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• For any two distinct users associated with any TP, k, j ∈ U ( b $^{)}$, b ∈ B ′ $_{1}$, such that R$_{k,b}$ R$_{k,}$$_{1}$ > R$_{j,b}$ $_{R$_{j,}$$_{1}$}$, we must have 

<img src='content_image/297392.jpg'>

• Slack ordering: For any two distinct users associated with any TP, k, j ∈ U ( b $^{)}$, b ∈ B ′ $_{1}$, such that w$_{k}$R$_{k,b}$ > w$_{j}$R$_{j,b}$ , we must have 

<img src='content_image/297393.jpg'>

Similarly, for any two distinct users, k, j ∈ U $^{′}$, such that w$_{k}$R$_{k,}$$_{1}$ > w$_{j}$R$_{j,}$$_{1}$ , we must have 

<img src='content_image/297394.jpg'>

• For any user k ∈ U ( b $^{)}$, b ∈ B ′ $_{1}$, such that γ$_{k,b}$ > 0 , if there exists any other user j ∈ U ′ with θ$_{j,}$$_{1}$ > 0 and R$_{j}$ > R min j , we must have 

<img src='content_image/297395.jpg'>

Similarly, for any user k ∈ U ( b $^{)}$, b ∈ B ′ $_{1}$, such that θ$_{k,}$$_{1}$ > 0 , if there exists any other user j ∈ U ( b ) with γ$_{j,b}$ > 0 and R$_{j}$ > R min j , we must have 

<img src='content_image/297399.jpg'>

Letting ˆ O (Γ , { Γ$_{b}$ } ) denote the optimal value of (3) for the given budgets, we embark to solve (3) and characterize ˆ O (Γ , { Γ$_{b}$ } ) . Towards that end, without loss of generality, in this section we assume a labeling of user indices such that for any two users k, j ∈ U ( b ) for any b ∈ B ′ $_{1}$, k < j ⇒ R$_{k,b}$ R$_{k,}$$_{1}$ ≥ R$_{j,b}$ $_{R$_{j,}$$_{1}$}$. Further, without loss of generality, the user indices in U ( b ) for each b ∈ B ′ $_{1}$, are assumed to be consecutive. Then, upon appyling primal decomposition on (3), we see that if we fix the share of the Macro resource that can be used by each TP b ∈ B ′ 1 as Z$_{b}$ , where ∑ b ∈B ′ $_{1}$Z$_{b}$ = Γ , (3) decouples into |B ′ $_{1}$| sub-problems. In particular, the problem at hand for TP b is given by (9). Let ˆ O ( Z$_{b}$, Γ$_{b}$ , b ) denote the optimal solution of (9) for each TP b ∈ B ′ 1 so that 

