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

Abstract : 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.
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Cited literature [24 references]

https://hal.inria.fr/hal-01183934
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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. ⟨hal-01183934⟩

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