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The matrix has entries (c) i = for all (i, c) ∈ D. Thus, λ + ∈ Λ implies that it is possible to find multi-commodity flows to support the augmented traffic rate matrix associated with (c) adding a new stream of rate to each of the source queues Ui (t) (for (i, c) ∈ D). Data from any new stream of rate is simply treated as if it has the same commodity as the source queue it enters. To simplify analysis, we assume that D = D. This is equivalent to the following assumption. “No Trapping Nodes” Assumption: If it is possible to send commodity c data to a particular node i (so that a commodity c queue exists for that node), then it is possible to support a non-zero communication rate from node i to the destination of commodity c (possibly by using multi-hop paths).

However, the fact that the above policy exists plays a crucial role in the analysis of a more practical stabilizing strategy presented in Section 4. 3. The capacity of ON-OFF downlink. Consider a simple example of a two-queue wireless downlink that transmits data to two downlink users 1 and 2 over two different channels. 3. The capacity of one hop networks 39 (for i ∈ {1, 2}). All packets are assumed to have fixed lengths, so that the queue backlog is measured in units of packets. Let U1 (t) and U2 (t) respectively represent the current backlog of packets waiting for transmission to user 1 and user 2, respectively.

The theory of Lyapunov drift has a long history in the field of discrete stochastic processes and Markov chains (see, for example, [10, 106]). The first application of the theory to the design of dynamic control algorithms for radio networks appeared in [143], where a general algorithm was developed to stabilize a multi-hop packet radio network with configurable link activation sets. The concepts of maximum weight matching and differential backlog scheduling, developed in [143], play important roles in the dynamic control strategies we present in later sections.

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