all users in U ( b $^{)}$. Again invoking Proposition 1 we can explicitly detail h ( Z$_{b}$, b ) , as For any given Γ$_{b}$ , we let S ( Z$_{b}$, Γ$_{b}$ , b ) , b ∈ B ′ $_{1}$, Z$_{b}$ ≥ ¯ h (Γ$_{b}$ , b ) denote the slope of the function ˆ O ( Z$_{b}$, Γ$_{b}$ , b ) at Z$_{b}$ . In particular, S ( Z$_{b}$, Γ$_{b}$ , b ) = lim$_{δ}$$_{→}$$_{0}$$_{+}$ ˆ O ( Z$_{b}$ + δ, Γ$_{b}$ ,b ) − ˆ O ( Z$_{b}$, Γ$_{b}$ ,b ) δ . Henceforth, without loss of generality, we assume h (1 , b ) ≤ 1 & ¯ h (1 , b ) ≤ 1 , ∀ b ∈ B ′ $_{1}$. Proposition 2. For any fixed Γ$_{b}$ ≥ h (1 , b ) , ˆ O ( Z$_{b}$, Γ$_{b}$ , b ) is continuous, non-decreasing, piecewise linear and concave in Z$_{b}$ ∈ [ ¯ h (Γ$_{b}$ , b ) , 1] . For any fixed Z$_{b}$ ≥ ¯ h (1 , b ) , ˆ O ( Z$_{b}$, Γ$_{b}$ , b ) is continuous, non-decreasing, piecewise linear and concave in Γ$_{b}$ ∈ [ h ( Z$_{b}$, b ) , 1] . Proof. We only prove the first claim since proof for the second one follows along similar lines. The continuity and non-decreasing properties are straightforward to verify. It can be shown that the conditions stated in Proposition 1 provide necessary and sufficient conditions to determine an optimal set of allocation fractions for the problem in (9). To verify the other two properties, we start at Z$_{b}$ = ¯ h (Γ$_{b}$ , b ) . Then, if ¯ h (Γ$_{b}$ , b ) = 0 , the slack at the pico TP b , Γ$_{b}$ − ∑ j ∈U ( b ) R min j R$_{j,b}$ must be distributed among users in the decreasing order { w$_{k}$R$_{k,b}$ }$_{k}$$_{∈U}$ ( b ) subject to their respective maximum rate limits (cf. slack ordering in Proposition 1). On the other hand, when ¯ h (Γ$_{b}$ , b ) > 0 there is no slack at the pico TP for this Z$_{b}$ . The next key observation we use is the one in Proposition 1 pertaining to the order in which macro resources are assigned to users in U ( b $^{)}$. Following our labelling, we see that when Z$_{b}$ = ¯ h (Γ$_{b}$ , b ) > 0 either user ˜ k + 1 (when Ξ$_{b}$ > 0 ) or user ˜ k (when Ξ$_{b}$ = 0 ) is the user with the largest index in U ( b ) to be assigned a positive resource by TP b . Let user k ′ be this user so that users k ∈ U ( b ) : k < k ′ are assigned resource only by TP b at this Z$_{b}$ = ¯ h (Γ$_{b}$ , b ) . The slope of ˆ O ( Z$_{b}$, Γ$_{b}$ , b ) can be determined as S ( ¯ h (Γ$_{b}$ , b ) , Γ$_{b}$ , b ) = max $^{{}$max { w$_{k}$R$_{k,b}$R$_{k}$$_{′}$$_{,}$$_{1}$ R$_{k}$$_{′}$$_{,b}$ : k ∈ U ( b ) & k < k ′ $^{}}$, max { w$_{k}$R$_{k,}$$_{1}$ : k ∈ U ( b ) & k ≥ k $^{′}$} $^{}}$. Then, as Z$_{b}$ is increased to Z$_{b}$ + δ , for any arbitrarily small δ > 0 , the slack is put to the user yielding the slope S ( ¯ h (Γ$_{b}$ , b ) , Γ$_{b}$ , b ) (i.e., offering the maximum bang-per-buck). If such a user is some ˆ k ≥ k ′ h ( Z$_{b}$, b ) =    R min ˘ k − $_{1}$− ˜ Z$_{b}$R$_{˘}$ k − 1 , 1 R$_{˘}$ k − 1 ,b + ∑ j ∈U ( b $^{)}$: j< ˘ k − 1 R min j R$_{j,b}$ , ˘ k − 1 ∈ U ( b ) 0 , Else ˘ k = min$^{}$   k : k ∈ U ( b ) & ∑ j ∈U ( b $^{)}$: j ≥ k R min j R$_{j,}$$_{1}$ ≤ Z$_{b}$$^{}$   , ˜ Z$_{b}$ = Z$_{b}$ − ∑ j ∈U ( b $^{)}$: j ≥ ˘ k R min j R$_{j,}$$_{1}$ (12)