Statoinary Markov processes
1. Situation diagram the work of the continuous Markov system in Fig. There is a state change rate in units per hour. Find
a) the finite probability of each state.
b) System Availability
c) the MTTF value of the system when
a. State 1 is Normal, State 2 is Alternate and State 3 is Failure (100 hours).
b. Status 1 is working state, state 2 and state 3 are fail state (50 hours).
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2. If assume collection of modules have exponentially distributed lifetimes (age of component doesn't matter in...
2. If assume collection of modules have exponentially distributed lifetimes (age of component doesn't matter in failure probability) and modules fail independently, overall failure rate of collection is sum of failure rates of modules. Calculate MTTF of a disk subsystem with • 10 disks, each rated at 1,000,000 hour MTTF • 1 SCSI controller, 500,000 hour MTTF • 1 power supply, 200,000 hour MTTF • 1 fan, 200,000 MTTF • 1 SCSI cable, 1,000,000 hour MTTF 1) Failure Rate? 2)...
Markov chains
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2. (10 points) Consider a continuous-time Markov chain with the transition rate matrix -4 2 2 Q 34 1 5 0 -5 (a) What is the expected amount of time spent in each state? (b) What is the transition probability matrix of the embedded discrete-time Markov chain? (c) Is this continuous-time Markov chain irreducible? (d) Compute the stationary distribution for the continuous-time Markov chain and the em- bedded discrete-time Markov chain and compare the two
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Need 2-8 - CHAPTER QUESTIONS Reliability 245 1. Definere blir ne reliability in your own words....
Need 2-8
- CHAPTER QUESTIONS Reliability 245 1. Definere blir ne reliability in your own words. Describe the key elements and why they are important 2. Describe the three phases of the life syde curve. Drew the curve and label it in detail (the axes, phases, type of product failure, etc.). 3. Determine the failure rate for the following You have tested circuit boards for failures during a 300-hour continuous use test. Four of the 25 boards tailed. The first...
Consider the following Markov chain with the following transition diagram on states (1,2,3 2 1/3 1 1/4 2 3 s this Mar...
Consider the following Markov chain with the following transition diagram on states (1,2,3 2 1/3 1 1/4 2 3 s this Markov chain irreducible? 1 marks (a) (b) Find the probability of the Markov chain to move to state 3 after two time steps, providing it starts in state 2 [3 marks 14 Find the stationary distribution of this Markov chain [4 marks (c) (d) Is the stationary distribution also a limiting distribution for this Markov chain? Explain your answer...
1. Consider a Markov chain (X) where X E(1.2,3), with state transition matrix 1/2 1/3 1/6 0 1/4 (a) (6 points) Sketch the associated state transition diagram (b) (10 points) Suppose the Markov ch...
1. Consider a Markov chain (X) where X E(1.2,3), with state transition matrix 1/2 1/3 1/6 0 1/4 (a) (6 points) Sketch the associated state transition diagram (b) (10 points) Suppose the Markov chain starts in state 1. What is the probability that it is in state 3 after two steps? (c) (10 points) Caleulate the steady-state distribution (s) for states 1, 2, and 3, respee- tively
1. Consider a Markov chain (X) where X E(1.2,3), with state transition matrix...
6. [10 pts.] Suppose a computer system works in two modes of operation: a sleep mode, A', where i...
6. [10 pts.] Suppose a computer system works in two modes of operation: a sleep mode, A', where it is under-utilized and working mode A where it is adequately utilized. Every hour the computer system changes its state according to the following state diagram: 0.2 0.1 a. Finish labeling the transition on the state diagram. b. Give the corresponding probability matrix. c. What are the probabilities of being in the states A and A' after 3 hours of work. Suppose...
Homework Assignment 3.5 Summer 2018 Question 3: Continuous-time Markov Chains (a) A facility has three that...
Homework Assignment 3.5 Summer 2018 Question 3: Continuous-time Markov Chains (a) A facility has three that are identical. Each machine fails independently with an exponential distribution with a rate of 1 every day; repairs on any machine are also exponentially distributed with a rate of 1 every 12 hours. Create a continuous-time Markov chain to model this (identify the rates, and the transition probabilities) (b) Now, assume the facility above has three machines, but one of them is of Type...
11. Consider a Markov process with transition matrix State 1 State 2 State 1 0.2 0.11...
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Question 1 A Markov process has two states A and B with transition graph below. a) Write in the two missing probabilities. (b) Suppose the system is in state A initially. Use a tree diagram to find t...
Question 1 A Markov process has two states A and B with transition graph below. a) Write in the two missing probabilities. (b) Suppose the system is in state A initially. Use a tree diagram to find the probability B) 0.7 0.2 A that the system wil be in state B after three steps. (c) The transition matrix for this process is T- (d) Use T to recalculate the probability found in (b.
Question 1 A Markov process has two...
2. (10 points) Consider a continuous-time Markov chain with the transition rate matrix -4 2 2 Q 34 1 5 0 -5 (a) What is the expected amount of time spent in each state? (b) What is the transition probability matrix of the embedded discrete-time Markov chain? (c) Is this continuous-time Markov chain irreducible? (d) Compute the stationary distribution for the continuous-time Markov chain and the em- bedded discrete-time Markov chain and compare the two
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Need 2-8
- CHAPTER QUESTIONS Reliability 245 1. Definere blir ne reliability in your own words. Describe the key elements and why they are important 2. Describe the three phases of the life syde curve. Drew the curve and label it in detail (the axes, phases, type of product failure, etc.). 3. Determine the failure rate for the following You have tested circuit boards for failures during a 300-hour continuous use test. Four of the 25 boards tailed. The first...
Consider the following Markov chain with the following transition diagram on states (1,2,3 2 1/3 1 1/4 2 3 s this Markov chain irreducible? 1 marks (a) (b) Find the probability of the Markov chain to move to state 3 after two time steps, providing it starts in state 2 [3 marks 14 Find the stationary distribution of this Markov chain [4 marks (c) (d) Is the stationary distribution also a limiting distribution for this Markov chain? Explain your answer...
1. Consider a Markov chain (X) where X E(1.2,3), with state transition matrix 1/2 1/3 1/6 0 1/4 (a) (6 points) Sketch the associated state transition diagram (b) (10 points) Suppose the Markov chain starts in state 1. What is the probability that it is in state 3 after two steps? (c) (10 points) Caleulate the steady-state distribution (s) for states 1, 2, and 3, respee- tively
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6. [10 pts.] Suppose a computer system works in two modes of operation: a sleep mode, A', where it is under-utilized and working mode A where it is adequately utilized. Every hour the computer system changes its state according to the following state diagram: 0.2 0.1 a. Finish labeling the transition on the state diagram. b. Give the corresponding probability matrix. c. What are the probabilities of being in the states A and A' after 3 hours of work. Suppose...
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