Network utility maximization in uplink multiuser wireless LANs

Abstract

Multiuser Multiple-Input Multiple-Output (MU-MIMO) enables several simultaneous transmission from the stations toward a multiple antennas access point of a WLAN and thereby boosts the network throughput. However, this throughput enhancement greatly depends on which stations are transmitting concurrently, which in turn necessitates the design of an efficient scheduling algorithm. In this paper, we model uplink MU-MIMO scheduling as a network utility maximization problem that is a non-convex problem due to the wideband channel aware transmission rate function and scheduling constraints. Relying on the zero duality gap, a stochastic learning algorithm is designed to solve the dual of that problem. Determination of stochastic subgradient in the stochastic learning algorithm involves an NP- complete problem, hence imperialist competitive algorithm is exploited to solve this problem. The simulation results substantiate significant throughput improvement and airtime fairness superiority of this scheme compared to the existing ones. The results also show the proposed scheme is not able to guarantee the optimal throughput due to employing imperialist competitive algorithm.

Introduction

There has been an increasing appetite for higher transmission rates in the WLANs. In order to cope with this demand, the upcoming standard for WLANs, which is known as IEEE 802.11ax [1], is committed to employ wide range of physical layer techniques including Orthogonal Frequency Division Muliple Access (OFDMA), MU-MIMO and high-density modulation. MU-MIMO enables the AP to concurrently communicate with multiple STAs in either of downlink (DL) or uplink (UL) cases, and accordingly to fully utilize the channel capacity. IEEE 802.11ac standard supports DL MU-MIMO and UL MU-MIMO will be included in IEEE 802.11ax. The performance of the UL MU-MIMO is determined by the set of STAs that transmit concurrently toward AP. The goal of this paper is to develop a scheduling algorithm for the UL MU-MIMO enabled WLANs that determines the concurrent STAs such that, while maintaining fairness among STAs, the total utility perceived by them is maximized.

Most of the resource allocation problems of the communication networks can be captured by the model which is called Network Utility Maximization (NUM) [2]. In the context of this model, utility function is a measure of the degree of user contentment given the allocated resources. This model originally is designed for a multi-hop wired network, and its seminal form is expressed as

$$\begin{array}{cc}& \text{max}\sum _{i\in I}{U}_i\left(x_i\right)\hfill \\ & \text{s.t.}\mathbf{A}\mathbf{x}⩽\mathbf{C}\text{and}\mathbf{x}⩾\mathbf{0},\hfill \end{array}$$

where I is set of the network nodes. The x_i is the allocated data rate to user i and U_i(x_i) is utility that user i receives by the allocated rate x_i. Moreover, the optimization variable x is a vector with the allocated rates of all users and C is the vector containing the fixed capacities of all links. Finally, A is a routing matrix, whose element of kth-column and lth-row is 1 if link k is part of the route l and 0 otherwise. In case of concave utility function, the problem formulated in (1) is a convex optimization problem, and thanks to the zero duality gap, its dual can be solved by means of the (sub)gradient descent algorithm to reach the solution. However, extending this framework to the wireless networks is not straightforward, mainly due to the non-fixed wireless channel capacity. Hence, a practical approach to tackle this limitation is to conduct optimization according to the expected value of the channel capacity. Suppose the capacity of the wireless channel is given by function f(θ, φ(θ)), where θ and φ(θ) are time-varying channel parameter and allocated resource corresponding to this parameter. Therefore, the aforementioned maximization can be performed based on the $\mathbb{E}\left[f\right(\theta ,\phi \left(\theta \right)\left)\right]$ instead of C. In general, the function f is not convex. Accordingly the problem is not convex, and thereby the duality gap is not zero irrespective of the concavity of the utility function. However, it has been shown in [3, Theorem 1] that, under the specific circumstance, the duality gap is zero for an optimization problem in the general form of

