I am reviewing some queueing problems from Gross and Harris, and had a question on problem 5.40 part b.
The problem is stated as follows: Part B: Show that the stationary output of a $ G/M/1 $ queue is Poisson if and only if $G$ is exponential. Hint: The idle time in an arbitrary interdeparture period (called the virtual idle time) has a CDF given by $ F(u) = A(u) + \int_{u}^{\infty} e^{-\mu (1 - r_{0})(t - u)} d A(t) $. Use the fact that each departure time is a sum of a virtual idle time and a service time.
I think that I should use the following approach:
- From the given virtual idle time CDF, create a virtual idle time LST. Note that in the CDF, $A(u)$ is the arrival time CDF
- Given 1, create LST of virtual idle time $+$ service time by multiplying LSTs, where service time LST is $ \frac{\mu}{\mu+s} $
- Show that the LST from 2 is an exponential LST
However, I am getting a bit stuck on creating the LST from $F(u)$.