Algorithm for Dynamic Power Control and Scheduling in IEEE 802.11ax Infrastructure Networks

Abstract—With the growth of the number of wireless networks and connected devices, the negative effect of the interference becomes more dramatic. One of the most promising ways to reduce interference is the simultaneous adjustment of transmission powers together with the scheduling of transmission by various network devices. When developing such solutions for wireless networks operating in the unlicensed spectrum, an important requirement is to take into account the regulatory restrictions on the transmission power, as well as the prohibition of data transmission if the power of the signal and noise detected in the channel exceeds a certain regulated threshold. In this paper, we propose an algorithm for optimizing the transmission powers of various stations and scheduling transmissions with above-mentioned restrictions. The proposed algorithm is based on the branch and bound method and allows finding the optimal solution with a specified accuracy in terms of the chosen utility function.

APPENDIX

Proof of Statement 1. Let γ' = f^–1(v). Since the rate vector components are non-decreasing functions of the corresponding signal-to-noise ratios, then it follows from v ≽ u that γ' ≽ γ. It was proved in [4, Lemma 2] that, if γ' ≽ γ, then x' ≽ x. Therefore,

$$\sum\limits_j {{{b}_{{ij}}}} x_{j}^{'} \geqslant \sum\limits_j {{{b}_{{ij}}}} {{x}_{j}} > \hat {c}.$$

The statement is proven.

Proof of Statement 2. A similar statement was proved in [4] for signal-to-noise ratio vectors in the presence of a constraint on the maximum transmit power, but in the absence of constraints (5). If the signal-to-noise ratio vector γ' is attainable, then any signal-to-noise ratio vector γ is also attainable such that γ ≼ γ'.

Let γ = f^–1(u) and γ' = f^–1(v). Since the function f is non-decreasing, the inverse function f^–1 is also non-decreasing, and f^–1(γ) ≼ f^–1(γ'). The data rate vector v is attainable; therefore, the corresponding signal-to-noise ratio vector γ' is also attainable. Based on the statement proved in [4], the vector γ is attainable; therefore, the corresponding vector u is also attainable in the absence of constraints (5).

Since v is attainable, constraints (5) are satisfied for all indices i such that the corresponding transmission power $x_{i}^{'}$ is greater than zero. Then, according to Corollary 1, these constraints are also satisfied for the corresponding indices of the vector u. For any other index j, $x_{j}^{'}$ = 0 is fulfilled; therefore, x_j = 0 since x' ≽ x while the fulfillment of the constraints (5) for index j is not required.

The converse statement is a direct consequence of what was proved.

Proof of Statement 3. Since [p, q]\[h, q] = {y ∈ [p, q] | y$\notin$ [h, q]}, then for ∀y ∈ [p, q]\[h, q], there is an index i such that y_i$\leqslant$h_i. In this case, for any index j, y_j$\leqslant$q_j is fulfilled. Thus, there exists an index i such that y ≼ s_i.

The $\hat {U}$ monotonously depends on the transmission rates; therefore, $\hat {U}$(y) $\leqslant$$\hat {U}$(s_i) $\leqslant$ max_{j = 1, …, N}$\hat {U}$(s_j).

Consequently, it is true for ∀y ∈ [p, q]\[h, q] so that $\hat {U}$(y) $\leqslant$ max_{i = 1, …, N}$\hat {U}$(s_i).

The statement is proven.