The upper bound on the first term of the sum in (A.1) is derived as follows. where the first equality in (F.4) follows from (F.2), and the equality in (F.5) follows from (F.1). Note that for each j ∈ I$_{0}$, The inequality in (F.6) follows from the fact that for j ∈ I$_{0}$, Pr( P$_{1}$$_{j}$ ≤ x ) ≤ x for all x ≥ 0 . Combining (F.7) with (F.5) we obtain an upper bound for the first term of E ( ∑ j ∈ I$_{0}$ R$_{j}$ max( R, 1) ) = ∑ j ∈ I$_{0}$ m ∑ r =1 1 r Pr ( j ∈ R $_{1}$, P$_{1}$$_{j}$ ≤ rq$_{1}$ m ∑ m s =1 1 s , P$_{2}$$_{j}$ ≤ r ( q − q$_{1}$ ) |R 1 | , C ( j ) r ) ≤ j ∈ I$_{0}$ m r =1 1 r Pr P$_{1}$$_{j}$ ≤ rq$_{1}$ m ∑ m s =1 1 s , C ( j ) r ) (F.3) = ∑ j ∈ I$_{0}$ m ∑ r =1 r ∑ l =1 1 r p$_{jrl}$ = j ∈ I$_{0}$ m l =1 m ∑ r = l 1 r p$_{jrl}$ (F.4) ≤ j ∈ I$_{0}$ m l =1 m r = l 1 l p$_{jrl}$ ≤ ∑ j ∈ I$_{0}$ m ∑ l =1 1 l m r =1 p$_{jrl}$ = ∑ j ∈ I$_{0}$ m ∑ l =1 1 l Pr ( P$_{1}$$_{j}$ ∈ ( ( l − 1) q$_{1}$ m ∑ m s =1 1 s , lq$_{1}$ m ∑ m s =1 1 s ]) , (F.5) m ∑ l =1 1 l Pr ( P$_{1}$$_{j}$ ∈ ( ( l − 1) q$_{1}$ m ∑ m s =1 1 s , lq$_{1}$ m ∑ m s =1 1 s ]) = m ∑ l =1 1 l [ Pr P$_{1}$$_{j}$ ≤ lq$_{1}$ m ∑ m s $_{=1}$1 /s ) − Pr ( P$_{1}$$_{j}$ ≤ ( l − 1) q$_{1}$ m ∑ m s $_{=1}$1 /s )] = m l =1 1 l Pr ( P$_{1}$$_{j}$ ≤ lq$_{1}$ m m s $_{=1}$1 /s − m − 1 ∑ l =0 1 l + 1 Pr ( P$_{1}$$_{j}$ ≤ lq$_{1}$ m ∑ m s $_{=1}$1 /s ) = m − 1 ∑ l =1 ( 1 l − 1 l + 1 ) Pr ( P$_{1}$$_{j}$ ≤ lq$_{1}$ m ∑ m s $_{=1}$1 /s ) + 1 m Pr ( P$_{1}$$_{j}$ ≤ q$_{1}$ ∑ m s $_{=1}$1 /s ) ≤ m − 1 l =1 1 l + 1 ( q$_{1}$ m ∑ m s $_{=1}$1 /s ) + q$_{1}$ m m s $_{=1}$1 /s (F.6) = ( q$_{1}$ m ∑ m s $_{=1}$1 /s ) m l =1 1 l = q$_{1}$ m . (F.7) 41