content_image/301147.jpg content_image/301148.jpg content_image/301149.jpg
<img src='content_image/301147.jpg'>

the original problem in (3) can be expressed as 

<img src='content_image/301148.jpg'>

A straightforward approach to determine the optimal Macro resource share among the TPs is to optimize { Z$_{b}$ } using the generic subgradient method. However, we will show that exploiting the structure of the problem at hand leads directly to a very simple algorithm. First, let us define the function, ¯ h : I R$_{+}$ × B ′ 1 → I R$_{+}$ , such that ¯ h (Γ$_{b}$ , b ) for any given TP b and corresponding budget, Γ$_{b}$ , yields the minimum Macro resource needed (in addition to the available budget Γ$_{b}$ for the pico TP b ) accommodate the minimum rates of all users in U ( b $^{)}$. In particular, we can invoke Proposition 1 to explicitly detail ¯ h (Γ$_{b}$ , b ) after recalling the labelling we have adopted, as 

<img src='content_image/301149.jpg'>

In (11) we use the convention that ˜ k + 1 reurns the user with the lowest index in U ( b ) whenever ˜ k is null on account of ∑ j ∈U ( b $^{)}$: j ≤ k R min j R$_{j,b}$ > Γ$_{b}$ for all k ∈ U ( b $^{)}$. In a similar manner we define h : I R$_{+}$ × B ′ 1 → I R$_{+}$ , such that h ( Z$_{b}$, b ) for any given TP b and a given Macro budget, Z$_{b}$ , yields the additional minimum resource needed by TP b to accommodate the minimum rates of 

