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Unformatted text preview: 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 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 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 , what is the longrun average profit rate? 2. (17 points total) Suppose we have a single server queue with a Poisson arrival process, say N ( t ) with rate > . Customers service times are exponential with rate > . 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 > . Customers that do not begin service before their patience time is up, leave without service. Let { Q ( t ) ,t } 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 { i ,i } and death rates { i ,i 1 } . (b) (4 points) Is { Q ( t ) ,t } uniformizable? Explain why. (c) (10 points) Recall that a solution to A = 0 for a birth and death process is n = n 1 i =0 i n i =1 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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This note was uploaded on 03/01/2010 for the course ORIE 361 taught by Professor Lewis,m. during the Spring '07 term at Cornell University (Engineering School).
 Spring '07
 LEWIS,M.

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