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Unformatted text preview: IOE 316 Fall 2007 Homework 6 SOLUTIONS Due MONDAY December 10, 2007 NOON (12pm) Note: To turn in your homework, you can either slide it under instructors office door (1783 IOE) by noon, or bring it to Arleighs office hours that morning. No late homework will be accepted. 1. (15 points) In the summer, an air conditioner is in one of three states: (0) off, (1) low, or (2) high. While off, transitions to low occur after an exponential time with expected time 3 minutes. While in the low state, transitions to off or high are equally likely and transitions out of the low state occur at rate 2 per minute. When the system is in the high state, it makes a transition to the low state with probability 2/3 or to the off state with probability 1/3. The time spent in the high state is an exponential random variable with a mean of 2 minutes. Model this air conditioning system using a continuous time Markov chain (CTMC). (a) (5 points) Draw the CTMC rate diagram, clearly label your states and transition rates ( q ij s) (b) (5 points) Find transition rates out of the states ( v i s) and transition probabilities ( P ij s). (c) (5 points) Find CTMC limiting probabilities (make sure to write out the balance equa- tions). SOLUTION: Let the states 0, 1 and 2 represent the air conditioner being, off, low, and high, respectively. (a) (5 points) Using the answers to part (b), we calculate q 01 = v P 01 = 1 / 3 q 10 = v 1 P 10 = 1 q 12 = v 1 P 12 = 1 q 20 = v 2 P 20 = 1 / 6 q 21 = v 2 P 21 = 1 / 3 This gives us the rate transition diagram below. 1 (b) (5 points) From the given data, v = 1 / 3 and P 01 = 1. Also, v 1 = 2, and P 10 = P 12 = 1 / 2. Finally, v 2 = 0 . 5, P 21 = 2 / 3 and P 20 = 1 / 3. (c) (5 points) The balance equations are given by: P 1 + P 2 / 6 = P / 3 P / 3 + P 2 / 3 = 2 P 1 P 1 = P 2 / 2 P + P 1 + P 2 = 1 The unique solution to this system of equations is P = 4 7 , P 1 = 1 7 , P 2 = 2 7 . 2. (15 points) A processor in a parallel processing computer can work on up to four tasks at once. When the processor is working on one or more tasks, the task completion rate is three tasks per millisecond. When there are three or fewer tasks assigned to the processor, tasks are assigned at the rate of two tasks per millisecond. The processor is unreliable in the sense that any time the processor is working, it may reboot and discard all of its tasks. In any statethat any time the processor is working, it may reboot and discard all of its tasks....
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