Wireless Ad Hoc and Sensor Networks

448 Wireless Ad Hoc and Sensor Networksgiven to show the performance guarantee. Theorem 9.3.6 guarantees proportionalfairness for all flows, whereas Theorem 9.3.7 guarantees that theminimal throughput is achieved by each flow. Finally, Theorem 9.3.8 tiesboth provisions from Theorem 9.6 and Theorem 9.7, thus guaranteeingoverall performance of the proposed scheme. The proofs closely followthose given by Regatte and Jagannathan (2004).REMARK 5In fact, the weight update (Equation 9.13) ensures that the actual weightassigned to a packet at each node converges close to its target value.REMARK 6The value φ is finite for each flow at a given node. Let ̃φ ijij be the weighterror defined as ̃φij = φij − ˆ φij. Note that from now on, the weight of a packetfor flow f at node l is denoted as φ f, l and it can be related to φ f at eachnode as φ σ φ .f, l=f fTHEOREM 9.3.6For any interval [ t1, t2]in which flows f and m are backlogged during the entireinterval, the difference in the service received by two flows at a cluster head isgiven asmaxmaxWf( t1, t2) φf, l −Wm( t1, t2)φm,l ≤ lf φf,l+ lmφ ml ,(9.34)REMARK 7Notice that Theorem 9.3.6 holds regardless of the service rate of the clusterhead. This demonstrates that the algorithm achieves fair allocation ofbandwidth and thus meets a fundamental requirement of a fair schedulingalgorithm for integrated services networks.REMARK 8Weights can be selected initially and may not get updated. If this occurs,it becomes the self-clocked fair queuing (SFQ) scheme (Golestani 1994).One can still use the proposed backoff interval selection scheme. However,it is demonstrated in Chapter 7 that the ADFS increases throughput whileensuring fairness compared with SFQ.THEOREM 9.3.7 (THROUGHPUT GUARANTEE)If Q is the set of flows served by an ADFS node following FC service modelwith parameters ( λ( t1, t2), ψ( λ)), and ∑n∈Qφn, l≤λ( t1, t2), then for all intervals