during this time shows that an average of 80 guests arrive each hour (although there is no upward limit on the number that could arrive at any given time). It takes an average of 3.2 minutes for each frontdesk clerk to register each guest. Donna is considering three plans for improving guest service by reducing the length of time guests spend waiting in line. The first proposal would designate one employee as a quick-service clerk for guests registering under corporate accounts, a market segment that fills about 30% of all occupied rooms. Because corporate guests are preregistered, their registration takes just 2 minutes. With these guests separated from the rest of the clientele, the average time for registering a typical guest would climb to 3.8 minutes. Under Plan 1, noncorporate guests would choose any of the remaining four lines. The second plan is to implement a single-line system. All guests could form a single waiting line to be served by whichever of five clerks became available. This option would require sufficient lobby space for what could be a substantial queue. The third proposal involves using an automatic teller machine (ATM) for check-ins. This ATM would provide a service just slightly better than a clerk, 3.0 minutes. Given that initial use of this technology might be minimal, Shader estimates that 20% of customers, primarily frequent guests, would be willing to use the machines. (This might be a conservative estimate if the guests perceive direct benefits from using the ATM, as bank customers do. Citibank reports that some 95% of its Manhattan customers use its ATMs.) Donna would set up a single queue for customers who prefer human check-in clerks. This would be served by the five clerks, although Donna is hopeful that the machine will allow a reduction to four. Discussion Questions 1. Determine the average amount of time that a guest spends checking in under the current mode of operation. How would this change under each of the stated options? 2. Which option do you recommend? 3. What other considerations or limitations should be considered/evaluated? Page 1 of 1 WorkfileContents.docx Waiting Lines M/M/s Data Arrival rate (l) Service rate (m) Number of servers(s) Server cost $/time) Waiting cost ($/time) Probabilities Number 80 18.75 5 Results Average server utilization(r) 0.85333 Average number of customers in the queue(Lq) 3.84836 Average number of customers in the system(L) 8.11503 Average waiting time in the queue(Wq) 0.0481 Average time in the system(W) 0.10144 Probability (% of time) system is empty (P0) 0.00823 Cost - based on waiting 0 Cost - based on system 0 Probability Cumulative Probability 0 0.008233 0.008233 1 0.035127 0.043360 2 0.074939 0.118299 3 0.106579 0.224878 4 0.113685 0.338563 5 0.097011 0.435574 6 0.082783 0.518356 7 0.070641 0.588997 8 0.060280 0.649278 9 0.051439 0.700717 10 0.043895 0.744612 11 0.037457 0.782069 12 0.031963 0.814032 13 0.027275 0.841307 14 0.023275 0.864582 15 0.019861 0.884443 16 0.016948 0.901392 17 0.014463 0.915854 18 0.012341 0.928196 19 0.010531 0.938727 20 0.008987 0.947714 term2 0 1.000000 1 4.26666667 1 2 9.10222222 5.2666666667 3 12.9453827 14.3688888889 4 13.8084082 27.3142716049 5 11.783175 41.1226798354 6 8.37914668 52.9058548587 7 5.10728941 61.285001542 8 2.72388768 66.3922909489 9 1.29132453 69.1161786326 10 0.55096513 70.4075031641 11 0.21370769 70.9584682976 12 0.07598496 71.1721759857 13 0.02493865 71.2481609415 14 0.00760035 71.2730995936 15 0.00216188 71.2806999448 16 0.0005765 71.2828618224 17 0.00014469 