Abstract
Chapter 1 introduced the θ t -framework, a convenient formalism for the study of point processes and stochastic processes which are jointly stationary. In Chapter 2, it was shown that the house is not empty but shelters a number of stationary queueing systems. In the stable case, coupling arguments often show convergence in variation of an initially non-stationary state to the stationary state obtained by a construction of the Loynes type. Thanks to the existence of construction points, from which secondary processes (such as the congestion process in G/G/1/∞) can be obtained from the stationary state (the workload process in G/G/1/∞), the convergence in variation of the secondary processes is a consequence of the same phenomenon for the stationary state. This state of things allows one to obtain the basic formulas of queueing theory concerning limit processes directly on the stationary system, in the θ t -framework. This separates the task of obtaining formulas from that of obtaining limit distributions, and enables one to give short and elementary proofs of the classical formulas and results of queueing theory.
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Baccelli, F., Brémaud, P. (2003). Formulas. In: Elements of Queueing Theory. Applications of Mathematics, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-11657-9_3
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DOI: https://doi.org/10.1007/978-3-662-11657-9_3
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