Network utility maximization in uplink multiuser wireless LANs
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 aswhere I is set of the network nodes. The xi is the allocated data rate to user i and Ui(xi) is utility that user i receives by the allocated rate xi. 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 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 ofwhere φ denotes the set of admissible resource allocations, and also sets χ and 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.
Section snippets
System model
The model envisaged in this paper considers an Infrastructure Basic Service Set (IBSS) where N single antenna STAs communicate with an M-antenna AP under the saturated traffic. There is a frequency selective wideband channel at uplink that is divided into L narrowband frequency flat data subcarriers. The channel state in subcarrier l , 1 ≤ l ≤ L from the single antenna of ith STA to the M antennas of AP is represented by where [ · ]Tis the transpose operator. The AP can
Problem formulation and overview of the proposed scheme
The perceived QoS of an STA in a WLAN can be quantified as a function of its experienced long-term throughput. The proposed scheme considers the following α-fair utility function [13]where x denotes the long-term throughput, and α is a fairness parameter based on which different fairness criteria are satisfied. For example, the proportional and max-min fairness can be achieved when and α → ∞, respectively.
The ergodic limit of instantaneous throughput
NUM in the dual domain
The non-convex constraints of the problem modeled as (16) in the previous section make it a none-convex optimization problem. However, it is in the form of (2), and thereby its duality gap is zero. Therefore solving the dual of the (16) leads to the intended solution. In order to transform the original problem to the dual domain, is defined as the non-negative Lagrange multipliers set, corresponding to the N inequality constraints of the problem (16). According to this multipliers
Simulation results
In this section, we evaluate the performance of the proposed scheme by implementing it in MATLAB®. The channel was modeled based on the TGn channel models [18]. Specifically, we used profile E (large office, highly frequency selective) in our simulations. In order to make that model appropriate for multiuser scenarios, we modified some of its parameters, including angle of arrival (AoA) and angle of departure (AoD), according to the IEEE 802.11ac task group recommendation. The system was
Conclusion
This paper modeled scheduling of concurrent STAs in the uplink multi user WLANs as a network utility maximization problem which is a non-convex problem. In spite of that, considering the zero duality gap the dual of this problem is solved by means of a stochastic learning algorithm. Determination of stochastic subgradients involves dealing with an NP-complete problem. The imperialist competitive algorithm is employed to solve this problem with significantly less computational complexity than
Acknowledgments
This work was partly supported by the National Research Foundation of Korea under grant 2015R1D1A1A01057611, and the Yonsei University Research Fund under grant 2016-52-0033.
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