Stochastic Models(Renewal-Reward Processes)
Stochastic Models(Renewal-Reward Processes)
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(1 point) Consider a production line with two stations in series. The items produced at station 1 are passed to station 2 where they are assembled. There is a buffer space before station 2. Each station can handle only one item at a time. The production time of an item at station 1 is exponentially distributed with mean , while the assembly time of an item at station 2 is exponentially distributed with mean . The production times and the assembly times are assumed to be mutually independent. Suppose that presently an item is in production at station 1, while station 2 has items including the one in assembly (1 is under assembly and other are in the buffer). What is the probability that station 2 becomes available before the current production at station 1 is completed. (Hint: You may consider how to use memoryless properties). You just need to give me the formulation without too much simplification. (i.e., you may give me a formulation that looks pretty complicated.)
(1 point) For one specific machine, the machine lifetime satisfies a Weibull Distribution with mean = 10 months and standard deviation = 3 months based on historical data analysis. The failed machine will be immediately replaced with a brand new one upon the failure and the replacement time is negligible. Each machine will cost $100,000 dollars and the system is designed to run for 5 years. What is the average total cost for this machine in the system life cycle by using the approximation of the renewal function on slide 24 of Lecture 3.
(1 point) For a renewal process let be the second moment of the number of renewals up to time t. Please use the Key Renewal Theorem to verify
(1 point) Consider a production line with two stations in series. The items produced at station 1 are passed to station 2 where they are assembled. There is a buffer space before station 2. Each station can handle only one item at a time. The production time of an item at station 1 is exponentially distributed with mean , while the assembly time of an item at station 2 is exponentially distributed with mean . The production times and the assembly times are assumed to be mutually independent. Suppose that presently an item is in production at station 1, while station 2 has items including the one in assembly (1 is under assembly and other are in the buffer). What is the probability that station 2 becomes available before the current production at station 1 is completed. (Hint: You may consider how to use memoryless properties). You just need to give me the formulation without too much simplification. (i.e., you may give me a formulation that looks pretty complicated.)
(1 point) For one specific machine, the machine lifetime satisfies a Weibull Distribution with mean = 10 months and standard deviation = 3 months based on historical data analysis. The failed machine will be immediately replaced with a brand new one upon the failure and the replacement time is negligible. Each machine will cost $100,000 dollars and the system is designed to run for 5 years. What is the average total cost for this machine in the system life cycle by using the approximation of the renewal function on slide 24 of Lecture 3.
(1 point) For a renewal process let be the second moment of the number of renewals up to time t. Please use the Key Renewal Theorem to verify
(Hint: you need to have a renewal equation first).
(1 point) A shuttle to the airport arrives at a hotel with independent interarrival times uniformly distributed between 10 and 20 minutes. If you want to take the shuttle to the airport at 4 PM and have no information about the operation of the shuttle before that…
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