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, Contents 1 Introduction 3

.. The-controlled-queueing-model,

M. The,

, Optimization via stochastic dynamic programming 8

O. Policies,

B. Equations and .. ,

.. Optimal,

, 4 Smoothed and uniformized models 10

A. , , p.11

O. Definition, , p.11

.. Bellman-equations, , p.12

, 5 Structural properties of smoothed and uniformized models 13

P. Definition, , p.13

I. Implications, , p.14

.. Identities-for-difference-operators, , p.15

T. Invariant-properties-for-operator,

, Invariant property for operator

N. ,

, Optimality results 21

.. Optimality-results-in-the-approximated-models, 21 6.1.1 Overview of the propagation framework 22 6.1.2 Study of the sign of ? q T (u), p.25

.. Optimality-in-the-base-model, , p.26

, Methodological comments, vol.27

, Intractability of structural properties propagation framework in the Base Model, p.27

.. Intractability-of-sample-path-approaches, , p.29

.. Submodularity-study, 30 7.3.2 Details on submodularity in approximated models . . . . . . . . . . . . . . 30

.. , Study of the change of structural properties, p.31

.. , Limits of truncated non-smoothed models, p.33

, A Proof of Proposition, vol.1

A. Theorem, , p.38