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Consider a set of n machines and a single repair facility to service these machines. Suppose that when machine i , i = 1,2, โ€ฆ n, fails, it requires an exponentially distributed amount of work with rate ๐œ‡๐‘– to repair it. The repair facility divides its efforts equally among all failed machines in the sense that whenever there are k failed machines each one receives work at a rate of 1/k per unit time. If there are a total of r working machines, including machine i , then i fails at an instantaneous rate ๐œ†๐‘–/r

(a) Find the instantaneous transition rates

The answer is as following, enter image description here

Okay, I can understand that qS,S-i = ยตi / |S|. As far as I know, because the repair facility divides its efforts equally among all failed machines in the sense, we need to make ยตi divided by |S|.

But I cannot understand that qS,S+j = ๐œ†๐‘–.

In my opinion, since the sentence-"i fails at an instantaneous rate ๐œ†๐‘–/r" should imply that the rate at which process makes a transition from state S into state S+j is ๐œ†j/r, then qS,S+j should be
๐œ†j/|$S^c$|.

Do I misunderstand something?

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