Question

The Burger Dome waiting line model studies the waiting time of customers at its fast-food restaurant. Burger Dome's single-server waiting line system has an arrival rate of 0.75 customers per minute and a service rate of 1 customer per minute. Adapt the Black Sheep Scarves spreadsheet shown below to simulate the operation of this waiting line. Make sure to use the random values for both interarrival and service times generated in the worksheet_12-23. Assuming that customer arrivals follow a Poisson probability distribution, the interarrival times can be simulated with the cell formula - (1/4)*LN(RANDO), where i = 0.75. Assuming that the service time follows an exponential probability distribution, the service times can be simulated with the cell formula -v*LN(RANDO), where u = 1. Run the Burger Dome simulation for 1000 customers. Discard the first 100 customers and collect data over the next 900 customers. The analytical model indicates an average waiting time of 3 minutes per customer. What average waiting time does your simulation model show? Round your answer to 3 decimal places. H А B c D E F G 1 Black Sheep Scarves with One Quality Inspector 2 3 3 Interarrival Times (Uniform Distribution) 4 + Smallest Value 0 5 Largest Value 5 6. 7 Service Times (Normal Distribution) 8 Mean 2 Standard Dev 0.5 10 11 12 Simulation 13 14 Interarrival Arrival Service Waiting Service Completion 15 Customer Time Time Start Time Time Time Time 16 1 1.4 1.4 1.4 0.0 2.3 3.7 17 2 1.3 2.7 3.7 1.0 1.5 5.2 18 3 4.9 7.6 7.6 0.0 2.2 9.8 19 4 3.5 11.1 1 + 11.1 0.0 13.6 20 40 5 0.7 11.8 13.6 1.8 18 1.0 15.4 1011 996 0.5 2496.8 2498.1 1.3 0.6 2498.7 1012 997 0.2 2497.0 2498.7 1.7 2.0 2500.7 1013 998 2.7 1010 2499.7 2500.7 1.0 1.8 2502.5 1014 999 3.7 2503.4 2503.4 0.0 2.4 2505.8 1015 1000 4.0 2507.4 2507.4 0.0 1.9 2509.3 1016 11 1017 Summary Statistics 1018 Number Waiting 549 1019 Probability of Waiting 12 0.6100 1020 Average Waiting Time 1.59 1021 Maximum Waiting Time 13.5 1022 Utilization of Quality Inspector 0.7860 1023 Number Waiting > 1 Min 393 1024 Probability of Waiting > 1 Min 0.4367 2.5 Time in System 2.3 2.5 2.2 2.2 2.5 3.6 3.0 1.9 3.7 2.8 2.4 1.9

Answer 1

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H 2 1 14 16 5.0 A B с D E F G Black Sheep Scarves with One Quality Inspector 2 3 InterArrival Time (Poisson distribution) 4 Mean (1) 0.75 5 6 7 Service Time (exponential distribution) 8 Mean (u) 9 10 11 12 Simulation 13 Interarrival Arrival Service Waiting Service Completion 15 Customer Time Time Start Time Time Time Time 1 0.1 0.08 0.08 0.00 0.56 0.65 17 2 5.13 5.13 0.00 3.16 8.29 18 3 1.4 6.51 8.29 1.78 0.02 8.31 19 4 3.0 9.50 9.50 0.00 1.00 10.50 20 1.9 11.37 11.37 0.00 2.06 13.43 1011 996 0.4 1288.50 1295.94 7.44 0.06 1296.00 1012 997 3.4 1291.88 1296.00 4.12 1.40 1297.41 1013 998 0.1 1292.01 1297.41 5.39 1.34 1298.75 1014 999 0.4 1292.40 1298.75 6.35 0.42 1299.17 1015 1000 0.4 1292.78 1299.17 6.39 0.88 1300.05 1016 1017 Summary statistics 1018 Number Waiting 702 1019 Probability in Waiting 0.7800 1020 Average Waiting Time 2.82 1021 Maximum Waiting Time 15.5 1022 Utilisation of Server 0.8022 1023 Number Waiting > 1 minute 574 1024 Probability Waiting > 1 minute 0.6378 1025 Average Time in System 3.85 1026 Time in System 0.56 3.16 1.80 1.00 2.06 7.50 5.52 6.73 6.77 7.27

