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  "body": "\n## Summary\n\nA continuous-time Markov chain is a process with values in a countable set that jumps between states at random times and has the [[Markov property|0B77KZCQ51]].\nIt stays in each state for an [[exponentially distributed|0E99F51GVB]] time, and then jumps according to fixed probabilities.\nIts law is described by a rate matrix $Q$, and on a finite set its transition probabilities form the [[matrix exponential|020M04FN1H]] $e^{tQ}$.\nQueues, birth–death processes and the Poisson process are standard examples.\n\n## Explanations\n\n### L1 human\n\nA discrete-time [[Markov chain|0G48PYM96Q]] moves at each tick of a clock.\nA continuous-time chain instead waits a random amount of time in each state before it moves.\nThe waiting time has no memory: having waited a while does not make a jump more or less likely soon.\nFor example, a machine that breaks down at rate $a$ and is repaired at rate $b$ is a chain on two states, working and broken.\nIn the long run it is working a fraction $b/(a + b)$ of the time.\n",
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