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...� f �� qf(t) − q ⋆� f (ˆxf(t) − x ⋆ f )� ≤ − δ Lγ �q(t) − q⋆�I�q(t)−q⋆ �≥cLγ + ζI�q(t)−q⋆ �≤cLγ, where δ, ζ and c are positive constants which are independent of L. Proof: This statement is proved in =-=[12]-=- for a large class of utility functions and for the case of a single transmitter transmitting to many receivers. Here, we consider the multi-hop scenario and further generalize the utility functions. ...

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...ralized to networks with different interference structures. In addition to the works mentioned above, our paper is related to recent work combining flow control with routing and scheduling, including =-=[13, 24, 7, 18, 8]-=-. While these papers also propose algorithms achieving fair allocations without violating stability, the routing-scheduling component of these algorithms are based on the centralized control approach ...

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... should be thought of as the network-designated capacity for a flow. We note that the concept of shadow queues here is different from the notion of virtual queues used in [12] for the Internet and in =-=[5]-=- for wireless networks. In networks with virtual queueing systems, the arrival rates to both the real and virtual queues are the same, but the virtual queue is drained at a slower rate than the real q...

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...Another notion for fairness is to maximize the sum of the certain functions of the rates (utilities) of all users. Recently, algorithms have been proposed for attaining this goal in wireless networks =-=[3]-=-, [9], [12], [23]. In Section II, we describe the fairness objective and the network model, and present conditions that are necessary and sufficient for a bandwidth allocation to be maxmin fair. In Se...

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...e Max-Weight algorithm will do this. As already mentioned, if the congestion control and scheduling are performed in isolation then adverse effects can occur. In recent years a number of papers [24], =-=[11]-=-, [23], [18] have addressed this issue by considering algorithms for jointly combining congestion control and scheduling. In particular, each of these algorithms defines a network control that determi...

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...ement. The central idea of considering the maximization of the sum of the user utilities is due to [30]. See also [34, 40]. Combining this objective with the scheduling appears in [44, 45] as well as =-=[12, 13]-=-. For a related survey, see [8, 51]. Randomized versions of MW by Tassiulas [53] and its variant by Giaccone, Prabhakar and Shah [16] provide a simpler (centralized) implementation of MW for input-que...

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...es. In this case, we design a related algorithm that maximizes time average profit. The Lyapunov optimization technique we use [10][11][12] is related to the primal-dual and fluid-model techniques in =-=[13]-=-[14][15][16]. The work in [10][11][12] establishes a general [O(1/V ), O(V )] performance-congestion tradeoff for stochastic network optimization problems with i.i.d. (and more general ergodic) proces...

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... (n, c) (so that zero throughput for a given stream implies zero utility for that stream). Such utility functions are often used as a quantitative measure of network fairness [33] [10] [18] [16] [13] =-=[29]-=- [2] [1], and different choices of gnc(r) lead to different fairness properties [34]. We further assume that all utility functions have finite right derivatives. The optimal network utility g ∗ is def...