EXCEL FORMULA:

E H A B D 1 Black Sheep Scarves with One Quality Inspector 2 3_InterArrival Time (Poisson distribution) 4 Mean (A) 0.75 5 6 7_Service Time (exponential distribution) 8 Mean (u) 1 9 10 11 12 Simulation 13 14 Interarrival Arrival Service Waiting 15 Customer Time Time Start Time Time 16 1 =-(1/2)* LN(RANDO) =B16 =C16 =D16-C16 17 2 =-(1/2)*LN(RANDO) =C16+B17 =MAX(G16,C17) =D17-C17 18 3 =-(1/2)*LN(RAND() =C17+B18 =MAX(G17, C18) =D18-C18 19 4 I=-(1/2)*LN(RANDO) =C18+B19 =MAX( G18,C19) =D19-C19 205 I=-(1/2)*LN(RANDO) =C19+B20 =MAX(G19, C20) =D20-C20 1011 996 =-(1/2)*LN(RANDO) =C1010+B1011 =MAX(G1010,C1011) =D1011-C1011 1012 997 =-(1/2)*LN(RAND() =C1011+B1012 =MAX(G1011, C1012) =D1012-C1012 1013 998 =-(1/2)*LN(RAND() =C1012+B1013 =MAX(G1012,C1013) =D1013-C1013 1014 999 =-(1/2)*LN(RANDO) =C1013+B1014 =MAX(G1013,C1014) =D1014-C1014 1015 1000 =-(1/2)*LN(RANDO) =C1014+B1015 =MAX(G1014,C1015) =D1015-C1015 1016 1017 Summary statistics 1018 Number Waiting =COUNTIF(E116:E1015,">0") 1019 Probability in Waiting =E1018/COUNT(E116:E1015) 1020 Average Waiting Time FAVERAGE(E116:E1015) 1021 Maximum Waiting Time =MAX(E116:E1015) 1022 Utilisation of Server ESUM(F116:F1015)/(G1015-G115) 1023 Number Waiting > 1 minute =COUNTIF(E116:E1015,">1") 1024 Probability Waiting > 1 minute =E1023/COUNT(E116:E1015) 1025 Average Time in System EAVERAGE(H116:H1015) 1026 Service Completion Time Time Time in System =-u*LN(RANDO) =D16+F16 =G16-C16 =-u*LN(RAND() =D17+F17 =G17-C17 =-u*LN(RANDO) =D18+F18 =G18-C18 =-u*LN(RAND() =D19+F19 =G19-C19 =-u*LN(RANDO) =D20+F20 =G20-C20 =-u*LN(RANDO) =D1011+F1011 =G1011-C1011 =-u*LN(RAND() =D1012+F1012 =G1012-C1012 =-u*LN(RAND() =D1013+F1013 =G1013-C1013 =-u*LN(RAND() =D1014+F1014 =G1014-C1014 =-u*LN(RANDO) =D1015+F1015 =G1015-C1015

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Simulation model is following:

H 1 A B с D E F 1 Black Sheep Scarves with One Quality Inspector 2 3 InterArrival Time (Poisson distribution) 4 Smallest Value o 5 Largest Value 5 6 7 Service Time (Normal distribution) 8 Mean (u) 2 9 Std Dev (0) 0.5 10 11 12 Simulation 13 14 Interarrival Arrival Service Waiting Service Completion 15 Customer Time Time Start Time Time Time Time 16 0.4 0.40 0.40 0.00 1.93 2.33 17 2 0.9 1.30 2.33 1.03 2.33 4.66 18 3 2.1 3.40 4.66 1.26 2.09 6.76 19 0.7 4.10 6.76 2.66 2.23 8.98 20 5 3.6 7.70 8.98 1.28 1.06 10.04 1011 996 1.9 2489.90 2494.57 4.67 2.44 2497.01 1012 997 2.8 2492.70 2497.01 4.31 1.98 2498.99 1013 998 1.1 2493.80 2498.99 5.19 1.78 2500.78 10141 999 4.2 2498.00 2500.78 2.78 1.83 2502.61 1015 1000 4.7 2502.70 2502.70 0.00 1.95 2504.65 1016 1017 Summary statistics 1018 Number Waiting 538 1019 Probability in Waiting 0.5978 1020 Average Waiting Time 1.60 1021 Maximum Waiting Time 13.4 1022 Utilisation of Server 0.7805 1023 Number Waiting > 1 minute 386 1024 Probability Waiting > 1 minute 0.4289 1025 Average Time in System 3.57 1026 4 Time in System 1.93 3.36 3.36 4.88 2.34 7.11 6.29 6.98 4.61 1.95

