By Eitan Altman, Augustin Chaintreau
This booklet constitutes the refereed complaints of the second one Euro-NF foreign convention, NET-COOP 2008 held in Paris, France, in September 2008. The thirteen revised complete papers provided have been conscientiously reviewed and chosen from a complete of 27 submissions. The papers are equipped in topical sections on economics and peer-to-peer networks; routing and measurements; scheduling; tcp and congestion keep an eye on; in addition to instant networks.
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Extra resources for Network Control and Optimization: Second EuroFGI Workshop, NET-COOP 2008 Paris, France, September 8-10, 2008, Revised Selected Papers
2 The Stochastic Model A ﬁle is represented as a set F of size n, the elements of which are called chunks. Consider a large networked “swarm” of N nodes called peers. Each peer possesses a certain (possibly empty) subset A of F . As time goes by, this peer interacts with other peers, the goal being to enlarge his set A until, eventually, the peer manages to collect all n chunks of F . The interaction between peers can either be a download or a swap; in both cases, chunks are being copied from peer to peer and are assumed never lost.
Picking two peers labelled A and B (provided that A ∼ B) from the population is done in xA xB ways. Thus the rate of a swap A B is proportional to xA xB . So if we fix two chunks i ∈ A \ B, j ∈ B \ A and specify that A = A + j, B = B + i, then the chance of picking i from A \ B and j from B \ A is 1/|A \ B||B \ A|. Thus, ⎧ xA xB ⎨ , the rate of the swap (A B) (A , B ) equals γ (SR) |A \ B||B \ A| ⎩ a long as A A , B B , A − A ⊂ B, B − B ⊂ A, where γ > 0. 4 Deriving the Markov Chain Rates Having deﬁned the rates of each individual interaction we can easily deﬁne rates P(F ) q(x, y) of a Markov chain in continuous time and state space Z+ as follows.
7) β δ β β Furthermore, there exists a λ0 > 0 such that for all λ < λ0 (7) holds. 6 Conclusions and Open Problems We proposed a stochastic model of a BitTorrent-like network and showed the existence of an ODE limit, along with a diﬀusion approximation. Several simulations  test the suitability of the model. Proofs of some of the results presented in this paper can be found in . One can look at the ODE limit and, more speciﬁcally, its equilibria in order to obtain crude information about the stationary distribution of the original model.
Network Control and Optimization: Second EuroFGI Workshop, NET-COOP 2008 Paris, France, September 8-10, 2008, Revised Selected Papers by Eitan Altman, Augustin Chaintreau