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The Central Grounds Garage has installed a card-operated gate, which, unfortunately, is vulnerable to absent-minded faculties. In particular, each day, a car crashes the gate

The Central Grounds Garage has installed a card-operated gate, which, unfortunately, is vulnerable to absent-minded faculties. In particular, each day, a car crashes the gate with probability p, in which case a new gate must be installed. Also, a gate that has survived for m days must be replaced as a matter of periodic maintenance. What is the steady-state expected frequency of gate replacements?

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