71.2834383231 18 3.430E-005 71.2835830135 19 7.702E-006 71.2836173105 20 1.643E-006 71.2836250123 21 3.338E-007 71.2836266553 22 6.474E-008 71.2836269891 23 24 25 26 27 28 29 30 P0(s) -1.306122449 -8.031372549 -30.6601169591 -207.1261234568 80.3398297045 29.0047385189 13.0796436031 5.8369021794 2.4553353769 0.9609856979 0.3491264212 0.11790769 0.0371224211 0.0109320119 0.0030212576 0.0007861373 0.000193173 4.49523491419409E-005 9.9321563782026E-006 2.08861760055453E-006 4.18943960674311E-007 8.03188421562492E-008 -3.2667 -0.3617 -0.0614 -0.0056 0.00823 0.01221 0.01345 0.01384 0.01397 0.01401 0.01402 0.01403 0.01403 0.01403 0.01403 0.01403 0.01403 0.01403 0.01403 0.01403 0.01403 0.01403 Rho(s) 4.266667 2.133333 1.422222 1.066667 0.853333 0.711111 0.609524 0.533333 0.474074 0.426667 0.387879 0.355556 0.328205 0.304762 0.284444 0.266667 0.25098 0.237037 0.224561 0.213333 0.203175 0.193939 Lq(s) -5.57279 -5.46817 -6.33937 -18.4305 3.848362 0.871637 0.274553 0.092355 0.030924 0.010021 0.003102 0.000913 0.000254 6.7E-005 1.7E-005 4.0E-006 9.1E-007 2.0E-007 4.0E-008 7.9E-009 1.5E-009 2.7E-010 L(s) -1.30612 -1.2015 -2.07271 -14.1638 8.115029 5.138303 4.541219 4.359022 4.29759 4.276687 4.269769 4.267579 4.266921 4.266734 4.266684 4.266671 4.266668 4.266667 4.266667 4.266667 4.266667 4.266667 Wq(s) -0.06966 -0.06835 -0.07924 -0.23038 0.048105 0.010895 0.003432 0.001154 0.000387 0.000125 3.9E-005 1.1E-005 3.2E-006 8.4E-007 2.1E-007 5.0E-008 1.1E-008 2.4E-009 5.0E-010 9.9E-011 1.9E-011 3.4E-012 W(S) -0.01633 -0.01502 -0.02591 -0.17705 0.101438 0.064229 0.056765 0.054488 0.05372 0.053459 0.053372 0.053345 0.053337 0.053334 0.053334 0.053333 0.053333 0.053333 0.053333 0.053333 0.053333 0.053333 Waiting Lines M/M/s Data Arrival rate (l) Service rate (m) Number of servers(s) Server cost $/time) Waiting cost ($/time) Probabilities Number 64 20 5 Results Average server utilization(r) 0.64 Average number of customers in the queue(Lq) 0.51299 Average number of customers in the system(L) 3.71299 Average waiting time in the queue(Wq) 0.00802 Average time in the system(W) 0.05802 Probability (% of time) system is empty (P0) 0.03715 Cost - based on waiting 0 Cost - based on system 0 Probability Cumulative Probability 0 0.037150 0.037150 1 0.118881 0.156032 2 0.190210 0.346241 3 0.202891 0.549132 4 0.162312 0.711445 5 0.103880 0.815324 6 0.066483 0.881808 7 0.042549 0.924357 8 0.027232 0.951588 9 0.017428 0.969017 10 0.011154 0.980171 11 0.007139 0.987309 12 0.004569 0.991878 13 0.002924 0.994802 14 0.001871 0.996673 15 0.001198 0.997871 16 0.000766 0.998637 17 0.000491 0.999128 18 0.000314 0.999442 19 0.000201 0.999643 20 0.000129 0.999771 term2 0 1.000000 1 3.2 1 2 5.12 4.2 3 5.46133333 9.32 4 4.36906667 14.78133333 5 2.79620267 19.1504 6 1.49130809 21.94660267 7 0.68174084 23.43791076 8 0.27269634 24.1196516 9 0.0969587 24.39234793 10 0.03102678 24.48930663 11 0.00902597 24.52033341 12 0.00240693 24.52935939 13 0.00059247 24.53176631 14 0.00013542 24.53235879 15 2.889E-005 24.53249421 16 5.778E-006 24.5325231 17 1.088E-006 24.53252888 18 1.934E-007 24.53252997 19 3.257E-008 24.53253016 20 5.210E-009 24.53253019 21 7.940E-010 24.5325302 22 1.155E-010 24.5325302 23 24 25 26 27 28 29 30 P0(s) -1.4545454545 -8.5333333333 -81.92 21.8453333333 7.7672296296 3.1956601905 1.2558383906 0.4544938938 0.150453151 0.0456276223 0.0127289367 0.0032821721 0.0007859351 0.0001755479 3.6724789521521E-005 7.22254193923246E-006 1.33983386698805E-006 2.35163209722545E-007 3.91607932927557E-008 6.20291262180993E-009 9.36709501540736E-010 