Wireless Ad Hoc and Sensor Networks

88 Wireless Ad Hoc and Sensor Networksaccepted notion of fairness is the max-min fairness criterion, and it isdefined asτ⎛ M l ⎞a− ⎜i + MR⎝⎜∑τ∑⎟i i ⎠⎟= 1 = 1Fair share = MRp+, (3.14)τN − Mlwhere MR p is the minimum rate for a source p, ∑ i=1 MR is the sum ofminimum rates of l active sources, a is the total available bandwidth onthe link, N is the number of active sources, τ is the sum of bandwidthτof M active sources bottlenecked elsewhere. In order to achieve fairness,the feedback rate is selected for adjustable sources asmM∑ i ∑i= 1 i= m+1( k)uk ( ) = u + max( u( k), Fairshare(k))i(3.15)where ui( k); i= 1,...,M are source rates and for the last M sources, the feedbackis used to alter their rates. Note, that in many cases, the number ofcontrollable sources is not accurately known and hence the proposedscheme can also be used to alter some real-time traffic as the real-time video,for instance, is compressed and can tolerate bounded delays that are small.The proposed methodology can be described as follows: the NN at theswitch estimates the network traffic; this estimated value together with celllosses is used to compute the rate, uk ( ), by which the source has to reduceor increase its rate from the original intended rate. If uk ( ) is a positive value,it indicates that the buffer space is available and no congestion is predicted,and the sources can increase their rates by an amount given by Equation3.14 or Equation 3.15. On the other hand, if u( k)is negative, it indicates thatthere are cell losses at the switch, and the sources have to decrease theirrates using Equation 3.14 or Equation 3.15. In any event, the NN-basedcontroller should ensure performance and stability while learning.3.3 Traffic Rate Controller Design for ATM NetworksIn the case of two-layer NN, the tunable NN weights enter in a nonlinearfashion. Stability analysis by Lyapunov’s direct method is performed fora family of weight-tuning algorithms for a two-layer NN. Assume, therefore,that the approximation error is bounded above |( ε k)|≤ ε with theN