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...indow control algorithm must be stable, in addition to being fair and efficient, at the flow level. The emerging theory of large-scale networks under end-to-end control, e.g., [31], [42], [35], [50], =-=[46]-=-, [76], [44], [41], [6], [51], [63], [34], [19], [39], [33], [64], [53], [8], [52], [73], [11], [56] (see also, e.g., [43], [40], [30], [57] for recent surveys), forms the foundation of the flow-level...

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..., shows that the network as a whole converges to an operating point that minimizes P r 1 d 2 r xr . This utility function called the potential delay was introduced as a possible fairness criterion in =-=[13]-=-. The throughput under this utility function is given by 1=d r p p r ; where p r is the loss probability seen by User r which is consistent with the models in [14, 6, 7]. While we use this simplified ...

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...of these flows. The Br must satisfy the capacity constraints: � xrBr ≤ Cl, for l ∈L. (11) r∋l The issue of what constitutes a fair bandwidth allocation has been discussed in a number of recent papers =-=[20, 26, 24]-=-. A significant range of fairness notions can be expressed as the solution to the following optimization problem [26]: Maximize � B 1−α r wrxr , subject to constraints (11), 1 − α r (12) where α �= 1 ...

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...cs but not the utility function. The relation (17) between equilibrium source rate and loss probability reduces to the well known relation (see e.g. [40, 41]): xi = a τi √ qi when the probability qi is small, or equivalently, when the window τixi is large compared with √ 2.1 The value of the constant a, around 1, has been found empirically to depend on implementation details such as TCP variant (e.g., Reno vs. NewReno vs. SACK) and whether delayed acknowledgment is implemented. Equating U ′i(xi) with qi, the utility function of TCP Reno becomes: Ui(xi) = − a 2 τ2i xi . This version is used in [15, 42]. We now turn to link algorithm, i.e. how loss probabilities pl(t) are generated as a function of link rate yl. Both DropTail and RED produce losses as a function of the state of the queue, so the 1This corresponds to replacing (1 − qi(t)) in (16) by 1, as done in [22]. 11 question involves modeling of the queue dynamics as a function of the input rate yl. In the context of flow models as we are discussing, it is natural to model queues as integrating the excess capacity; this option is discussed below, and in the case of RED one can then easily relate queues to loss (or marking) probability. ...

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...ion, thereby drastically shortening simulation times. Such models are typically derived via an analogy between network connections, or flows, and fluids in pipes while ignoring the packet granularity =-=[24]-=-. Several authors have proposed theoretical models for TCP traffic [9, 16, 25]. Building on these results, SimGrid implements an efficient algorithm for simulating TCP flows competing over multipath r...