Fairness and QoS Guarantees of WiMAX OFDMA Scheduling with Fuzzy Controls
© Chao-Lieh Chen et al. 2009
Received: 1 July 2008
Accepted: 29 December 2008
Published: 15 February 2009
A fairness and QoS guaranteed scheduling approach with fuzzy controls (FQFCs) is proposed for WiMAX OFDMA systems. The controllers, respectively, adjust priority and transmission opportunity (TXOP) for each WiMAX connection according to QoS requirements and service classes. The FQFC provides intra- and interclass fairness guarantees by making connections within the same class achieve equal degree of QoS while at the same time making those without QoS requirements equally share the remaining resources. Even in dynamic environments such as mobile WiMAX networks with time-variant traffic specifications, the FQFC fairly guarantees delay, throughput, and jitter, which are seldom achieved at the same time by state-of-the-art solutions.
IEEE 802.16 standard (WiMAX) [1, 2] is one of the most popular standards for fixed and mobile broadband wireless access systems to provide last mile access. Due to various users with diverse QoS requirements and wireless communication technologies, the resource scheduler plays an important role to provide fairness and QoS guarantees. As summarized in , a resource scheduler in wireless multimedia networks needs to possess the following features: efficient link utilization, delay bound, low implementation complexity, throughput, scalability, and fairness.
For WiMAX and OFDMA systems, various scheduling algorithms have been proposed for achieving QoS guarantees. For example, Liu et al.  proposed a priority-based scheduler which assigns each connection a priority updated according to QoS parameters and channel state and then assigns time slots to connections according to the order of priority values. The method has low implementation complexity because the scheduler simply updates the priority of each connection per frame and allocates time slots to the connection with the highest priority. However, it does not consider fairness and jitter issues which are important metrics for real-time applications. To maintain low implementation complexity under considering fairness and jitter, we use the priority-based scheduling scheme for initial priority assignment and afterward, the proposed a fairness and QoS guaranteed scheduling approach with fuzzy controls (FQFC) mechanism takes care of the scheduling job using fuzzy control approach. Many algorithms have been proposed to deal with the fairness problem, and can be briefly divided into two categories. The first category is to reduce the resource allocation problem into an optimization problem. Based on the optimization theory, for example, [5, 6] have good performance on spectrum efficiency and system utilization. They formulate the cross-layer optimization problem to maximize the average utility of all active users subject to certain constraints. However, in addition to implementation complexity, these methods still suffer some problems. To achieve optimal spectrum efficiency, the optimization approaches may, on the other hand, fail to provide QoS guarantees. Moreover, the relation between traffic specifications and network state is uncertain. Uncertainty and dynamics in mobile environment make exact modeling of objective function and constraints impossible when performing the optimization steps. In this paper, the FQFC adopts fuzzy control technique to deal with the modeling problem. The reason we use fuzzy control is to tackle uncertainty and dynamics in wireless communication environment. Among soft computing methods, inference based on probability theory is also widely used for modeling uncertainty and dynamics. However, the controller based on probability must rely on statistical observations to perform inference. Correctness of statistical information is based on the law of large number. In case that gathering large amount of statistic information in short time is difficult, it will be infeasible. Moreover, inference based on probability usually assumes some specific probability model for result of feedback observation to follow. As shown in , a single model usually fails to represent the behaviors of dynamic environments such as mobile wireless networks with sudden bursts or changes.
The second category is a utility-based scheduler. A utility function is a measure of relative satisfaction from users' requirements. The schemes in [8, 9] apply utility functions to maximize the total utility of all connections. Utility-based optimization approaches did guarantee QoS of some connections but also starve others. On the other hand, some approaches such as [10, 11] propose utility-fair bandwidth adaptation schemes for multiclass traffic in wireless networks. Rather than achieving resource fairness, the bandwidth adaptation schemes make sure that all connections can obtain similar utility values to achieve the so-called utility fairness. These schemes are effective in both achieving utility fairness and increasing network resource utilization. However, the utility-fair schemes may fail to provide QoS guarantees since it does not consider the priority of the connections.