EXCEL FORMULA:

F G H 1 A B D E Black Sheep Scarves with One Quality Inspector 2 3 InterArrival Time (Poisson distribution) 4 Smallest Value o 5 Largest Value 5 6 7 Service Time (Normal distribution) 8 Mean (u) 2 9 Std Dev (0) 0.5 10 11 12 Simulation 13 14 Interarrival Arrival Service Waiting 15 Customer Time Time Start Time Time 16 1 =ROUND($B$4+RAND()*($B$5-$B$4),1) =B16 =C16 =D16-C16 17 2 =ROUND($B$4+RAND()*($B$5-$B$4),1) =C16+B17 =MAX(G16,C17) =D17-C17 18 3 =ROUND($B$4+RAND()*($B$5-$B$4),1) =C17+B18 =MAX( G17, C18) =D18-C18 194 =ROUND($B$4+RANDO*($B$5-$B$4),1)=C18+B19 =MAX(G18, C19) =D19-C19 205 =ROUND($B$4+RAND()*($B$5-$B$4),1) =C19+B20 =MAX(G19,C20) =D20-C20 1011 996 =ROUND($B$4+RAND()*($B$5-$B$4),1) =C1010+B1011 =MAX(G1010, C1011) =D1011-C1011 1012 997 =ROUND($B$4+RAND()*($B$5-$B$4),1) =C1011+B1012 =MAX(G1011, C1012) =D1012-C1012 1013 998 =ROUND($B$4+RANDO*($B$5-$B$4),1)=C1012+B1013 =MAX(G1012, C1013) =D1013-C1013 1014 999 =ROUND($B$4+RAND()*($B$5-$B$4),1) =C1013+B1014 =MAX(G1013,C1014) =D1014-C1014 1015 1000 =ROUND($B$4+RAND()*($B$5-$B$4),1) =C1014+B1015 =MAX(G1014, C1015) =D1015-C1015 1016 1017 Summary statistics 1018 Number Waiting =COUNTIF(E116:E1015,">0") 1019 Probability in Waiting =E1018/COUNT(E116:E1015) 1020 Average Waiting Time =AVERAGE(E116:E1015) 1021 Maximum Waiting Time =MAX(E116:E1015) 1022 Utilisation of Server =SUM(F116:F1015)/(G1015-G115) 1023 Number Waiting > 1 minute =COUNTIF(E116:E1015,">1") 1024 Probability Waiting > 1 minute =E1023/COUNT(E116:E1015) 1025 Average Time in System =AVERAGE(H116:H1015) 1026 Service Completion Time Time Time in System =NORMINV(RAND(),$B$8,$B$9) =D16+F16 =G16-C16 =NORMINV(RAND() $B$8,$B$9) =D17+F17 =G17-C17 =NORMINV(RAND() $B$8,$B$9) =D18+F18 =G18-C18 =NORMINV(RAND() $B$8,$B$9) =D19+F19 =G19-C19 =NORMINV(RAND(),$B$8,$B$9) =D20+F20 =G20-C20 =NORMINV(RAND() $B$8,$B$9) =D1011+F1011 =G1011-C1011 =NORMINV(RAND(),$B$8,$B$9) =D1012+F1012 =G1012-C1012 =NORMINV(RAND() $B$8,$B$9) =D1013+F1013 =G1013-C1013 =NORMINV(RAND() $B$8,$B$9) =D1014+F1014 =G1014-C1014 =NORMINV(RAND() $B$8,$B$9) =D1015+F1015 =G1015-C1015