The $M/M/1$ queue have all the properties of the countable state continuous time markov chain.
Is any general queue also a countable state CTMC?
2026-03-28 07:05:17.1774681517
Are queues CTMC?
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$M/M/1$ and its relatives are the queues which are markov chains.
This is since these queues have exponentially distributed interarrival times and service times.
$M/D/1$ is not a markov chain but there exists an imbedded discrete time markov chain whose properties provide information about the process.
$M/G/1, G/G/1$ etc are not markov chains.
But if either the interarrival times or service times are exponentially distributed then the general theory of markov chains still provides a method for studying the queue.