You have been tasked with examining the queueing system of a continuously operating ride at a theme park. However, there is not a position where you can fully see the entire queue. In particular, if there are 9 or fewer riders, you cannot determine the exact number of riders. You can always see the 10th rider in line (if there are 10 or more riders). Similarly, you cannot see if there are 31 or more riders in line. You can always see the 30th rider in line. You do know, however, that there can be at most 40 riders in line. The data below provides the fraction of time that you observed k = 10, 11, . . . , 30 for each individual k. These frequencies total .9 (90% of the time). You observed that .04 of a fraction of time (4%), there were 9 or fewer riders in line and that .06 of a fraction of time (6%), there were 31 or more riders in line. It is well known that riders arrive to the line at a rate of 30 per hour and the average ride time is 10 minutes. (a) Determine an appropriate, data-driven lower bound on the expected number of riders currently at this ride (both in line and riding it). Please show all relevant calculations (i.e., if you calculated other characteristics of the queueing system such as the expected number of riders in the queue) done to determine this lower bound. Provide the logic as to why this is an appropriate lower bound. (b) Determine an appropriate, data-driven upper bound on the expected number of riders currently at this ride (both in line and riding it). Please show all relevant calculations (i.e., if you calculated other characteristics of the queueing system such as the expected number of riders in the queue) done to determine this lower bound. Provide the logic as to why this is an appropriate lower bound. (c) The theme park believes that it costs them, in rider goodwill, $20 per hour a rider spends interacting with this ride. Provide a range of the possible costs associated with the ride currently. If the theme park believes these costs are too high, provide at least one option for them to implement to decrease these costs. Theme Park Ride Observations: k: 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 P_{kq}: 0.005 0.005 0.010 0.020 0.025 0.035 0.050 0.060 0.060 0.080 0.100 0.080 0.060 0.080 0.060 0.050 0.050 0.030 0.020 0.010 0.010 0.9

You have been tasked with examining the queueing system of a continuously operating ride at a theme park. However, there is not a position where you can fully see the entire queue. In particular, if there are 9 or fewer riders, you cannot determine the exact number of riders. You can always see the 10th rider in line (if there are 10 or more riders). Similarly, you cannot see if there are 31 or more riders in line. You can always see the 30th rider in line. You do know, however, that there can be at most 40 riders in line. The data below provides the fraction of time that you observed k = 10, 11, . . . , 30 for each individual k. These frequencies total .9 (90% of the time). You observed that .04 of a fraction of time (4%), there were 9 or fewer riders in line and that .06 of a fraction of time (6%), there were 31 or more riders in line. It is well known that riders arrive to the line at a rate of 30 per hour and the average ride time is 10 minutes. (a) Determine an appropriate, data-driven lower bound on the expected number of riders currently at this ride (both in line and riding it). Please show all relevant calculations (i.e., if you calculated other characteristics of the queueing system such as the expected number of riders in the queue) done to determine this lower bound. Provide the logic as to why this is an appropriate lower bound. (b) Determine an appropriate, data-driven upper bound on the expected number of riders currently at this ride (both in line and riding it). Please show all relevant calculations (i.e., if you calculated other characteristics of the queueing system such as the expected number of riders in the queue) done to determine this lower bound. Provide the logic as to why this is an appropriate lower bound. (c) The theme park believes that it costs them, in rider goodwill, $20 per hour a rider spends interacting with this ride. Provide a range of the possible costs associated with the ride currently. If the theme park believes these costs are too high, provide at least one option for them to implement to decrease these costs. Theme Park Ride Observations: k: 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 P_{kq}: 0.005 0.005 0.010 0.020 0.025 0.035 0.050 0.060 0.060 0.080 0.100 0.080 0.060 0.080 0.060 0.050 0.050 0.030 0.020 0.010 0.010 0.9

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter3: Solving Equation And Problems

Section3.6: Problem Solving: Using Charts

Problem 2OE

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