Question

A company has three machines. Each day, independent of each other, a machine breaks down with probability p. Each night, there is one repairperson who can repair at most one machine. Let Xn

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be the number of machines available at the beginning of the nth day. Find the transition matrix for the chain {Xn } n=0,1,2,….
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.

Answer 1

To find the transition matrix for the chain {Xn} n=0,1,2,..., we need to consider the possible transitions between the states of the system and assign appropriate transition probabilities. The transition matrix is a table that represents these probabilities.

Step-by-step explanation:

To find the transition matrix for the chain {Xn} n=0,1,2,..., we need to consider the possible transitions between the states of the system. In this case, there are three possible states for Xn: 0 machines available, 1 machine available, and 2 machines available.

Let's denote the transition probabilities as follows:

• P(Xn+1 = 0 | Xn = 0) = (1-p)
• P(Xn+1 = 1 | Xn = 0) = p
• P(Xn+1 = 0 | Xn = 1) = (1-p)
• P(Xn+1 = 1 | Xn = 1) = p
• P(Xn+1 = 1 | Xn = 2) = p
• P(Xn+1 = 2 | Xn = 2) = (1-p)

Using these transition probabilities, we can construct the transition matrix:

Xn+1 = 0Xn+1 = 1Xn+1 = 2Xn = 0(1-p)p0Xn = 1(1-p)ppXn = 200(1-p)