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Queues, stores, and tableaux

Part of: Special processes Equilibrium statistical mechanics Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Moez Draief ,
Jean Mairesse  and
Neil O'Connell
Moez Draief*
Affiliation:
Université Paris 7
Jean Mairesse*
Affiliation:
Université Paris 7
Neil O'Connell*
Affiliation:
The University of Warwick
*
∗Postal address: LIAFA, Université Paris 7, case 7014, 2 place Jussieu, 75251 Paris Cedex 05, France.
∗Postal address: LIAFA, Université Paris 7, case 7014, 2 place Jussieu, 75251 Paris Cedex 05, France.
∗∗∗∗Postal address: Mathematics Institute, The University of Warwick, Coventry CV4 7AL, UK. Email address: noc@maths.warwick.ac.uk
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Abstract

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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.

Keywords

Single-server queuestorage modelBurke's theoremnoncolliding random walktandem of queuesRobinson–Schensted–Knuth algorithm

MSC classification

Primary: 60K25: Queueing theory 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60C05: Combinatorial probability 82B41: Random walks, random surfaces, lattice animals, etc.
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 4 , December 2005 , pp. 1145 - 1167
Copyright
© Applied Probability Trust 2005 

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