In this paper, our objective is to provide efficient control for both QoS and fairness guarantees of WiMAX OFDMA scheduling. For QoS guarantees, we address the problem of head-of-line (HOL) delay and jitter control for real time applications and throughput control for nonreal-time applications. The FQFC scheduler assigns each connection a priority and TXOP, and adjusts them according to channel quality, QoS requirements, and service classes. Due to uncertainty and dynamics of the environment, it is difficult to find out the mapping between priority and QoS requirements. For fuzzy inference, it is the simplest way to model a complex system when there is few and uncertain information available. In the field of controller design, fuzzy controller is one of the most popular approaches. Moreover, fuzzy control has been widely used in researches on communication networks such as [7, 12–17]. However, there are few articles talking about using fuzzy control for WiMAX. In this paper, the FQFC model is developed for WiMAX OFDMA systems and is proved that both fairness and QoS are guaranteed. Other fuzzy control methods [14–17] may be proved to achieve certain degree of QoS. However, fairness is seldom assured in these and state-of-the-art approaches. Then, we define two types of fairness including intraclass and interclass fairness. To achieve intraclass fairness, we set up a reference goal to each connection according to the QoS requirements, and make the connections achieve the goal by priority scheduling and TXOP allocation. If each connection can achieve its QoS requirement, intraclass fairness is guaranteed. For achieving interclass fairness, the FQFC does not allocate superfluous resources out of what required. Compared to state-of-the-art methods, connections of high-priority classes release more resources to lower priority ones. Thus, the FQFC makes the connections without QoS requirements evenly share the remaining resources. Based on the priority scheduling and TXOP allocation methods, the FQFC provides both intraclass and interclass fairness with QoS guarantees and featuring low implementation complexity.
This paper is organized as follows. In Section 2, we introduce background including network configuration, MAC QoS, PHY resource allocation, and fairness descriptions. Section 3 describes the FQFC mechanism and depicts the design of the fuzzy controllers for each service class. In Section 4, we investigate the mechanism performance of QoS and fairness through simulations. Finally, we conclude the paper in Section 5.
2.1. Network Configuration
WiMAX specifies two communication modes which form different topologies—point-to-multipoint (PMP) and mesh modes. In PMP mode, a base station (BS) centrally allocates downlink (from BS to SS) and uplink (from SS to BS) resources to subscriber stations (SSs). All SSs are only allowed to communicate with a BS. In mesh mode, SS can act as a router to assist its neighbor to relay data. In the 802.16 standard, this mode is optional and is not discussed in this paper. Hence, we focus on proposing a downlink scheduling algorithm to provide QoS guarantees in PMP mode.
IEEE 802.16 WiMAX PHY adopts the orthogonal frequency-division multiple access (OFDMA) technology based on OFDM modulation. The OFDMA technology allows multiple users transmitting packets at the same OFDMA symbol via different subchannels, such that wireless resources are utilized ultimately.
2.2. Scheduling Services in MAC Layer
IEEE 802.16 MAC protocol is connection-oriented; each connection is assigned a connection ID (CID) and a single scheduling service determined by a set of QoS parameters. Four scheduling services in the 802.16 standard are supported: unsolicited grant service (UGS), real-time polling service (rtPS), nonreal-time polling service (nrtPS), and best effort (BE). The UGS supports real-time constant bitrate data streams, such as voice over IP (VoIP) without silence suppression. The QoS parameters of UGS service are minimum reserved traffic rate, the tolerated jitter, maximum latency, and request/transmission policy. The rtPS supports real-time variable-rate data streams, such as MPEG video or VoIP with silence suppression. The QoS parameters of rtPS are maximum latency, request/transmission policy, minimum reserved traffic rate, and traffic priority. The nrtPS supports delay-tolerant variable-rate data streams, such as FTP. The QoS parameters of nrtPS are minimum reserved traffic rate, request/transmission policy, and traffic priority. The BE supports best-effort data streams. The QoS parameter is request/transmission policy. In IEEE 802.16e , an additional service class called extended real-time polling service (ertPS) has superior efficiency than both UGS and rtPS. It supports real-time variable-rate data streams, such as VoIP with silence suppression. The QoS parameters of ertPS are minimum reserved traffic rate, maximum latency, request/transmission policy, and the tolerated jitter. Hence, considering the QoS requirements of the four class services, we calculate the reference goal as traffic specification (TSPEC) according to these QoS parameters.
