QUESTION 4 (LEARNING OUTCOME 4) A certain machine is in one of the three possibl

Question

QUESTION 4 (LEARNING OUTCOME 4) A certain machine is in one of the three possible states: (a) working, (b) under repair, (c) in an idle mode when there are no tasks to process. Being in a working mode, it can break down and get into a repair state at any hour with the probability 0.1. It can finish a task and enter an idle mode at any hour with the probability 0.2. Once machine is under repair, it will be repaired and start to process a task any hour with the probability 0.5 or wait for new task (i.e. idling) with also the (i.e. working) at the probability 0.3. Being in an idle mode, the machine receives a new task at any hour with the probability 0.8 and enters a working mode. The machine in an idle mode will never break directy. a) Describe the system as a Markov process and write down the transition matrix T of the markov chain; 4 MARKS) If the system is in working state at the beginning, what is the probability that the system will be in an idle mode after 2 hours? b) (5 MARKS) c) What is the probability that the system wll be in each state after a long time 6 MARKS) running?

Explanation / Answer

a)

There are 3 states working, repair, idle.

Each pair of transition is given.

The total transition probabilities for any row is 1.

Hence if 2 probabilities are given for a row, then then the third cell probability is 1 - cell1 prob - cell2 prob

b)

Matrix multiply the initial transition state with the transition probabilities.

Initial transition state is

select theree consecutive cells in a row and press f2.

+mmult(initital transition range, transition probability range)

press ctrl+shift +enter

This will give the state at 1st hour.

Do it once more to get the states at 2nd hour.

The probability that the machine will be in idle mode is 0.21

c)

If you go on doing this, it will end up at this equilibrium state

Working Repair Idle Working 0.7 0.1 0.2 1 Repair 0.5 0.2 0.3 1 Idle 0.8 0 0.2 1

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