Consider a time reversible continuous-time Markov chain having infinitesimal transition rates qij and limiting probabilities {Pi }.
Question:
Consider a time reversible continuous-time Markov chain having infinitesimal transition rates qij and limiting probabilities {Pi }. Let A denote a set of states for this chain, and consider a new continuous-time Markov chain with transition rates q
∗
ij given by
where c is an arbitrary positive number. Show that this chain remains time reversible, and find its limiting probabilities.
36. Consider a system of n components such that the working times of component i, i = 1, . . . , n, are exponentially distributed with rate λi . When a component fails, however, the repair rate of component i depends on how many other components are down. Specifically, suppose that the instantaneous repair rate of component i, i = 1, . . . , n, when there are a total of k failed components, is αkμi .
(a) Explain how we can analyze the preceding as a continuous-time Markov chain. Define the states and give the parameters of the chain.
(b) Show that, in steady state, the chain is time reversible and compute the limiting probabilities.
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