The transition state diagram for an $M/M/1$ queue is shown as enter image description here

The only explanation I have for the transition probabilites is that arrivals occur at rate $\lambda$ according to a Poisson process and move the process from state $i$ to $i+1$ and service completes (so departures occur) at rate $\mu$ and so this moves the process from state $i$ to $i-1$. I thought that transition diagrams show the probabilities of moving between two states so surely this explanation of how those values were obtained is not enough. How would I derive these? Or am I missing something obvious?

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    $\begingroup$ Ib this diagram the quantities on the arrows are rates. There are similar diagrams where those quantities are probabilities. $\endgroup$ – Ethan Bolker Mar 24 at 22:10
  • $\begingroup$ Understood! Are there different names for such diagrams? $\endgroup$ – user499701 Mar 24 at 22:16
  • $\begingroup$ I don't know of any. You just have to tell from context. $\endgroup$ – Ethan Bolker Mar 24 at 22:19

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