You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
We formulate throughput maximizing, max-min fair, weighted max-min fair, and proportionally fair scheduling problems for cognitive radio networks managed by a centralized cognitive base station. We propose a very general scheduling model accomplishing goals such as making frequency, time slot, and data rate allocation to secondary users with possibly multiple antennas, in a heterogenous multi-channel and multi-user scenario. Moreover, our schedulers ensure that reliable communication between the cognitive base station and secondary users are maintained, no collisions occur among secondary users, and primary users in the service area of the cognitive base station are not disturbed. Two distinctive features of our fair schedulers are that they provide joint temporal and throughput fairness, and take throughput values experienced by secondary users in the recent past, referred to as window size, into account and use this information in the current scheduling decision. We also propose a heuristic algorithm for our fair schedulers and demonstrate through simulations that our proposed heuristic yields very close solutions to the values obtained from the optimization softwares. Furthermore, we make extensive simulations to evaluate our schedulers' performance in terms of both total throughput and fairness for varying number of secondary users, frequencies, antennas, and window size.

Cited Abstract(s):
[1]: 

[2]: The fair allocation of resources among different nodes is one of the critical problems in wireless mesh networks. Existing solutions mainly focus on rate-limitation policies or distributed fair MAC schemes at the potential expense of total network utilization. This paper investigates a special starvation problem among TCP flows that are different hops away from the BS, as well as its recently proposed solution, the ‘Minimum Content Window’ policy based on IEEE 802.11e. It is found that the aggregate throughput degrades sharply because the effect of this policy on the TCP congestion mechanism has been overlooked. This paper proposes a priority-based congestion control by using ‘Cross-Layer Explicit Congestion Notification’. Analysis and simulation results demonstrate that our scheme can improve the fairness of TCP flows while the aggregate throughput is at least 20 higher than the ‘Minimum Content Window’ policy. Copyright © 2011 John Wiley & Sons, Ltd.

[3]: This paper addresses the problem of fairness in wireless ad-hoc networks. Usually, the problem of fairness in wireless ad-hoc networks is addressed in a classic approach inherited from wired networks. The common assumption is that nodes flows have pre-assigned fair shares. The task becomes then to modify wired networks' fair queueing algorithms to address wireless network nature. Wired networks have efficient means of allocating fiar shares through admission control and additionally, the fair shares remain constant throughout the session duration due to the static nature of the nodes. In ad-hoc networks, it is meaningless to assume statically pre-assigned fair shares, since on one hand, not only the nodes move, but also the routers are mobile, and on the other hand, contention is location dependent, such that in terms of absolute guarantees, fairness would at most mean "avoiding starvation": thus applying rate proportional fiar queueing algorithms is beyond the original goal of such algorithm, viz. flow isolation protection and bandwidth guarantee. In this work we argue in favor of multi-level scheduling for wireless ad-hoc networks with max-min fair allocation of the fair shares at the lower-most layer (MAC layer). This paper, mainly lays down the theoretical framework by which one can calculate the fair shares that would achieve max-min fairness in an ad-hoc network. We design distributed algorithms that allow each node to determine its max-min per-link fair share in a global ad-hoc network without knowledge of the global topology of the network. The results are then used in conjunction with a novel practical scheduling algorithm for IEEE 802.11 to show how fairness is achieved

[4]: Emerging spread spectrum high-speed data networks utilize multiple channels via orthogonal codes or frequency-hopping patterns such that multiple users can transmit concurrently. In this paper, we develop a framework for opportunistic scheduling over multiple wireless channels. With a realistic channel model, any subset of users can be selected for data transmission at any time, albeit with different throughputs and system resource requirements. We first transform selection of the best users and rates from a complex general optimization problem into a decoupled and tractable formulation: a multiuser scheduling problem that maximizes total system throughput and a control-update problem that ensures long-term deterministic or probabilistic fairness constraints. We then design and evaluate practical schedulers that approximate these objectives.

[5]: Link-layer fairness models that have been proposed for wireline and packet cellular networks cannot be generalized for shared channel wireless networks because of the unique characteristics of the wireless channel, such as location-dependent contention, inherent conflict between optimizing channel utilization and achieving fairness, and the absence of any centralized control. In this paper, we propose a general analytical framework that captures the unique characteristics of shared wireless channels and allows the modeling of a large class of system-wide fairness models via the specification of per-flow utility functions. We show that system-wide fairness can be achieved without explicit global coordination so long as each node executes a contention resolution algorithm that is designed to optimize its local utility function. We present a general mechanism for translating a given fairness model in our framework into a corresponding contention resolution algorithm . Using this translation, we derive the backoff algorithm for achieving proportional fairness in wireless shared channels, and compare the fairness properties of this algorithm with both the ideal proportional fairness objective, and state-of-the-art backoff-based contention resolution algorithms. We believe that the two aspects of the proposed framework, i.e. the ability to specify arbitrary fairness models via local utility functions, and the ability to automatically generate local contention resolution mechanisms in response to a given utility function, together provide the path for achieving flexible service differentiation in future shared channel wireless networks.

[6]:
There are various efforts in the literature that addressed fairness in the general domain of wireless networks [2] [1] [6] . Authors in [3] proposed a set of algorithms for the assignment of max-min fair shares to nodes in a wireless ad hoc network. The work in [4] proposed schedulers that provide deterministic and probabilistic fairness by decoupling throughput optimization and fairness guarantees as two distinct blocks. Authors in [5] proposed a fair framework for ad hoc wireless networks via a contention resolution algorithm. However, none of these works takes into account the unique features of the CRN concept such as the requirement to ensure PU protection; therefore, they are unsuitable for implementation in CRNs.