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set { R$_{u,}$$_{1}$ $_{R$_{u,b}$}$}$_{u}$$_{∈U}$ ( b ) and recall that these ratios are all strictly positive and distinct. Then, defining µ$_{N}$$_{b}$$_{+1}$$_{,b}$ = ∞ , we have 0 < µ$_{1}$$_{,b}$ < µ$_{2}$$_{,b}$ < · · · < µ$_{N}$$_{b}$$_{,b}$ < µ$_{N}$$_{b}$$_{+1}$$_{,b}$ = ∞ . Next, we define a function h : I R$_{++}$ × B ′ 1 → I R$_{+}$ as 

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Similarly, we define function g : I R$_{++}$ × B ′ 1 → I R$_{+}$ such that, g ( λ, b ) , ∀ λ > 0 , b ∈ B ′ 1 equals 

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The following result is proved in the appendix. 

Theorem 1. The optimal objective value of (16) is given by 

where ˆ λ ∈ (0 , ∞ ) is the unique solution to the relation 

which can be determined via bisection search. 

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Corollary 3. Suppose that the optimal ˆ λ satisfying (20) is given. Then, if ˆ λ ∈ (( m − 1) µ$_{m,b}$, mµ$_{m,b}$ ) for some m = 1 , · · · , N$_{b}$ , the optimal solution comprises of assigning an identical resource share γ$_{k,b}$ = µ$_{m,b}$/ ˆ λ with θ$_{k,}$$_{1}$ = 0 for all users k ∈ U ( b ) : R$_{k,}$$_{1}$ R$_{k,b}$ < µ$_{m,b}$ , whereas all users k ∈ U ( b ) : R$_{k,}$$_{1}$ R$_{k,b}$ > µ$_{m,b}$ are assigned θ$_{k,}$$_{1}$ = 1 / ˆ λ with γ$_{k,b}$ = 0 . The user k ∈ U ( b ) : R$_{k,}$$_{1}$ R$_{k,b}$ = µ$_{m,b}$ is assigned θ$_{k,}$$_{1}$ = m/ ˆ λ − 1 /µ$_{m,b}$ with γ$_{k,b}$ = 1 − ( m − 1) µ$_{m,b}$/ ˆ λ . On 