$$\begin{array}{ccc}& & \text{max}\sum _{i\in I}{U}_i\left(x_i\right)\hfill \\ & & \text{s.t.}\mathbf{x}⩽\mathbb{E}\left[f\right(\theta ,\phi \left(\theta \right)\left)\right],\hfill \\ & & \mathbf{x}\in \chi ,\mathit{\phi }\in \Phi \hfill \end{array}$$

where φ denotes the set of admissible resource allocations, and also sets χ and $\Phi$ are compact. In other words, in case of existence of a feasible rate set that strictly satisfies the inequality constraints of (2) and also if the joint probability distribution of channel parameters contains no positive probability point, the duality gap is zero. In the sequel we leverage this fact to solve the problem of this paper.

Since this NUM model is able to take into account both efficiency and fairness metrics during resource allocation, various problems of wireless networks have been formulated as an NUM problem. Hence this paper employs NUM to model the scheduling problem of UL MU-MIMO enabled WLANs. This scheduling algorithm is equipped with a rate adaptation algorithm that makes it fading aware. The UL MU scheduling problem has been previously studied in the context of cellular networks [4], [5], [6]; these works assume a narrow band frequency flat channel and Shannon capacity-achieving channel coding, which makes it possible to determine transmission rate of each STA through a continuous concave function of its channel state and transmission power. However, neither narrowband channel nor continuous rate function assumptions are true for WLANs, where the channel is a wideband Orthogonal Frequency-Division Multiplexing (OFDM) channel, and the transmission rate is a discrete non-concave function of OFDM subcarriers’ impulse responses. MIMOMate [7] has recently been proposed as a centralized algorithm to determine STAs of each concurrent transmission in a UL MU WLAN, with the goal of total throughput maximization and fair channel access opportunity to the STAs. MIMOMate partitions STAs into groups, each of which consists of a leader and some followers. Given a specific leader STA, its followers are determined such that the total throughput of the concurrent transmission of that leader and its followers is maximized. This partitioning problem is NP-complete when AP has more than two antennas, hence a greedy heuristic is exploited to solve it. To provide fair channel access, MIMOMate relies on the classical contention scheme between leaders, which is unable to maintain airtime based fairness in rate diverse situations and suffers from performance anomaly [8]. The contention-based channel arbitration scheme implicitly supposes all the contenders employ same transmission rate, hence by assigning equal transmission opportunities to them ensures all of them receive similar throughput. In practice, however, STAs adapt the transmission rate according to their channel condition and similar transmission rates assumption is not held. Accordingly, in multirate situations, STAs that transmit at lower rate occupy the channel for longer duration. This phenomenon known as performance anomaly reduces the total network utility, because a considerable portion of the channel airtime is devoted to the low rate transmissions. This can be remedied through bandwidth proportional-fair resource allocation. Under this fairness criterion, the STAs are allocated the same amount of channel airtime, irrespective of their channel state quality. This kind of fairness can be achieved by capturing the utility of STA as a logarithmic function, and performing network resource allocation according to this utility function.

Similar to the MIMOMate, we also take a centralized approach for scheduling the concurrent transmissions in an UL MU WLAN. Grounded on NUM, our approach is able to take into account various fairness criteria including proportional and max-min metrics, while maximizing total throughput. The contributions of this paper are threefold. First, we define a simple yet accurate formula for the throughput of an STA in an UL MU WLAN. Exploiting that, the scheduling of the concurrent streams is formulated as an NUM problem. Second, since the formulated problem is non-convex, leveraging the zero duality gap, we solve it in the dual domain using a stochastic learning method. Third, in order to avoid substantial computation cost in obtaining the stochastic subgradients, this paper uses an evolutionary algorithm whose performance is approved by the simulation results. The rest of the paper is organized as follows. In Section 2, we explain the system model of this paper. In Section 3, we elaborate the formulation of UL MU WLAN scheduling as an NUM problem, and overview the proposed scheme. The details of the stochastic learning based approach for solving the addressed problem of this paper is discussed in Section 4 . The simulation results are presented in Section 5, and finally we draw the conclusion in Section 6.