I have this assignment question and I am stumped on how to complete a Feller construction:

A system consisting of two components is subject to a series of shocks . The time be- tween consecutive shocks can be modelled as independent Exp(λ) random variables. A shock causes the failure of component 1 alone with probability p, of component 2 alone with probability q, of both components with probability r, or it has no effect with probability 1 − p − q − r. Failure/survival events are independent from shock to shock. The system fails when both components have failed. When the system fails it is repaired to full working order, i.e., both components working, and repair times have the Exp(μ) law. All repair times are independent and independent of the shock process and its outcomes. (a) Model this system as a Markov process by defining suitable states, showing that your model is a Markov process, and determine the generator matrix. Write down the transition probability matrix of the corresponding embedded Markov chain.

I keep track of each component so the states are { (0,0), (1,0), (0,1), (1,1)}.

I determined that the jump probability matrix (probability of going from state i to j given a shock occured as:

0       0       0       1
r+p+q   1-r-p-q 0       0
r+q+p   0       1-r-q-p 0
r       p       q       1-p-q-r

I am not sure how to proceed with this question now. Any suggestions or hints?

  • 1
    $\begingroup$ You apply formulas that are valid in the discrete setting although, in this problem, the time is continuous. Have you ever seen the definition of the generator of a continuous time finite Markov process? $\endgroup$ – Did Apr 9 '14 at 14:04
  • $\begingroup$ Apologies for not keeping the forum updated. This problem has been solved after a long time contemplating. The generator matrix is quite simply the rate matrix except the main diagonals = -ri (rate for that particular state), instead of 0 $\endgroup$ – Seeking Alpha Apr 18 '14 at 14:06
  • $\begingroup$ Indeed--a simple matter of reading the definitions. $\endgroup$ – Did Apr 18 '14 at 14:10

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