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# midtermS05second - IEOR 4106 Introduction to Operations...

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IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2005, Professor Whitt, Second Midterm Exam Chapters 5-6 in Ross, Thursday, March 31, 11:00am-1:00pm Open Book: but only the Ross textbook plus one 8 × 11 page of notes Justify your answers; show your work. 1. The IEOR Department Ricoh Printer ( 30 points) The Columbia IEOR Department has a versatile Ricoh printer that can rapidly print one- sided and two-sided copies, but unfortunately it often goes down. Ricoh is alternately up (working) and down (waiting for repair or under repair). The average up time (time until breakdown) is 4 days, while the average down time (time until repair) is 3 days. Assume continuous operation. Let X ( t ) = 1 if the Ricoh is working at time t , and let X ( t ) = 0 otherwise. (a) What do we need to assume about the successive up and down times in order to make the stochastic process { X ( t ) : t 0 } a continuous-time Markov chain (CTMC)? Henceforth assume that these extra assumptions are in place, so that indeed the stochastic process { X ( t ) : t 0 } is a CTMC. (b) Construct the CTMC; i.e., specify the model. (c) Assuming that Ricoh has been working continuously for 7 days, what is the probability that it will remain working at least 8 more days? (d) Suppose that Ricoh has been working continuously for 12 days. From that moment forward, let T be the time until the second breakdown. What is the expected value E [ T ]? (e) What is the long-run proportion of time that Ricoh is up?

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