Markov Chain with 2 machines and 2 repair men

Question

If we have 2 machines and 2 repair men. The machines each break down at different rates $\alpha_a$ for machine A and $\alpha_b$ for machine B. If we assume that the repairmen have different abilities, they each repair with exponential random times with parameters $\beta_1$ and $\beta_2$ respectively(naturally, this rate does not depend on which machine they repair).

Define a Markov chain, specify the state space and the transition matrix. Also determine the communicating classes.

My attempt: I am unsure of how to proceed for this case. Should it been machine at the same rate, and the repairmen with same abilities, I know how to solve this question. But now that the rates have been split, I am unsure how to construct a state space.

Answer 1

When $\beta_1 = \beta_2$, this would be a classic 4-state (AB, !AB, A!B, !A!B) case. For $\beta_1 \ne \beta_2$, you would have to expand the cases for the availability of the repairmen. This would mean however that the failure transitions have to be split on two state variants where either repairman 1 or 2 is available to repair. But if you need to adjust $\alpha_a$ and $\alpha_b$ for this, you might as well adjust the effective repair rates of the 4-state case by an effective repair rate weighed by the repairmen availability: $$\beta_{eff} = {{\beta_1 P\{1\;available\}+\beta_2 P\{2\;available\}}\over {P\{1\;available\}+P\{2\;available\}}}$$ If the repairmen availability is proportional to their repair rate, i.e. $P\{i\;available\}\propto \beta_i $: $$\beta_{eff} = {{\beta_1^2+\beta_2^2}\over {\beta_1+\beta_2}}$$

Answer 2

Guide:

One possibility,

• No broken machine. $1$ state.
• Machine $i_1$ is broken and it is being fixed by repairman $j_1$, machine $i_2$ is fine. $4$ such states.
• Machine $i_1$ is broken and it is being fixed by repairman $j_1$, machine $i_2$ is being fixed by repairman $j_2$. $2$ states.

With that, try to write down the transition rate.