Queues, stores, and tableaux

Abstract

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 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 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.