Wireless Ad Hoc and Sensor Networks

Distributed Fair Scheduling in Wireless Ad Hoc and Sensor Networks 321Hence, the aggregate length of packets served by the wireless node inthe interval [ v , v ], denoted by Lv ˆ( , v), is1 21 2n=−−j k 1∑k+nl fLv ˆ( 1 , v2 ) ≤ λ ( t1 , t2) +φ f,k+nn=0∑n∈Q∧n≠flmax jn + l f(7.75)Let Τ( w)be the time taken by the wireless node to serve packets withaggregate length w in its busy period. From the definition of FC servicemodel, we getw ψλ ( )Τ( w)≤ +λ( t , t ) λ( t , t )1 2 1 2(7.76)Because packet departs at system virtual time v 2 and all the packetsserved in the time interval [ v1, v2]are served in the same busy period ofthe wireless node, we getP fjn=−−j k 1 k+nlA pkf( f )+ ∑ + φn=0f,k+nj( f )+ ≥ d ( f )1 2A p k Τ( L ˆ( v , v )) T P∑n∈Q∧n≠fmaxlnλ ( t , t )1 2jl f ψλ ( )j+ + ≥ Td( Pf)λ( t , t )λ( t , t )1 2 1 2(7.77)(7.78)From Equation 7.64, we getjTd∑ ≥ ( Pjf )n∈Q∧n≠fmaxjlnl f ψλ ( )Ta( Pf, φ f,j)+ + +λ( t1, t2) λ( t1, t2)λ( t1, t2)(7.79)THEOREM 7.3.5If Q is the set of flows served by an ADFS node following EBF service modelwith parameters ( λ( t1, t2), B, ω, ψ( λ)), γ ≥ 0, and ∑n∈QRn( v) ≤λ( t 1 , t 2 ) for all v,jjthen the departure time of packet P f at the node, denoted by T ( P ), is given bydf( )≤ ( f,j)+∑ λn∈Q∧n≠f1 2⎛P TdP jf TaPjf , φ⎝⎜maxln( t , t )jl⎞f ψλ ( ) γ+ + + ⎟ ≥1−Beλ( t1, t2)λ( t1, t2) λ( t1, t2)⎠−ωγ(7.80)