Wireless Ad Hoc and Sensor Networks

320 Wireless Ad Hoc and Sensor NetworksmThen, from the definition of Rn( v) and FP ( n ) , we know that thecumulative length of such flow n packets served by wirelessnode in the virtual time interval [ v1, v2], denoted by Cn( v1, v2),is given asv2Cn( v1, v2) ≤ Rn( v)dv∫v1(7.68)Hence, aggregate length of packets in this set,given as∑ n∈QC ( v , v ), isn1 2∑n∈Qv2Cn( v1, v2) ≤ ∑ Rn( v)dvn∈Q∫v1(7.69)v2≤∫ λ( t , t ) dvv11 2(7.70)≤λ( t , t ).( v −v)1 2 2 1(7.71)k+nkn=− j k−1l fBut because v2 = S( P f ), from the definition of k , v2 − v1 = ∑ n = 0 .φ f k+nHence,,∑n∈Qn=− j k−1l fCn( v1, v2) ≤ λ( t1, t2)∑ φn=0k+nf,k+n(7.72)• This set consists of packets that have a start tag at most andfinish tag greater than v 2 . Formally, the set of packets of flow n,denoted by E n , in this set ismE = mv ≤ S( p )≤v ∧ F( p m)> v(7.73)n{ | 1 n 2 n 2 }Clearly, at most one packet of flow n can belong to this set. Furthermore,Ef = {} j . Hence, the maximum aggregate length of packets in thisset isv 2∑n∈Q∧n≠flmax jn + lf(7.74)