Queues, stores, and tableaux

Consider the single server queue with an infinite buffer and a FIFO discipline, either of type M/M/1 or Geom/Geom/1. Denote by A the arrival process and by s the services. Assume the stability condition to be satisfied. Denote by D the departure process in equilibrium and by r the time spent by the customers at the very back of the queue. We prove that (D, r) has the same law as (A, s) which is an extension of the classical Burke Theorem. In fact, r can be viewed as the departures from a dual storage model. This duality between the two models also appears when studying the transient behavior of a tandem by means of the RSK algorithm: the first and last row of the resulting semi-standard Young tableau are respectively the last instant of departure in the queue and the total number of departures in the store.