then the additional available Macro resource, δ , is directly assigned as slack to it. Otherwise, the additional available Macro resource is first assigned to user k $^{′}$, which frees up resource δR$_{k}$$_{′}$$_{,}$$_{1}$ R$_{k}$$_{′}$$_{,b}$ at the pico TP b (while maintaining the minimum rate of user k $^{′}$). This freed up pico resource is assigned as slack to the user ˆ k yielding the slope S ( ¯ h (Γ$_{b}$ , b ) , Γ$_{b}$ , b ) . As δ is increased the slack is continuously assigned to the user ˆ k till the slope changes and that user is not the one yielding the maximum bang-per-buck. This happens either if user ˆ k attains its maximum rate upon which it is removed from the candidate list of users which can be assigned slack, or if user k ′ is no longer assigned any resource by pico TP b . In the former case, the slope changes to S ( Z$_{b}$, Γ$_{b}$ , b ) = max $^{{}$max { w$_{k}$R$_{k,b}$R$_{k}$$_{′}$$_{,}$$_{1}$ R$_{k}$$_{′}$$_{,b}$ : k ∈ U ( b ) \ ˆ k & k < k ′ $^{}}$, max { w$_{k}$R$_{k,}$$_{1}$ : k ∈ U ( b ) \ ˆ k & k ≥ k $^{′}$} $^{}}$, whereas in the latter case the slope changes to S ( Z$_{b}$, Γ$_{b}$ , b ) = max $^{{}$max { w$_{k}$R$_{k,b}$R$_{k}$$_{′}$$_{−}$$_{1}$$_{,}$$_{1}$ R$_{k}$$_{′}$$_{−}$$_{1}$$_{,b}$ : k ∈ U ( b ) & k < k ′ − 1 $^{}}$, max { w$_{k}$R$_{k,}$$_{1}$ : k ∈ U ( b ) & k ≥ k ′ − 1 } $^{}}$. Thenceforth additional Macro resources are assigned as slack to the user yielding the new slope and the process continues. Note that at every change the slope decreases because either users are removed from candidate list or the gain term multiplying the weight of each user served exclusively by the pico reduces. This demonstrates the piecewise linearity and concavity. The same arguments can be applied when ¯ h (Γ$_{b}$ , b ) = 0 . In particular, we begin at Z$_{b}$ = ¯ h (Γ$_{b}$ , b ) = 0 after determining user k ′ that has the highest index among those that have been assigned a positive resource by the pico TP and after removing users that have achieved their maximum rates from the candidate pool. The subsequent process proceeds as before and we can deduce the piecewise linearity and concavity. Corollary 1. For any fixed Γ$_{b}$ ≥ h (1 , b ) , ˆ O ( Z$_{b}$, Γ$_{b}$ , b ) can be computed as where H (Γ$_{b}$ , b ) = 0 whenever ¯ h (Γ$_{b}$ , b ) > 0 and when ¯ h (Γ$_{b}$ , b ) = 0 , it yields the weighted sum rate obtained by distributing the excess pico resource as slack among users in U ( b ) in the decreasing order { w$_{k}$R$_{k,b}$ } subject to their respective maximum rate limits. We now propose Algorithm I to determine Macro allocations that optimize (10) (and so (3)). ˆ O ( Z$_{b}$, Γ$_{b}$ , b ) = ∑ k ∈U ( b ) w$_{k}$R min k + H (Γ$_{b}$ , b ) +$^{∫}$ Z$_{b}$ ¯ h (Γ$_{b}$ ,b ) S ( z$_{b}$, Γ$_{b}$ , b ) dz$_{b}$, ∀ Z$_{b}$ ∈ [ ¯ h (Γ$_{b}$ , b ) , 1] , (13)