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## Corollary 2. 

<img src='content_image/324809.jpg'>

To illustrate our results, in Fig. 2 we consider a macro TP serving |U $^{′}$| = 30 users, with |B$_{1}$| = 10 pico nodes assigned to it and where each user is associated with the pico in B$_{1}$ from which it sees the strongest signal strength. We obtained the user peak rates by emulating a realistic deployment (the details are defered to the simulation results section) and imposed no maximum rate limits (i.e., each R max u = ∞ ). A unit budget at each pico was assumed and the minimum rate of each user was chosen to be a scalar times its peak rate from the macro TP (with the scalar being identical for each user). We considered several values for this scalar and in each case plot ˆ O (Γ , { 1 } ) , Γ ∈ [ ∑ b ∈B ′ 1 ¯ h (1 , b ) , 1] (computed using Algorithm I). As predicted by Proposition 3, each curve is non-decreasing, piece-wise linear and concave and we verified that the obtained value matches the one obtained by solving (3) via a generic LP solver. Notice as the minimum rate requirements become more stringent, more macro resource is needed to satisfy them and the optimized utility value decreases. Before offering our next key result, which is proved in the appendix, we introduce some notation. For any two pico TPs b$_{1}$, b$_{2}$ ∈ B ′ $_{1}$, we let e$_{b}$$_{1}$ , e$_{b}$$_{2}$ define |B ′ $_{1}$| × 1 unit vectors that have a zero on all their entries except those corresponding to b$_{1}$, b$_{2}$ , respectively, which are both one. 

Proposition 4. For any non-negative scalars δ, ˜ δ, δ$_{b}$$_{1}$, ˜ δ$_{b}$$_{2}$ and budgets Γ , Γ such that Γ ≥ ∑ b ∈B ′ 1 ¯ h (Γ$_{b}$ , b ) , we have that 

<img src='content_image/324808.jpg'>

## B. Optimal allocation fractions of (2) 

As before we obtain the sets U $^{′}$, B ′ 1 from the given assoication and let N$_{b}$ = |U ( b $^{)}$| denote the cardinality or the number of users associated with TP b ∈ B ′ $_{1}$. Consider the PF system utility optimization problem (restricted to the user pool U $^{′}$) for the the given user association, given in (16). Note that (16) is a purely continuous optimization problem. Next, for each b ∈ B ′ 1 define µ$_{1}$$_{,b}$ = min$_{u}$$_{∈U}$ ( b ) { R$_{u,}$$_{1}$ $_{R$_{u,b}$}$} . Similary, let µ$_{k,b}$, k ∈ { 2 , · · · , N$_{b}$ } denote that k th smallest ratio in the 

