Wireless Ad Hoc and Sensor Networks

Distributed Fair Scheduling in Wireless Ad Hoc and Sensor Networks 315n+1k+mBecause Sp ( f ) < v2 , packet p f is guaranteed to have been transmittedby t 2 .= +Hence, W ( t , t ) ≥∑ = l , and the lemma follows.f 1 2 n n 0m k f nLEMMA 7.3.4In an ADFS-based wireless node, during any interval[ t1, t2]W ( t , t ) ≤φ( v − v ) + lmaxf 1 2 f,l 2 1 f(7.43)where v= v( t ) and v = v( t )1 12 2PROOF The steps in the proof follow similar to the method of Goyal et al.(1997).From the definition of ADFS, the set of flow f packets served in theinterval [ v1, v2]have service tag of at least v 1 and at most v 2 .Hence, the set can be partitioned into two sets such as following:• Set D consisting of packets that have service tag of at least andfinish time at most . Formally,v 2{ | 1 f 2 f 2 }kD= k v ≤ S( p )≤v ∧ F( p k)≤ vFrom Equation 7.7 and Equation 7.8, we concludev 1(7.44)∑l kf ≤ φ f, l( v2 −v1)k∈d(7.45)• Set E consisting of packets that have service tag at most andfinish time greater than v 2 . Formally,{ 1 f 2 f 2}kkE= k v ≤ S( p )≤v ∧ F( p )> vClearly, at most one packet can belong to this set. Hence,v 2(7.46)∑k∈Elkf≤ lmaxf(7.47)From Equation 7.45 and Equation 7.47 we conclude that Equation 7.43holds.