Queues, stores, and tableaux

TL;DR

This paper extends Burke's theorem to single-server queues with infinite buffers, revealing a duality between departure processes and storage models, and connects queue behavior to Young tableau structures.

Contribution

It proves a new law equivalence for departure and arrival processes in M/M/1 and Geom/Geom/1 queues, linking queue dynamics to dual storage models and Young tableaux.

Findings

(D,r) has the same law as (A,s) in these queues

r can be interpreted as departures from a dual storage model

Queue behavior relates to Young tableau via RSK algorithm

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 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 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.