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  "body": "\n## Summary\n\nRenewal theory studies processes that start afresh at random times, with independent and identically distributed waiting times between two restarts.\nIts central object is the renewal process, which counts the restarts up to time $t$.\nIts main results, such as the renewal theorems, describe the long-run rate of restarts.\nIt applies to the replacement of machine parts, to queues and to returns of a Markov chain to a state.\n\n## Explanations\n\n### L1 human\n\nA light bulb burns out at a random time and is replaced at once by a new bulb of the same kind.\nEach replacement is a renewal: the system starts over as if new.\nRenewal theory asks how many bulbs are used by time $t$.\nIf a bulb lasts $\\mu$ hours on average, then in the long run one uses about $t / \\mu$ bulbs in $t$ hours.\nThe Poisson process is the renewal process with exponential waiting times.\n",
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