Setting up Kolmogorov's Backward Equations

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Homework Help Overview

The discussion revolves around setting up Kolmogorov's backward equations in the context of a system involving two machines with exponential lifetimes and a repairman servicing them at an exponential rate. Participants are exploring how to incorporate the rates into the equations and the necessary components for the setup.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants are attempting to identify the states and transition rates relevant to the problem. There is a focus on understanding how to construct the transition rate matrix and apply it to the backward equations. Questions are raised about the specifics of the setup and the application of textbook expressions.

Discussion Status

The discussion is ongoing, with participants seeking clarification on foundational aspects such as states and transition rates before proceeding further. There is a request for elaboration on the use of matrices in solving transition probabilities, indicating a productive exchange of ideas.

Contextual Notes

Participants reference a specific textbook, "Introduction to Probability Models" by Ross, noting that it lacks detailed explanations for the current topic, which may impact their understanding and approach.

dmatador
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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.
 
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dmatador said:
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
 
Ray Vickson said:
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.
 
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
 

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