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.
Highlights
The main purpose of this paper is to clarify the interplay between two models of queueing theory
¤ D ¥ r¦ has the same law as ¤ A ¥ s¦. This joint Burke Theorem is new, it is similar in spirit to the results proved in [17, 12] for a variant: queues with unused services
£ £ £ rn is the demand met at slot n 1, see equation (9); it is the amount of P departing at slot n 1; The variables ¤ Dn¦ n and the function Q do not have a natural interpretation in this model
Summary
The main purpose of this paper is to clarify the interplay between two models of queueing theory. We assume that the random variables driving the dynamic are either exponentially or geometrically distributed, and we consider the models in equilibrium (under the stability condition) In this situation, it is well known that a Burke’s type Theorem holds: the departures and the arrivals have the same law [5, 19, 3]. © ¢ where R is the total number of departures from the last store in the tandem up to time slot N This identity is proved in two steps by showing that λK § minπ Π ∑ i j π u¤ i¥ j¦ § R, where Π is a different set of paths in the lattice. We obtain on the same Young tableau the total departures for the two tandem models
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