Wireless Ad Hoc and Sensor Networks

310 Wireless Ad Hoc and Sensor Networksand hence, the backoff interval is also updated at each node. Backoffinterval, BI ij , for ith flow jth packet with packet length L ij and weight φ ijis defined asBI SF L ijij = ⎢ ⎥⎢ρ* * ⎥⎣⎢φij⎦⎥(7.11)where SF is the scaling factor and ρ is a random variable with mean one.The collision handling mechanism is similar to the one in Vaidya et al.(2000). This results in fair allocation of the bandwidth.7.3.3 Fairness GuaranteeW ( t1, t2) W ( t1, t2)fmTo prove that ADFS is fair, we need to prove a bound on | − |φ fφmfor a sufficiently long interval [ t1, t2]in which both flows, f and m, are backlogged.To proceed, the following assumption is needed.ASSUMPTIONTo arrive at a fair scheduling scheme, we assume that there exists a weightvector for a jth flow, ith packet, at each node l asφ ijφ =ij⎡ φijl⎢⎢ .⎢⎢.⎢ .⎢⎣⎢φijm⎤⎥⎥⎥⎥⎥⎥⎦(7.12)REMARK 1In fact, the weight update (Equation 7.9) ensures that the actual weightfor the packet at each node converges close to its target value.REMARK 2φ ij is finite for each flow at each node.Let ̃φ be the weight error defined as ̃φ ij ij , given bỹφij = φij − ˆ φij(7.13)