Wireless Ad Hoc and Sensor Networks

446 Wireless Ad Hoc and Sensor Networks2 −1as long as Equation 9.20 holds, and E[|()|] ei k > ( 1−σkvmax) εN( σk vmax+ σ).This demonstrates that E(∆J) is negative outside a compact set U. Accordingto a standard Lyapunov extension (Jagannathan 2006, Zawodniok andJagannathan 2006), the throughput error Ee [ i( k)] is bounded for all l ≥ 0.It is required to show that αˆ i ( k)or, equivalently, ̃α i ( k)is bounded. Thedynamics in error in the parameters estimates areα̃i ( k+ 1) = ⎡21−σR α̃i ( k) ⎤i( k) σRi( k)[ kvei⎣ ⎦ − ( k)+ ε( k)](9.32)where the error, ei( k) , is bounded and estimation error, ε( l), is bounded.Applying the persistency of excitation condition (Jagannathan 2006), onecan show that ̃α i ( k)is bounded.9.3.2.2 Rate PropagationThis total incoming rate is then divided among the upstream nodes proportionallyto the sum of flow weights passing through a given node jas uij k =ui k n flows at j thnode n m flows a thti node( ) ( ) ⋅∑ [ ϕ ]/ ∑ [ ϕm], where uij( k)is the rate allocatedfor a transmitting node j at receiving node i, ui( k)is the rate selectedfor all incoming flows at the ith node, given by Equation 9.3, and ϕ n and ϕ mare preassigned weights of the nth and mth flows, respectively. Next, theselected rate, uij( k), is communicated to the upstream node j to mitigatecongestion. This feedback continues recursively to the nodes upstreamfrom the congested link, so that they will also reduce transmission ratesand thus prevent overflowing buffers. One can update the preassignedweights to guarantee weighted fairness and to improve throughput asdiscussed next.9.3.3 Fair SchedulingData packets at a receiver are first scheduled using the adaptive dynamicfair scheduling (ADFS) scheme (Regatte and Jagannathan 2004). Weightsthat correspond to the packet flows are used to build a schedule fortransmission. This algorithm ensures weighted fairness among the flowspassing a given node. The proposed scheme offers an additional featureof dynamic weight adaptation that further boosts the fairness and guaranteesperformance analytically as presented in this chapter.This ADFS feature increases throughput while ensuring fairness of theflows by adjusting per-packet weight during congestion. This feature,though utilized here, can be optional in the congestion control schemebecause it introduces additional overhead, although shown to be low inChapter 7, in the form of (1) extra bits in each packet to carry the weightand (2) additional calculations performed to evaluate fairness and update