# instantaneous transition rates in continuous-time Markov chain

Consider a set of n machines and a single repair facility to service these machines. Suppose that when machine i , i = 1,2, … n, fails, it requires an exponentially distributed amount of work with rate 𝜇𝑖 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 𝜆𝑖/r

(a) Find the instantaneous transition rates

The answer is as following, Okay, I can understand that qS,S-i = µi / |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 µi divided by |S|.

But I cannot understand that qS,S+j = 𝜆𝑖.

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

Do I misunderstand something?