*32. Let D denote the time between successive departures in a stationary queue withI mage. Show, by conditioning on whether or not a departure has left the system empty, that D is exponential with rate λ.

Hint: By conditioning on whether or not the departure has left the system empty we see that Image whereI mage represents the sum of two independent exponential random variables having rates μ and λ. Now use momentgenerating functions to show that D has the required distribution.

Note that the preceding does not prove that the departure process is Poisson. To prove this we need show not only that the interdeparture times are all exponential with rate λ, but also that they are independent.