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Problem 3: A company owns u vending machines located all over Toronto, which it resupplies from a central depot. Each vending machine is equipped with a sensor that alerts the company when it needs to be resupplied. On any given day, a machine raises an alert with probability p. The company's resupply policy is purely corrective, i.e., it only resupplies the vending machines when it receives an alert. At the end of each day, the company compiles a list of the machines that raised an alert during that day and schedules a resupply for the next day. The company wants to study the number of resupplies it needs to perform. Assume that a machine does not push an alert on the same day that it was resupplied, and that the company has enough capacity to resupply as many machines as necessary in a day. a) Model this system as a Markov chain. State your assumptions explicitly, if any. b) Give an expression for the expected number of resupplies that the company will perform on a day in the long term.

Problem 3: A company owns u vending machines located all over Toronto, which it resupplies from a central depot. Each vending machine is equipped with a sensor that alerts the company when it needs to be resupplied. On any given day, a machine raises an alert with probability p. The company's resupply policy is purely corrective, i.e., it only resupplies the vending machines when it receives an alert. At the end of each day, the company compiles a list of the machines that raised an alert during that day and schedules a resupply for the next day. The company wants to study the number of resupplies it needs to perform. Assume that a machine does not push an alert on the same day that it was resupplied, and that the company has enough capacity to resupply as many machines as necessary in a day. a) Model this system as a Markov chain. State your assumptions explicitly, if any. b) Give an expression for the expected number of resupplies that the company will perform on a day in the long term.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter12: Probability
Section12.CR: Chapter 12 Review
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Problem 3: A company owns u vending machines located all over Toronto, which it resupplies from a central depot. Each vending machine is equipped with a sensor that alerts the company when it needs to be resupplied. On any given day, a machine raises an alert with probability p. The company's resupply policy is purely corrective, i.e., it only resupplies the vending machines when it receives an alert. At the end of each day, the company compiles a list of the machines that raised an alert during that day and schedules a resupply for the next day. The company wants to study the number of resupplies it needs to perform. Assume that a machine does not push an alert on the same day that it was resupplied, and that the company has enough capacity to resupply as many machines as necessary in a day. a) Model this system as a Markov chain. State your assumptions explicitly, if any. b) Give an expression for the expected number of resupplies that the company will perform on a day in the long term.
Transcribed Image Text:Problem 3: A company owns u vending machines located all over Toronto, which it resupplies from a central depot. Each vending machine is equipped with a sensor that alerts the company when it needs to be resupplied. On any given day, a machine raises an alert with probability p. The company's resupply policy is purely corrective, i.e., it only resupplies the vending machines when it receives an alert. At the end of each day, the company compiles a list of the machines that raised an alert during that day and schedules a resupply for the next day. The company wants to study the number of resupplies it needs to perform. Assume that a machine does not push an alert on the same day that it was resupplied, and that the company has enough capacity to resupply as many machines as necessary in a day. a) Model this system as a Markov chain. State your assumptions explicitly, if any. b) Give an expression for the expected number of resupplies that the company will perform on a day in the long term.
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