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.



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