1. Consider two machines that are maintained by a single repairman. Each machine functions for an exponentially distributed amount of time with rate A before it fails. The repair times for eachunit are exponential with rate µ.Formulate a Markov chain model for this situation with state space indicating the number of machines that are in the repair shop: S={0,1,2}. Notice that you move from 0 to 1 if one of the twomachines breaks down. You move from 1 to 0 when the machine in the repair room is repaired.2. Same as above but now machines are repaired in the order in which they fail. Each machine functions for an exponentially distributed amount of time with rate A¡ before it fails. The repairtimes for each unit are exponential with rate Pj. The state space has nodes that are keep track of the machine that is at the repair shop (in case there is only one) and keeps track of whichmachine is worked on in case there are two machines at the repair shop. That is the state space is S={0, 1, 2, 12, 21}. Formulate a Markov chain model for this situation.

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Answered: 1. Consider two machines that are…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 2EQ: 2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed...

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Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.
2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.7. How many bacteria of each strain can coexist in the test tube and consume all of the food? Table 2.7 Bacteria Strain I Bacteria Strain II Bacteria Strain III Food A 1 2 0 Food B 2 1 3 Food C 1 1 1
1. Consider two machines that are maintained by a single repairman. Each machine functions for an exponentially distributed amount of time with rate A=2 before it fails. The repair times for each unit are exponential with rate u=1. The Markov chain model for this situation has state space indicating the number of machines that are in the repair shop: S={0,1,2}. Notice that you move from 0 to 1 if one of the two machines breaks down. You move from 1 to 0 when the machine in the repair room is repaired. 2. Same as above but now machines are repaired in the order in which they fail. Each machine functions for an exponentially distributed amount of time with rate Aj before it fails. A1 = 2, A 2 = 3. The repair times for each unit are exponential with rate u j, with u 1=.5 and u 2=1 . The state space has nodes that keep track of the machine that is at the repair shop (in case there is only one) and keeps track of which machine is worked on in case there are two machines at the repair shop.…
Suppose a math professor collects data on the probability that students attending a given class meeting will attend the next one. He finds that 95% of students who attended a given class meeting will attend the following class meeting and that 25% of students who do not attend attend a given class meeting will not attend the next one. Build a discrete dynamical system model using linear algebra. Be sure to state your transition matrix explicitly. What percentage of students does your model predict will be attending class meetings by the end of the semester (in the long run)?
In a college class, 70% of the students who receive an “A” on one assignment will receive an “A” on the next assignment. On the other hand, 10% of the students who do not receive an “A” on one assignment will receive an “A” on the next assignment. Find and interpret the steady state matrix for this situation.
In some country, 90% of the daughters of working women also work and 20% of the daughters of nonworking women work. Assume that these percents remains 0.9 0.2 unchanged from one generation to the next. The corresponding transition matrix is A = 0.1 0.8 Consider a typical group of women, of whom 45% currently work. Use A and A² to determine the proportion of working women in the next two generations. The proportion of working women in the first generation is %. (Type an integer or a decimal.) %. The proportion of working women in the second generation is (Type an integer or a decimal.)
A factory worker will quit with probability 1/2 during her first month, with probability 1/4 during her second month and with probability 1/8 after that. Whenever someone quits, their replacement will start at the beginning of the next month. Model the status of each position as a Markov chain with 3 states. Identify the states and transition matrix. Write down the system of equations determining the long-run proportions. Suppose there are 900 workers in the factory. Find the average number of the workers who have been there for more than 2 months.
You have a markov chain and you can assume that P(X0 = 1) = P(X0 = 2) = 1/2 and that the matrix looks like this (1/2 1/2 1/3 2/3) Find P(X2 = 1)
Please do question 1b, 1c and 1d with full working out. I'm struggling to understand what to write
Consider the two state switch model from the videos with state space S = {1,2} and transition rate matrix where A = 1 and u = 1.5. (a) If the system is in state 1, what is the mean time until it transitions to state 2? Number (b) Evaluate P11 (0) Number (C) Evaluate P11 (0.4) Number (d) Evaluate P21 (0.4) Number
Consider the two state switch model from the videos with state space S = {1,2} and transition rate matrix where X = 1 and u = 1.2. (a) If the system is in state 1, what is the mean time until it transitions to state 2? Number (b) Evaluate P11 (0) Number (c) Evaluate P11 (0.6) Number (d) Evaluate P21 (0.6) Number
A system consists of five components, each can be operational or not. Each day one operational component is used and it will fail with probability 20%. Suppose there are five repairmen available that can each work on one broken down component per day. Each repairman successfully fixes the component with probability 70% regardless of whether he has worked on it previous days. Model the system as a Markov chain Write down equations for determining long-run proportions. Suppose that you are interested in the average number of repairmen who are working per day. Explain how you would find it using your model. You are not asked to solve any equations here, instead describe how you would use the solutions.

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