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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Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter2: Matrices

Section2.5: Markov Chain

Problem 47E: Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.

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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)?
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 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.
Please show solutions and explain steps
In the B&K model of Example 18.5-1, suppose that the interarrival time at the checkout area is exponential with mean 5 minutes and that the checkout time per customer is also exponential with mean 10 minutes. Suppose further that will add a fourth counter. Counters 1,2, and 3 will open based on increments of two customers and counter 4 will open when there are 7 or more in the store. (a) The steady-state probabilities, for all . (b) The probability that a fourth counter will be needed. (c) The average number of idle counters.
There are two printers in the computer lab. Printer i operates for an exponential time withrate λi before breaking down, i = 1, 2. When a printer breaks down, maintenance is called to fix it,and the repair times (for either printer) are exponential with rate μ. (a) Can we analyze this as a birth and death process? Briefly explain your answer.(b) Model this as a continuous time Markov chain (CTMC). Clearly define all the statesand draw the state transition diagram.

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