$M/M^B/1$ Burke's theorem : what is the distribution of the output batch interarrival times?

Question

Setup: Take an M/M/1 queue:

• the inputs arrive according to a Poisson process at rate $\lambda$,
• the service time per item is distributed exponentially with mean $1/\mu$, $\mu > \lambda$
• the outputs are grouped : when $B$ items are processed, the batch is sent out.

Think of pieces of wood that arrive to a wood carving station, which makes wooden balls out of them. When the worker has 10 balls ready, he puts them in a box and passes over to the next work station.

If $B=1$ the Burke's theorem holds: the batches arrive at the Poisson rate $\lambda$.

Question : What is the general sized ($B>1$) batch output process ?

This is a problem of batched departures, or bulk departures, as opposed to the batch arrivals, references to which are abundant.