A characterization of max–min SIR-balanced power allocation with applications

Abstract

We consider a power-controlled wireless network with an established network topology in which the communication links (transmitter-receiver pairs) are corrupted by the co-channel interference and background noise. We have fairly general power constraints since the vector of transmit powers is confined to belong to an arbitrary convex polytope. The interference is completely determined by a so-called gain matrix. Assuming irreducibility of this gain matrix, we provide an elegant characterization of the maxmin SIR-balanced power allocation under such general power constraints. This characterization gives rise to two types of algorithms for computing the max-min SIR-balanced power allocation. One of the algorithms is a utilitybased power control algorithm to maximize a weighted sum of the utilities of the link SIRs. Our results show how to choose the weight vector and utility function so that the utility-based solution is equal to the solution of the max-min SIR-balancing problem. The algorithm is not amenable to distributed implementation as the weights are global variables. In order to mitigate the problem of computing the weight vector in distributed wireless networks, we point out a saddle point characterization of the Perron root of some extended gain matrices and discuss how this characterization can be used in the design of algorithms in which each link iteratively updates its weight vector in parallel to the power control recursion. Finally, the paper provides a basis for the development of distributed power control and beamforming algorithms to find a global solution of the max-min SIRbalancing problem.

Utility-based power control with QoS support

“…0 depends on the Lagrangian multipliers resulting from the power constraints s 2 S (inequality constraints). Note that we must have tðaÞ 6 ¼ 0 since otherwise there would be no positive solution uðaÞ [ 0 to (34), regardless of the choice of V. By Theorem 4 in the appendix, we see that there exists a positive vector uðaÞ satisfying (34) iff qðDðaÞVÞ ¼ qðV T DðaÞÞ\1; which is satisfied due to the second equality in (34) and the existence of es ðaÞ [ 0:…”

“…Since, for all a C 2, qðV T DðaÞÞ\1 andpðaÞ ¼ es ðaÞ 2 P þ belongs to a bounded subset of R K þþ ; it follows from the second equality in (34) and Theorem 4 that there exists a constant c 1 [ (0, 1) independent of a such that 1 1=ð1 À qðDðaÞVÞÞ 1=c 1 \ þ 1 for all a C 2. From this and (37), we then have kuðaÞk 1 kZðaÞtðaÞk 1 =c 1 þ kRðaÞtðaÞk 1 : By (27) and (36), the left-hand side is bounded (consider u k (a) for any k 2 B 6 ¼ ;).…”

“…(ii) The uniqueness property and (22) follow from [34]. Finally, if we had p l [ 0 for some l 2 K n A with v k;l [ 0; k 2 A; then, by (22), we could always increase min k2A ðSIR k ðpÞ=c k Þ by letting p l ¼ 0: Thus, by (21), we must have p l ¼ 0. h It is important to notice that the lemma states that p defined by (21) may have zero components, in which case p has no counterpart in S. This problem is circumvented in the proof of the following proposition by considering a sequence of positive power vectors that approach a given nonnegative power vector.…”

“…Proceeding essentially as in [34] shows that the point, where the half-line starting at the zero point in the direction of the vector c ¼ ðc 1 ; . .…”

Mixed-integer linear programming framework for max-min power control with single-stage interference cancellation

“…Problem (2) has attracted considerable interest in the recent past; see e.g. [4] and references therein.…”