2.3. Resource Allocations in PHY Layer
IEEE 802.16 OFDMA system defines two types of subcarrier permutations, distributed subcarrier permutation and adjacent subcarrier permutation. The former permutation type includes partially and fully used subcarriers (PUSC and FUSC) which are pseudo-randomly selected and grouped into subchannels, while the later includes adaptive modulation and coding (AMC), and only adjacent subcarriers are clustered to form subchannels. Dispersing noise and interference in fast changing environment, the PUSC and FUSC modes are suitable for mobile networks. For AMC mode, the BS allocates appropriate subchannels for connections with larger SNR to enhance system performance, and it is suitable for fixed or low mobility environment. To support mobile WiMAX, the FQFC scheduling and allocation are based on distributed subcarrier permutation.
In OFDMA, the basic allocation unit is a slot that composes of one subchannel along with an OFDMA symbol, such that the resource allocation becomes a two-dimensional problem. By using the distributed subcarrier permutation, all subchannels are the so-called equally adequate for all SSs , and our resource allocation is based on Raster algorithm , in which the frame is filled row by row, from left to right and from top to bottom, and efficiently reduces the burst numbers.
In wireless networks, the fairness definition is not straightforward. As described in , a fair resource allocation usually does not produce equal connection data rate because the diverse connections also suffer from diverse channel conditions, network states, and dynamics. The dynamics result from mobility and time-variant traffic specifications (TSPECs). Moreover, WiMAX needs to provide QoS guarantees for four classes of scheduling services. Therefore, for fairness, it is necessary to consider QoS guarantees for different class connections. We define two types of fairness described as follows.
Intraclass fairness: the connections within the same class achieve equal degree of QoS.
Interclass fairness: the connections with QoS requirements achieve exactly their demands, and those without QoS requirements equally share the remaining resources.
Hence, our objective is to achieve both intraclass and interclass fairness.
2.5. Fuzzy Controller
3. Design of the Proposed Scheduling Mechanism
3.1. Controller Design for ertPS & rtPS
3.1.1. Goal Delay Controller
3.1.2. Priority Controller for Real Time Services
In our design, we denote negative, zero, and positive forces with fuzzy singletons S, M, and L. The control actions of these singletons at the conclusion parts of fuzzy rules are as follows:
As we can see in (2), when the goal delay is closer to the deadline, the adaptation force of the priority is larger. We depict the priority initialization and controlled direction as follows.
where is the deadline, is the guard time before the deadline, and denotes the HOL delay. If , the larger denotes the higher satisfaction of delay requirement, which causes lower priority. If , the HOL delay has been over the guard time of deadline. The connection should get resources immediately to avoid packet losses. Hence, the priority is set as . When is zero, the connection is under deep fading and should not be scheduled.
(b) Priority Controller. Let the controller action be the priority . One of the input is the difference between the actual value of the observed HOL delay and the desired value , that is, . The universe of is . The variable has three linguistic values N, E, and P which represent fuzzy concepts "Negative," "Equal," and "Positive," respectively. The fuzzy sets N, E, and P are characterized by the membership functions shown in Figure 6.
The other input of the controller is the difference between two errors, which is defined as . Substituting to , we obtain . The universe of is . The linguistic values of , , , and also representing fuzzy concepts "Negative," "Equal," and "Positive," respectively, are characterized by the membership functions as shown in Figure 7, where and . Sign of these values constitutes four cases as shown in Figure 7. The membership functions are time-variant and change along with the variable .
The priority controller makes the delay fall in the tolerable range which is below the deadline. Hence, each connection in the real-time class achieves the QoS specification, while intraclass fairness is guaranteed. When the delay is below the tolerable range, the controller decreases the priority for releasing the resources. This scheme guarantees the jitter and interclass fairness at the same time.
where is a constant to normalize the loss rate according to the predefined precision. According to (5), if the connection drops packets due to out of the deadline, the priority controller allocates more resources by increasing the priority for achieving intraclass fairness. Even if all connections are in an extremely bad environment, they will suffer the same loss rate.
3.1.3. TXOP Controller
3.2. Controller Design for nrtPS
If the average throughput is lower than minimum reserved rate, the priority controller raises the priority to increase the throughput. Moreover, the controller needs to prevent large jitter from over-high priority. Besides, if the average throughput exceeds the upper bound, the controller decreases the priority to release the resource. We depict the controller design as follows.
