Wireless Ad Hoc and Sensor Networks

318 Wireless Ad Hoc and Sensor NetworksFrom Equation 7.56 and Equation 7.57 we obtainˆ λ( t1, t2)( t2 −t1) −ψ( λ)t2 ≤ t1++λ( t , t )1 2ψλ ( )( t , t )λ 1 2(7.58)≤ t 2W t t v v l(7.59)From Lemma 7.3.3, it is clear that( ,ˆ ) ≥φ( − ) −maxf 1 2 f,l 2 1 f(7.60)Since ˆt≤ t2 2, using Equation 7.52 it follows thatmax∑ 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(7.61)THEOREM 7.3.3If 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∈Qφn, l≤λ( t1, t2),then forall intervals [ t1, t2]in which flow f is backlogged throughout the interval,Wf ( t1, t2)is given as⎛lnP ∑n QWf( t1, t2) < φf, l( t2 −t1)−φ∈f,l⎝⎜λ ( t , t )ψλ ( ) γ−φ, −φ, −lλ( t , t ) λ( t , t )maxf l f l f1 2 1 21 2⎞⎠⎟ ≤max−Be ωγ(7.62)7.3.5 Delay GuaranteeGenerally, a network can provide a bound on delay only if its capacity isnot exceeded. The weight φ f,l can also mean the rate assigned to a packetof flow f at node l. Let the rate function for flow f at virtual time v,denoted by Rf ( v),be defined as the rate assigned to the packet that hasstart tag less than v and finish tag greater than v. Formally,Rf( )⎧ jj⎪ φ f,l if ∃∋ j S( p f ) ≤ v < F( p f )( v)= ⎨⎩⎪ 0 otherwise(7.63)