1.3514430903688E-010 -2.2 -0.2308 -0.0138 0.0273 0.03715 0.03977 0.0405 0.04069 0.04075 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 Rho(s) 3.2 1.6 1.066667 0.8 0.64 0.533333 0.457143 0.4 0.355556 0.32 0.290909 0.266667 0.246154 0.228571 0.213333 0.2 0.188235 0.177778 0.168421 0.16 0.152381 0.145455 Lq(s) -4.65455 -5.25128 -18.054 2.38573 0.512988 0.145261 0.042827 0.01233 0.003382 0.000875 0.000213 4.9E-005 1.0E-005 2.1E-006 4.1E-007 7.4E-008 1.3E-008 2.1E-009 3.2E-010 4.8E-011 6.9E-012 9.4E-013 L(s) -1.45455 -2.05128 -14.854 5.58573 3.712988 3.345261 3.242827 3.21233 3.203382 3.200875 3.200213 3.200049 3.20001 3.200002 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 Wq(s) -0.07273 -0.08205 -0.28209 0.037277 0.008015 0.00227 0.000669 0.000193 5.3E-005 1.4E-005 3.3E-006 7.6E-007 1.6E-007 3.3E-008 6.3E-009 1.2E-009 2.0E-010 3.2E-011 5.1E-012 7.5E-013 1.1E-013 1.5E-014 W(S) -0.02273 -0.03205 -0.23209 0.087277 0.058015 0.05227 0.050669 0.050193 0.050053 0.050014 0.050003 0.050001 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 Waiting Lines M/M/s Data Arrival rate (l) Service rate (m) Number of servers(s) Server cost $/time) Waiting cost ($/time) Probabilities Number 80 18.75 5 Results Average server utilization(r) 0.85333 Average number of customers in the queue(Lq) 3.84836 Average number of customers in the system(L) 8.11503 Average waiting time in the queue(Wq) 0.0481 Average time in the system(W) 0.10144 Probability (% of time) system is empty (P0) 0.00823 Cost - based on waiting 0 Cost - based on system 0 Probability Cumulative Probability 0 0.008233 0.008233 1 0.035127 0.043360 2 0.074939 0.118299 3 0.106579 0.224878 4 0.113685 0.338563 5 0.097011 0.435574 6 0.082783 0.518356 7 0.070641 0.588997 8 0.060280 0.649278 9 0.051439 0.700717 10 0.043895 0.744612 11 0.037457 0.782069 12 0.031963 0.814032 13 0.027275 0.841307 14 0.023275 0.864582 15 0.019861 0.884443 16 0.016948 0.901392 17 0.014463 0.915854 18 0.012341 0.928196 19 0.010531 0.938727 20 0.008987 0.947714 term2 0 1.000000 1 4.26666667 1 2 9.10222222 5.2666666667 3 12.9453827 14.3688888889 4 13.8084082 27.3142716049 5 11.783175 41.1226798354 6 8.37914668 52.9058548587 7 5.10728941 61.285001542 8 2.72388768 66.3922909489 9 1.29132453 69.1161786326 10 0.55096513 70.4075031641 11 0.21370769 70.9584682976 12 0.07598496 71.1721759857 13 0.02493865 71.2481609415 14 0.00760035 71.2730995936 15 0.00216188 71.2806999448 16 0.0005765 71.2828618224 17 0.00014469 71.2834383231 18 3.430E-005 71.2835830135 19 7.702E-006 71.2836173105 20 1.643E-006 71.2836250123 21 3.338E-007 71.2836266553 22 6.474E-008 71.2836269891 23 24 25 26 27 28 29 30 P0(s) -1.306122449 -8.031372549 -30.6601169591 -207.1261234568 80.3398297045 29.0047385189 13.0796436031 5.8369021794 2.4553353769 0.9609856979 0.3491264212 0.11790769 0.0371224211 0.0109320119 0.0030212576 0.0007861373 0.000193173 4.49523491419409E-005 9.9321563782026E-006 2.08861760055453E-006 4.18943960674311E-007 8.03188421562492E-008 -3.2667 -0.3617 -0.0614 -0.0056 0.00823 0.01221 0.01345 0.01384 0.01397 0.01401 0.01402 0.01403 0.01403 0.01403 0.01403 0.01403 0.01403 0.01403 0.01403 0.01403 0.01403 0.01403 Rho(s) 4.266667 2.133333 1.422222 1.066667 0.853333 0.711111 0.609524 0.533333 0.474074 0.426667 0.387879 0.355556 0.328205 0.304762 0.284444 0.266667 0.25098 0.237037 0.224561 0.213333 0.203175 0.193939 Lq(s) -5.57279 -5.46817 -6.33937 -18.4305 3.848362 0.871637 0.274553 0.092355 0.030924 0.010021 0.003102 0.000913 0.000254 6.7E-005 1.7E-005 4.0E-006 9.1E-007 2.0E-007 4.0E-008 