Th`ese de Doctorat de l'université Paris VI Pierre et Marie Curie Mlle ...

Optimum Bandwidth Allocation Algorithm
If we assume that the utility function Uk(x) is known or can be estimated for every greedy
source k, then we can illustrate the Optimum Bandwidth Allocation (OBA) problem for-
mulation. The decision variable f n k , k ∈ Kg, represents the amount of extra-bandwidth
that is allocated to each greedy connection during the n−th update interval. We maximize
the total network extra-revenue considering the following mathematical model:
Maximize
Uk(srk + f n k ) − Uk(srk) (5.2)
k∈Kg
s.t.
f n k · a l k ≤ Rl, ∀l ∈ L (5.3)
k∈Kg
srk + f n k
f n k
≤ Oln−1
k , ∀k ∈ Kg (5.4)
≥ 0, ∀k ∈ Kg
(5.5)
The objective function (5.2) is the total network extra-revenue.
Constraint (5.3) represents capacity constraints expressed for each link of the graph.
Constraint (5.4) imposes that, for every update interval n, the total load allocated to each
greedy source k does not exceed the total load offered to the network by k, thus avoiding
to waste extra-bandwidth.
If sources offered load is unknown or difficult to estimate, we can consider an alternate
formulation to the OBA problem by simply dropping constraint (5.4).
If all users utility functions are differentiable and strictly concave, then the objective func-
tion (5.2) is differentiable and strictly concave. Since the feasible region (5.3), (5.4) and
(5.5) is compact, a maximizing value of f n k
exists and can be found using Lagrangian meth-
ods. Further, if the objective function can be approximated using piece-wise linear concave
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