We consider the design of optimal strategies for joint power adaptation, rate adaptation and scheduling in a multi-hop wireless network. Most existing strategies control either power and scheduling, or rates and scheduling, but not all three together as we do. We assume the underlying physical layer is in the linear regime (the rate of a link can be approximated by a linear function of the signal-to-interference-and-noise ratio), like in time hopping UWB (TH-UWB) and low gain CDMA systems, and that it allows fine-grained rate adaptation, like in 802.11a/g, HDR/CDMA, TH-UWB. The goal is to find properties of the power control in an optimal joint design. Our main finding is that optimal power control is simple 0-PMAX power control, i.e. when a node is sending it uses the maximum transmitting power allowed. We consider both high rate networks where the goal is to maximize rates under power constraints and low power networks where the goal is to minimize average consumed power while meeting minimum rate constraints. We prove analytically that in both scenarios the optimal can always be attained with 0-PMAX power allocation. Moreover, we prove that, when maximizing rates, and if power constraints are on peak and not average, 0-PMAX is the only optimal power control strategy, and any other is strictly suboptimal.
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