Wireless Ad Hoc and Sensor Networks

Distributed Fair Scheduling in Wireless Ad Hoc and Sensor Networks 317Let v1 = v( t1) and let Lv ˆ( 1 , v 2 ) denote the aggregate length of packetsserved by the wireless node in the virtual time interval [ v1, v2]. Then, fromLemma 7.3.4, we conclude∑Lv ˆ( , v ) ≤ φ ( v , v ) + l1 2 nl , 2 1n∈Q∑n∈Qmaxn(7.50)Because ∑n∈Qφn, l≤λ( t1, t2),∑ˆ( , ) ( , )( , )maxLv1 v2 ≤ λ t1 t2 v2 v1+ l nn∈Q(7.51)Defineasv 2v v t tlnmaxn∈Qψλ ( )= + − − −λ( t , t ) λ( t , t )2 1 2 1∑1 2 1 2(7.52)Then, from Equation 7.51, we conclude∑ ∈⎛lnLv ˆ( , v ) ( t , t )n Q1 2 ≤ λ ⎜1 2 v1+ t2 −t1−⎜λ ( t1,t2)⎝max⎞ψλ ( )− −v⎟1λ( t , t ) ⎟ + 1 2⎠∈∑n Qmaxl n(7.53)≤λ( t , t )( t −t) −ψ( λ)1 2 2 1(7.54)Let ˆt 2 be such that vt (ˆ 2)= v2. Also, let Τ( w)be the time taken by the wirelessnode to serve packets with aggregate length w in its busy period.Then,tˆ ≤ t +Τ( L ˆ( v , v ))2 1 1 2(7.55)≤ t + Τ( λ( t , t )( t −t) −ψ( λ))1 1 2 2 1(7.56)From the definition of FC service model, we getw ψλ ( )Τ( w)≤ +λ( t , t ) λ( t , t )1 2 1 2(7.57)