STAT_333_Assignment_4

# STAT_333_Assignment_ - 2 Repair times for either machine are exponential with rate μ a Define a continuous-time Markov chain to analyze this

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STAT 333 Assignment 4 Due: Tuesday, July 28 at the beginning of class Type I Problems 1. #21, page 349 of the text, 9 th edition. 2. #34, page 351 of the text, 9 th edition. 3. #59, page 356 of the text, 9 th edition. 4. #72, page 359 of the text, 9 th edition. Type II Problems 1. Calls to 911 follow a Poisson process with an average of 3 calls per minute. Unfortunately, 15% of 911 calls are for non-emergencies. a. Discuss briefly whether you think the properties of stationary and independent increments would hold for this situation b. Find the probability of 5 calls in 2 minutes. c. Find the probability of 6 emergency calls in 5 minutes. d. Given that 10 calls have occurred in 3 minutes, find the probability that 5 were in the first minute. e. Given that 10 calls have occurred in 3 minutes, find the probability that 7 of them were emergency calls. 2. Suppose two machines are maintained by a single mechanic. Machine 1 works for an exponential time with rate μ 1 before breaking down, and machine 2 with rate μ
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Unformatted text preview: 2 . Repair times for either machine are exponential with rate μ. a. Define a continuous-time Markov chain to analyze this problem. Define the state space and specify the generator matrix Q. b. If machine 1 works for an average of 5 hours, machine 2 for 4 hours, and a repair takes an average of 1 hour, solve for the equilibrium distribution and find the proportion of time neither machine is working. 3. A barbershop with a single barber has room for at most two customers. Potential customers arrive at a Poisson rate of 3 per hour, and the service times are independent exponential random variables with mean 15 minutes. a. Find the average number of customers in the shop. b. Find the proportion of potential customers that enter the shop. c. If the barber could cut his mean service time down to 10 minutes, how much of an increase in business (in customers per hour) would he see?...
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## This note was uploaded on 01/16/2011 for the course STAT 333 taught by Professor Chisholm during the Spring '08 term at Waterloo.

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