The district of Vitkovice in city of Ostrava in Czechia

The district of Vitkovice in city of Ostrava in Czechia is famous for being a highly industrialized area in Central Europe. Karlshann-Kirtscher is a company that focuses on steel production and has a large iron/steel factory in Vitkovice. The factory has four modern furnaces that can take a charge of iron of up to 350 tons and convert it into steel in less than 40 minutes compared to 10–12 hours in an open-hearth furnace. The production planning department of the factory knows that during any day, each furnace that is working at the beginning of the day has a 1/3 chance of breaking down. If a furnace breaks down during the day, it is a special repair team tries to repair it and the furnace and will be working two days after it breaks down. (Thus, if a furnace breaks down during Monday, it will be working at the beginning of day Wednesday). Based on this knowledge answer the following questions:

1. Is it possible to define the define the above-mentioned process as a Markov chain? Explain in detail.

2. Define the state space for this Markov Chain. Explain what each state refers to.

3. Construct the one-step probability transition matrix for this Markov chain. Explain how you calculate with each transition probability (each Pij).

4. Suppose that 2 furnaces are in working condition on July 24th, 2020. What is the probability that there all of the furnaces are broken down on July 27th, 2020? Display all of your calculations.

5. Now compute the corresponding transition matrix after 4, 5, and 10 transitions? What can you conclude about the values you find as the number of transitions increase? Explain in detail.