slopes) is identical in each case to ¯ h (Γ$_{b}$$_{1}$ , b$_{1}$ ) − ¯ h (Γ$_{b}$$_{1}$+ δ$_{b}$$_{1}$, b$_{1}$ ) . On the other hand, the improvement obtained in the former case is no greater than the latter case. This follows from the facts that while the same set of slope values S ( Z$_{b}$$_{1}$, Γ$_{b}$$_{1}$ , b$_{1}$ ) , ∀ Z$_{b}$$_{1}$ ≥ ¯ h (Γ$_{b}$$_{1}$ , b$_{1}$ ) are used to replace existing ones in each case, in the former case the replaced slope values are at least as large while the width (or the macro resource spanned under the replaced slopes) is no greater, thereby resulting in a smaller improvement. Taking these observations together we have the desired result in (32). 

Let us now consider the case b$_{1}$ = b$_{2}$ . The method we employ is to reformulate the problem at hand into one involving only one pico TP ( b$_{1}$ ), albeit with an expanded pool of associated users, and the macro TP. Towards this end, define a set of pico TPs ˆ B$_{1}$ = B ′ 1 \ b$_{1}$ . Then, specializing the result in (14) to the set of pico TPs ˆ B$_{1}$ along with the set of users associated to those TPs, we notice that the weighted sum rate is the sum of three terms. The first two terms are the weighted sums of minimum rates that have to be met and the rates obtained upon assigning the pico resource as slack (if any), respectively. The third term pertains to the weighted sum rate obtained by assigning macro resource as slack. As the first two terms are invariant to changes in Γ$_{b}$$_{1}$ we focus on the third term. Note that the latter term is the area under rectangles of decreasing heights, where these heights are the distinct slope values of the curve ˆ O ( Z, { Γ$_{b}$ }$_{b}$$_{∈}$$_{ˆ}$ $_{B$_{1}$}$) , Z ∈ [ ∑ b ∈ ˆ B$_{1}$ ¯ h (Γ$_{b}$ , b ) , 1] . For each such rectangle let us define a virtual user, say ϑ and set the parameters of that virtual user as follows. Arbitrarily choose its weight, w$_{ϑ}$ , and peak rate from the macro TP, R$_{ϑ,}$$_{1}$ , to be any positive scalars such that w$_{ϑ}$R$_{ϑ,}$$_{1}$ equals the height of that rectangle and set R max ϑ to be equal to R$_{ϑ,}$$_{1}$ times the width of the rectangle. Further, set R min ϑ = 0 , R$_{ϑ,b}$$_{1}$ = 0 . Collecting all such virtual users in a set ˆ U , consider the problem of allocating the macro resource when we have just one pico TP b$_{1}$ but to which an expanded user pool U ( b$_{1}$ ) ∪ ˆ U is associated. By our construction of the virtual user set it can be readily seen that for any feasible pico budgets, this reformulated problem is equivalent to the original one (3). Moreover, in this reformulated problem the pico resource of TP b$_{1}$ is assigned to only users in U ( b$_{1}$ $^{)}$. We can invoke Proposition 2 on this reformulated problem along with the relations (29) and (31) to deduce that the claims in (33) and (32) indeed hold true. 

Proof of Theorem 1 : The problem in (16) is a convex optimization problem for which the 

