Setting up Kolmogorov's Backward Equations

[dmatador]

Consider two machines, both of which have an exponential lifetime with mean 1/λ. There is one repairman who can service machines at an exponential rate of μ.

How does one set up the Kolmogorov backward equations for such a scenario? I am not sure after finding the rates how to work those into the formula, or turn them into the formula, rather.

 

[Ray Vickson]

Consider two machines, both of which have an exponential lifetime with mean 1/λ. There is one repairman who can service machines at an exponential rate of μ.

How does one set up the Kolmogorov backward equations for such a scenario? I am not sure after finding the rates how to work those into the formula, or turn them into the formula, rather.

Show your work. In particular: what are the states? What are the transition rates? Once you have the transition rate matrix you can just apply the textbook expressions for the backward Komogorov equations.

RGV

 

[dmatador]

Show your work. In particular: what are the states? What are the transition rates? Once you have the transition rate matrix you can just apply the textbook expressions for the backward Komogorov equations.

RGV

So you use a matrix to solve the transition probabilities in the backwards equations? Can you elaborate on this. I am using Ross's book "Introduction to
Probability Models" and he doesn't explain how to do this, at least not in this section.

 

[Ray Vickson]

Before answering that, could you please tell me your answers to my two questions: (i) what are the states?; and (ii) what are the transition rates? If you don't have the correct answers to these two questions there would be no point in proceeding further now.

RGV