# Why are the transition probabilities for an $M/M/1$ queue $\lambda$ and $\mu$?

The transition state diagram for an $$M/M/1$$ queue is shown as

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?

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