Queues, stores, and tableaux

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