7.9E-009 1.5E-009 2.7E-010 L(s) -1.30612 -1.2015 -2.07271 -14.1638 8.115029 5.138303 4.541219 4.359022 4.29759 4.276687 4.269769 4.267579 4.266921 4.266734 4.266684 4.266671 4.266668 4.266667 4.266667 4.266667 4.266667 4.266667 Wq(s) -0.06966 -0.06835 -0.07924 -0.23038 0.048105 0.010895 0.003432 0.001154 0.000387 0.000125 3.9E-005 1.1E-005 3.2E-006 8.4E-007 2.1E-007 5.0E-008 1.1E-008 2.4E-009 5.0E-010 9.9E-011 1.9E-011 3.4E-012 W(S) -0.01633 -0.01502 -0.02591 -0.17705 0.101438 0.064229 0.056765 0.054488 0.05372 0.053459 0.053372 0.053345 0.053337 0.053334 0.053334 0.053333 0.053333 0.053333 0.053333 0.053333 0.053333 0.053333 Waiting Lines M/M/s Data Arrival rate (l) Service rate (m) Number of servers(s) Server cost $/time) Waiting cost ($/time) Probabilities Number 64 20 5 Results Average server utilization(r) 0.64 Average number of customers in the queue(Lq) 0.51299 Average number of customers in the system(L) 3.71299 Average waiting time in the queue(Wq) 0.00802 Average time in the system(W) 0.05802 Probability (% of time) system is empty (P0) 0.03715 Cost - based on waiting 0 Cost - based on system 0 Probability Cumulative Probability 0 0.037150 0.037150 1 0.118881 0.156032 2 0.190210 0.346241 3 0.202891 0.549132 4 0.162312 0.711445 5 0.103880 0.815324 6 0.066483 0.881808 7 0.042549 0.924357 8 0.027232 0.951588 9 0.017428 0.969017 10 0.011154 0.980171 11 0.007139 0.987309 12 0.004569 0.991878 13 0.002924 0.994802 14 0.001871 0.996673 15 0.001198 0.997871 16 0.000766 0.998637 17 0.000491 0.999128 18 0.000314 0.999442 19 0.000201 0.999643 20 0.000129 0.999771 term2 0 1.000000 1 3.2 1 2 5.12 4.2 3 5.46133333 9.32 4 4.36906667 14.78133333 5 2.79620267 19.1504 6 1.49130809 21.94660267 7 0.68174084 23.43791076 8 0.27269634 24.1196516 9 0.0969587 24.39234793 10 0.03102678 24.48930663 11 0.00902597 24.52033341 12 0.00240693 24.52935939 13 0.00059247 24.53176631 14 0.00013542 24.53235879 15 2.889E-005 24.53249421 16 5.778E-006 24.5325231 17 1.088E-006 24.53252888 18 1.934E-007 24.53252997 19 3.257E-008 24.53253016 20 5.210E-009 24.53253019 21 7.940E-010 24.5325302 22 1.155E-010 24.5325302 23 24 25 26 27 28 29 30 P0(s) -1.4545454545 -8.5333333333 -81.92 21.8453333333 7.7672296296 3.1956601905 1.2558383906 0.4544938938 0.150453151 0.0456276223 0.0127289367 0.0032821721 0.0007859351 0.0001755479 3.6724789521521E-005 7.22254193923246E-006 1.33983386698805E-006 2.35163209722545E-007 3.91607932927557E-008 6.20291262180993E-009 9.36709501540736E-010 1.3514430903688E-010 -2.2 -0.2308 -0.0138 0.0273 0.03715 0.03977 0.0405 0.04069 0.04075 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 0.04076 Rho(s) 3.2 1.6 1.066667 0.8 0.64 0.533333 0.457143 0.4 0.355556 0.32 0.290909 0.266667 0.246154 0.228571 0.213333 0.2 0.188235 0.177778 0.168421 0.16 0.152381 0.145455 Lq(s) -4.65455 -5.25128 -18.054 2.38573 0.512988 0.145261 0.042827 0.01233 0.003382 0.000875 0.000213 4.9E-005 1.0E-005 2.1E-006 4.1E-007 7.4E-008 1.3E-008 2.1E-009 3.2E-010 4.8E-011 6.9E-012 9.4E-013 L(s) -1.45455 -2.05128 -14.854 5.58573 3.712988 3.345261 3.242827 3.21233 3.203382 3.200875 3.200213 3.200049 3.20001 3.200002 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 Wq(s) -0.07273 -0.08205 -0.28209 0.037277 0.008015 0.00227 0.000669 0.000193 5.3E-005 1.4E-005 3.3E-006 7.6E-007 1.6E-007 3.3E-008 6.3E-009 1.2E-009 2.0E-010 3.2E-011 5.1E-012 7.5E-013 1.1E-013 1.5E-014 W(S) -0.02273 -0.03205 -0.23209 0.087277 0.058015 0.05227 0.050669 0.050193 0.050053 0.050014 0.050003 0.050001 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05