Joint Burke's Theorem and RSK Representation for a Queue and a Store

Moez Draief (1) , Jean Mairesse (1) , Neil O'Connell (2)

1 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
2 WMI - Warwick Mathematics Institute

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 $\mathcal{A}$ the arrival process and by $s$ the services. Assume the stability condition to be satisfied. Denote by $\mathcal{D}$ the departure process in equilibrium and by $r$ the time spent by the customers at the very back of the queue. We prove that $(\mathcal{D},r)$ has the same law as $(\mathcal{A},s)$ 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.

Keywords: Single server queue; storage model; Burke's theorem; non-colliding random walks; tandem of queues; Robinson-Schensted-Knuth algorithm

Domains: Data Structures and Algorithms [cs.DS]; Discrete Mathematics [cs.DM]; Combinatorics [math.CO]; Computational Geometry [cs.CG]

Moez Draief, Jean Mairesse, Neil O'Connell. Joint Burke's Theorem and RSK Representation for a Queue and a Store. Discrete Random Walks, DRW'03, 2003, Paris, France. pp.69-82, ⟨10.46298/dmtcs.3339⟩. ⟨hal-01183934⟩