Show that q₁ =00003/502/5for the Markov chain below. If the chain is equally likely to begin ineach of the states, what is the probability of being in state 2 aftermany steps?P=11/331/41/3 1/41/41/3 1/2 1/20003/113/115/11 and 9221/404000=5are steady-state vectors0 10 20 32/31/2400 1/3 1/2 5The vector q, is a steady-state vector becauseq/pPq₁9 P-1P¹q₁simplified fraction for each

Given information: q1=31131151100P=13141400131414001312120000023120001312

Answered: 0 0 0 are steady-state vectors 0 3/5 0…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 12EQ: 12. Robots have been programmed to traverse the maze shown in Figure 3.28 and at each junction...

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12. Robots have been programmed to traverse the maze shown in Figure 3.28 and at each junction randomly choose which way to go. Figure 3.28 (a) Construct the transition matrix for the Markov chain that models this situation. (b) Suppose we start with 15 robots at each junction. Find the steady state distribution of robots. (Assume that it takes each robot the same amount of time to travel between two adjacent junctions.)
A manufacturer has a machine that if it ran all day today has a proba- bility of 0.2 of breaking down sometime during the day tomorrow. When the machine breaks down, it goes offline for the remainder of the day and then a technician will spend the next day (after the breakdown) repairing it. A newly repaired machine only has a proability of 0.1 of breaking down sometime tomorrow. (a) Formulate the evolution of the status of the machine at the end of the day as a Markov Chain by identifying the three possible states at the end of the day and providing the transition probabilities between these states. (b) Determine the expected first passage times, µij , for all combinations of states i and j where i cannot equal to j (i.e., you don’t need to determine the recurrence times). You must provide the set of equations used to calculate these µij (micro with lowercase i and j). (c) Using your results from (b): (i) Identify the expected number of full days that the machine will remain…
According the Ghana Statistical Service data collected in 2020 shows that, 5% of individuals living within the city move to the rural areas during a one-year period, while 4% of individuals living in the rural areas move to the city a one-year period. Assuming that, this process is modeled by a Markov process with two states: city and rural areasa) i. Prepare the matrix of transition probabilitiesii. Compute the steady-state probabilities.iii. In a particular District, 40% of the population lives in the city, and 60% of the population lives in the suburbs. What population changes do your steady-state probabilities project for this metropolitan area?
Data collected from selected major metropolitan areas in the eastern United States show that 3% of individuals living within the city limits move to the suburbs during a one-year period, while 1% of individuals living in the suburbs move to the city during a one-year period. Answer the following questions assuming that this process is modeled by a Markov process with two states: city and suburbs. (a) Prepare the matrix of transition probabilities. To From City Suburbs City Suburbs (b) Compute the steady-state probabilities. (Enter your probabilities as fractions.) City?1= Suburbs?2=
(Note: Express your answers as decimal fractions rounded to 4 decimal places (if they have more than 4 decimal places).)Find the vector WW of stable probabilities for the Markov chain whose transition matrix appears below:
The computer center at Rockbottom University has been experiencing computer downtime. Let us assume that the trials of an associated Markov process are defined as one-hour periods and that the probability of the system being in a running state or a down state is based on the state of the system in the previous period. Historical data show the following transition probabilities. To From Running Down Running 0.80 0.20 Down 0.30 0.70 (a) If the system is initially running, what is the probability of the system being down in the next hour of operation? (b) What are the steady-state probabilities of the system being in the running state and in the down state? (Enter your probabilities as fractions.) Running?1=Down?2=

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