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On page 620, example 9.32 of Introduction to probability models, the average UP and DOWN times of a system that is composed of some components is given for a series system (all components should function for the system to function) and a parallel system (any component should function for the system to function). We can use these to get the rate at which system failures happen. Is there a way to get the distribution of the number of failures in a certain time period? Can we say it's going to be Poisson if the distribution of UP and DOWN times at the component level are exponential?

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