Wireless Ad Hoc and Sensor Networks

Distributed Fair Scheduling in Wireless Ad Hoc and Sensor Networks 319Let Q be the set of flows served by the node. Then, an FC or EBF nodewith average rate λ( t1, t2), is defined to have exceeded its capacity atvirtual time v if ∑ R > . If the capacity of a SFQ–based node isn∈Qn ( v) λ( t 1 , t 2 )not exceeded, then it guarantees a deadline to a packet based on itsjexpected arrival time. Expected arrival time of packet P f , denotedjby T( P , φ ), is defined asaff,j( ) ( f j−1)+⎧⎪max APjf , TaPj −1⎨f , φ ,⎩⎪f,j−1(7.64)0where Ta( Pf , φ f,0) =−∞.A deadline guarantee based on expected arrivaltime has been referred to as delay guarantee. Theorem 7.3.4 and Theorem7.3.5 establish the delay guarantee for FC and EBF service models, respectively,and follow the steps for proof similar to that of Goyal et al. 1997.φlj−1f⎫⎪⎬⎭⎪j ≥ 1THEOREM 7.3.4If Q is the set of flows served by an ADFS node following the FC service modelwith parameters ( λ( t1, t2), ψ( λ)), and ∑n∈QRn( v) ≤λ( t 1 , t 2 ) for all v , then thejjdeparture time of packet P f at the node, denoted by T ( P ), is given bydfjjd ( jf )≤ a ( f f,j)+ ∑ +n∈Q∧n≠f 2 1 2T P T P , φPROOF Let H be defined as follows:H mm | 0 S P m v A Pmaxl lnf ψλ ( )+λ( t1, t2) λ( t1,t ) λ( t , t ){ ( )}= > ∧ ( mf )= ( f )(7.65)(7.66)kkLet k ≤ jbe largest integer in H . Also, let v1 = v( A( P f )) and v2 = S( P f ).Observe that as the node virtual time is set to the maximum finish tagk jassigned to any packet at the end of a busy period, packets P f and P f areserved in the same busy period of a wireless node. From the definitionof ADFS, the set of flow f packets served in the interval [ v1, v2]have astart tag at least and at most . Hence, the set can be partitioned intotwo sets:v 1 v 2v 1• This set consists of packets that have start tag at least and finishtag at most v 2 . Formally the set of packets of flow n, denotedby D n , in this set ismD = mv ≤ S( p )≤v ∧ F( p m)≤ v(7.67)n{ | 1 n 2 n 2 }