Consider a machine that can be in one of three states: good condition, fair condition, or broken down. Suppose that a machine in good condition will remain this way for a mean time μ1 and then will go to either the fair condition or the broken condition with respective probabilities 3/4 and 1/4

Question

Consider a machine that can be in one of three states: good condition, fair condition, or broken down. Suppose that a machine in good condition will remain this way for a mean time [latex]μ_{1}[/latex] and then will go to either the fair condition or the broken condition with respective probabilities [latex]\frac{3}{4} and \frac{1}{4}[/latex]. A machine in fair condition will remain that way for a mean time [latex]μ_{2}[/latex] and then will break down. A broken machine will be repaired, which takes a mean time [latex]μ_{3}[/latex], and when repaired will be in good condition with probability [latex]\frac{2}{3}[/latex] and fair condition with probability [latex]\frac{1}{3}[/latex] What proportion of time is the machine in each state?

Accepted Answer

Letting the states be 1, 2, 3, we have by Equation (7.23)

[latex]\pi_i=\sum\limits_{j=1}^N \pi_j P_{j i}, \quad i=1,2, \ldots, N[/latex] (7.23)

that the [latex]π_{i}[/latex] satisfy
[latex]\begin{aligned}\pi_1+\pi_2+\pi_3 & =1, \\pi_1 & =\frac{2}{3} \pi_3, \\pi_2 & =\frac{3}{4} \pi_1+\frac{1}{3} \pi_3, \\pi_3 & =\frac{1}{4} \pi_1+\pi_2\end{aligned}[/latex]
The solution is
[latex]\pi_1=\frac{4}{15}, \quad \pi_2=\frac{1}{3}, \quad \pi_3=\frac{2}{5}[/latex]
Hence, from Equation (7.24)

[latex]P_i=\frac{\pi_i \mu_i}{\sum_{j=1}^N \pi_j \mu_j}, \quad i=1,2, \ldots, N[/latex] (7.24)

we obtain that [latex]P_i[/latex], the proportion of time the machine is in state i, is given by
[latex]\begin{aligned}& P_1=\frac{4 \mu_1}{4 \mu_1+5 \mu_2+6 \mu_3}, \& P_2=\frac{5 \mu_2}{4 \mu_1+5 \mu_2+6 \mu_3}, \& P_3=\frac{6 \mu_3}{4 \mu_1+5 \mu_2+6 \mu_3}\end{aligned}[/latex]
For instance, if [latex]\mu_1=5, \mu_2=2, \mu_3=1[/latex], then the machine will be in good condition[latex]\frac{5}{9}[/latex] of the time, in fair condition [latex]\frac{5}{18}[/latex] of the time, in broken condition [latex]\frac{1}{6}[/latex] of the time.

Question 7.27: Consider a machine that can be in one of three states: good ......

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