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FinalORIE361S08 - ORIE 361 Final Examination Spring 2008...

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ORIE 361 – Final Examination Spring 2008 Instructor: Mark E. Lewis May 7, 2008 This is a closed book, closed notes exam. However, you are allowed to use a single 8.5 X 11 inch sheet of paper with whatever formulas or notes you deem appropriate. No other resources are allowed . Please note that communication devices or calculators are NOT allowed in sight dur- ing the exam. You have 150 minutes to complete the exam. The total possible number of points is 100 with a 2 point bonus question. The completed exam must be turned in to me or the TAs by the end of the exam period on the day of the exam. NO EXCEPTIONS . The academic integrity code is to be adhered to at all times. Good luck! Name: Section: netid: Other than that given by the instructor or proctors, I have not been given nor received aid on this exam. Signature: 1
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ORIE 361 – Spring 2008, Final 2 1. (20 points total) The Markov chain of a manufacturing process goes as follows: A part to be manufactured will begin the process by entering step 1. After step 1, 20% of the parts must be reworked; i.e., returned to step 1, 10% of the parts are thrown away and 70% proceed to step 2. After step 2, 5% of the parts must be returned to the step 1, 10% to step 2, 5% are scrapped, and 80% emerge to be sold for a profit. (a) (5 points) Formulate a four state Markov chain with states 1,2,3,4 describing the path of a single part, where 3=a part that was scrapped and 4=a part that was sold for a profit. Give the transition matrix. (b) (15 points) Suppose parts arrive according to a Poisson process with rate λ and parts that are sold for profit can be sold for a profit of $R per part. Assuming that the whole manufacturing process can be completed at a rate (strictly) faster than λ , what is the long-run average profit rate? 2. (17 points total) Suppose we have a single server queue with a Poisson arrival process, say N ( t ) with rate λ > 0 . Customers service times are exponential with rate μ > 0 . However, each customer has its own independent and unknown time that they are willing to wait until their service begins. Assume that this time is exponential with rate β > 0 . Customers that do not begin service before their “patience” time is up, leave without service. Let { Q ( t ) , t 0 } denote the process describing the number of customers in the system at time t . Consider the following questions. (a) (3 points) Q ( t ) is a birth and death process. State the birth rates { λ i , i 0 } and death rates { μ i , i 1 } . (b) (4 points) Is { Q ( t ) , t 0 } uniformizable? Explain why. (c) (10 points) Recall that a solution to ηA = 0 for a birth and death process is η n = Π n - 1 i =0 λ i Π n i =1 μ i Under what conditions does a stationary distribution exist? Explain how you came to your conclusion. (be specific...but choose the most relaxed conditions you can) 3. (18 points total, 2 points each) Please answer the following questions with “True” or “False”.
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