I have the problem below.

There are n identical machines. They are all operational at time 0. The lifetime of each one is an exponential random variable with rate L. There are r repairmen (1 ≤ r ≤ n), but only one works on a machine at a time. When a machine fails, a repairman immediately begins to fix it. The repair time is an exponential random variable with rate M. Let X(t) be the number of failed machines at time t. What is the limiting distribution of this Markov chain?

I was able to find the rate matrix, but it is awkward because I had to define k = min{r, i} where i is the number of failed machines. So my rate matrix looks like:

0 | nL | 0 | ...

kM| 0 | (n-1)L | ...

0 | kM | 0 | ...

Sorry for the bad formatting, I'm new and not sure how to make matrices. But how do I find the limiting distribution from here? Thanks.


pi P = pi , and pi_0+pi_1+ ... +pi_n=1

where P is your matrix and pi=(pi_0 pi_1 ... pi_n)

Here's what I started off with





pi_0+pi_1+ ... +pi_n=(pi_0)(1+1/kM{1+(n-1)/kM-(n-1)nL-(n-2)(n-1)L-...2*3L-1*2L})=1

Hopefully I didn't make any mistakes there with the simplification.

Sorry this doesn't look very neat. Newb here as well.


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