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?