Wireless Ad Hoc and Sensor Networks

316 Wireless Ad Hoc and Sensor NetworksBecause unfairness between two flows in any interval is maximumwhen one flow receives maximum possible service and the other minimumservice. Theorem 7.3.1 follows directly from Lemma 7.3.3 andLemma 7.3.4.THEOREM 7.3.1For 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 an ADFS wirelessnode is given asmax maxWf( t1, t2) Wm( t1, t2)l f lm− ≤ +φ φ φ φmlf, lml ,f,l,(7.48)REMARK 3If E ij = 0 at each node, then the proposed ADFS will become a DFS scheme(Vaidya et al. 2000).REMARK 4In Theorem 7.3.1, no assumption was made about the service rate of thewireless node. Hence, this theorem holds regardless of the service rate ofthe wireless node. This demonstrates that ADFS achieves fair allocationof bandwidth and thus meets a fundamental requirement of fair schedulingalgorithms for integrated services networks.7.3.4 Throughput GuaranteeTheorem 7.3.2 and Theorem 7.3.3 establish the throughput guaranteed toa flow by an ADFS FC and EBF service model, respectively, when appropriateadmission control procedures are used.THEOREM 7.3.2If Q is the set of flows served by an ADFS node following the FC service modelwith parameters ( λ( t1, t2), ψ( λ)), and ∑n∈Q φnl , ≤λ( t1, t2), then for all intervals[ t1, t2]in which flow f is backlogged throughout the interval, Wf ( t1, t2)is given asmax∑ lnn ∈ Qf 1 2 f, l 2 1 f,lλ( t1, t2)W ( t , t ) ≥φ( t −t) −φ−φf,lψλ ( )−lλ( t , t )1 2maxf(7.49)PROOF The steps in the proof follow similar to that of Goyal et al. 1997.