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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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Consider a time-homogeneous markov chain (Xt: t = 0,1, 2, ...) with states (1,2,3}. what is P[X1 a, X4 = d | XO = i0]?
Consider the migration (Markov) matrix A = 0.7 0 0.2 0.1 0.4 0.2 0.2 0.6 0.6 Suppose that, initially, there are 96 residents in location 1, 82 residents in location 2, and 151 residents in location 3. Assume time is measured in years. Find the population in each location after 1 year. Find the population in each location after 2 years.
The figure above illustrates a continuous-time Markov process. Suppose the system is currently instate 2. After a small amount of time Δ, what is the probability that the system is in each state?
3. suppose that the manufacturer keeps a spare machine that only is used when the primary machine is being repaired. During a repair day, the spare machine has a probability of 0.1 of breaking down, in which case it is repaired the next day. Denote the state of the system by (x, y), where x and y, respectively, take on the values 1 or 0 depending upon whether the primary machine (x) and the spare machine (y) are operational (value of 1) or not operational (value of 0) at the end of the day. [Hint: Note that (0, 0) is not a possible state.] (a) Construct the (one-step) transition matrix for this Markov chain. (b) Find the expected recurrence time for the state (1, 0).
Suppose qo = [1/4, 3/4] is the initial state distribution for a Markov process with the following transition matrix: %3D [1/2 1/2] M = [3/4 1/4] (a) Find q1, 92, and q3. (b) Find the vector v that qo M" approaches. (c) Find the matrix that M" approaches.
Anne and Barry take turns rolling a pair of dice, with Anne going first. Anne’s goal is to obtain a sum of 3, while Barry’s goal is to obtain a sum of 4. The game ends when either player reaches his goal,and the one reaching the goal is the winner. Define a Markov Chain to model the problem.
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.
A state vector X for a three-state Markov chain is such that the system is as likely to be in state 2 as in state 3 and is five times as likely to be in state 1 as in 2. Find the state vector X.
Cs401 3b
What are the difficulties in estimating the following model? Use as much detail as possible in answering this question while considering the Gauss-Markov assumptions and OLS estimator. Economic productivity = β0 + β1Unemployment + β2Innovation + θiControls + ei Where unemployment is the average unemployment rate of a country and innovation is an index of R&D performance.
A Markov system with two states satisfies the following rule. If you are in state 1 then of the time you change to state 2. If you are in state 2 then of the time you remain in state 2. At time t = 0, there are 100 people in state 1 and no people in the other state. state 1 Write the transition matrix for this system using the state vector v = state 2 T = Write the state vector for time t = 0. Vo Compute the state vectors for time t = 1 and t = 2. Vị = V2
(c) Suppose there is an epidemic in which every month a quarter of those who are well become sick and 1/8 of those who are sick recover and acquire immunity for life. i. Find the matrix that describe this Markov process ii. Find the steady state of this process. Briefly justify your answer.

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