Transforms Derivation of Burke’s Theorem
In an M/M/1, when the server is busy, jobs depart at rate μ. However, when the
M/M/1 is idle, then no jobs depart. Thus, the interdeparture times are either distributed Exp(μ) (when the server is busy), or Exp(λ) + Exp(μ) (when idle) –
this latter term comes from having to wait for an arrival and then for that arrival
to depart. It is not at all clear how having interarrival times switch between
these modes could form a Poisson(λ) departure process.
Let T denote the time between departures. Prove that T ∼ Exp(λ) by deriving
its Laplace transform via conditioning.