A company has two machines. During any day, each machine that is working at the

Question

A company has two machines. During any day, each machine that is working at the beginning of the day has a chance of breaking down. If a machine breaks down during the day, it is sent to a repair facility and will be working two days after it breaks down. (Thus, if a machine breaks down during day 3, it will be working at the beginning of day 5). Letting the state of the system be the number of machines working at the beginning of the day, formulate a transition probability matrix for the situation.

Explanation / Answer

The aim is to form markov chain for the given information,

For the given information there are two states,1)Sunny 2)Cloudy

As state of weather for tomorrow depends on the stste that is today so Markov chain is formed, Also as probabilities do not change with time,so a stationary Markov chain is formed.

The trasition probability matrix for the two state markov chain is,

P= sunny cloudy

sunny 0.9 0.1

cloudy 0.2 0.8

The above transition matrix is explained as,

Suppose today is sunny day,then the probability that tomorrow being sunny day is 0.9. Hence entry is 0.9 in row sunny and column sunny.

If it is not a sunny day then it is cloudy day,hence  probability that today being sunny day and tomorrow being cloudy day is (1-0.9)= 0.1

Hence entry is 0.1 in row sunny and column cloudy.

Similarly for today being cloudy and tomorrow being cloudy or sunny.

Hence Markov chain is shown below,

P= sunny cloudy

sunny 0.9 0.1

cloudy 0.2 0.8

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