3.2.1. Priority Controller
3.2.2. TXOP Setting
3.2.3. TXOP Adaptation for Fairness
For intraclass fairness, all nrtPS connections should have the same throughput ratio of average throughput with respect to minimum reserved rate. Via setting the upper bound in (13), we control the average throughput within the range between the minimum reserved rate and the upper bound, and we make the throughput ratio of all nrtPS connections the same. For interclass fairness, the average throughput will not exceed the upper bound. Hence, we can release more resources to the connections without QoS requirements.
3.3. Controller Design for BE
3.3.1. Priority Setting
3.3.2. TXOP Setting
In this paper, we also perform priority adaptation. Therefore, the overhead, especially the complexity, will be slightly higher than that of the priority-only method. Since in centralized PMP mode, all traffic flows are managed by base stations which have much more powerful computing ability than SSs, the additional computation overhead will not give any sensibly negative effect. Moreover, the proposed controllers do not use any control/management packets for fairness and QoS purposes. There is no additional network overhead caused by the proposed FQFC.
4. Evaluations and Simulation Results
We first introduce intraclass and interclass fairness criteria and then according to these criteria, we evaluate the performances of the fairness.
4.1. Fairness Criteria
The descriptions of fairness indices are as follows.
4.1.1. Intraclass Fairness Index
Intraclass fairness means that the connections within the same class achieve equal QoS guarantees. Because the connections in different service classes have different QoS requirements, we define respective intraclass fairness indices for real-time, nrtPS, and BE classes.
4.1.2. Interclass Fairness Index
According to the definition of interclass fairness, the interclass fairness has two folds: (1) the connections with QoS requirements achieve the demands; (2) the connections without QoS requirements equally share the remaining resources.
where is a tunable parameter which determines the tolerable range. and are the average state and the QoS goal of class , respectively. In the real-time class, the average state is the mean loss rate, and its goal loss rate is zero. In the nrtPS class, the QoS parameter is the average throughput, and the goal is the minimum reserved rate. The smaller the difference between the average state and the QoS goal is, the larger requirement indicator is. When the mean allocated resources for a class are away from the requirement, no matter above or below the goal, the requirement indicator decreases. When the allocated resources reach the requirements exactly, not only the QoS is guaranteed but also the remaining resources are most preserved at the same time.
In (23), is the number of classes with QoS requirements, and is the weighting factor of class , which determines the importance of the class. and are the weighting factors of the classes with and without QoS requirements, respectively. In contrast to the indices of intraclass fairness, a larger value indicates better interclass fairness.
4.2. Simulation Configuration
OFDMA FFT size
Number of subchannels
Number of OFDMA symbols for DL
Scenario 1. We set 20 real-time connections. The QoS requirements of real-time connection are the loss rate, deadline, and required jitter. The traffic rates of connections are 8 connections in 1 Mbps, 10 connections in 500 kbps, and 2 connections in 250 kbps. This scenario is to verify the guarantees of maximum latency, the tolerated jitter, and the intraclass fairness in real-time class. It is difficult to find out the mapping between priority and QoS requirements. We prove that the FQFC can efficiently control the delay.
Scenario 2. We set 10 real-time connections and 10 nrtPS connections. The QoS parameter of nrtPS connection is the minimum reserved rate. The traffic rates are 2 real-time connections in 1 Mbps, 8 real-time connections in 500 kbps, 5 nrtPS connections in 1 Mbps, and 5 nrtPS connections in 500 kbps. This scenario is to verify the guarantees of minimum reserved rate and fair resource allocation of the FQFC scheme.
Scenario 3. We set 10 real-time connections, 10 nrtPS connections, and 10 BE connections. BE connection has no QoS requirement. The traffic rates are 1 real-time connection in 1 Mbps, 9 real-time connections in 500 kbps, 3 nrtPS connections in 750 kbps, 2 nrtPS connections in 500 kbps, 5 nrtPS connections in 1 Mbps, and 10 BE connections in 100 kbps. This scenario is to verify the fair resource allocation of FQFC.
Scenario 4. In this scenario, we simulate the wireless link degrades. This will cause the modulation to change. The experiment is designed to test the robustness of the FQFC whether it can efficiently track the goal delay when the channel quality degrades. The simulated network consists of 1 BS and 10 SS (numbered from 1 to 10). In the downlink, each SS with number has 1 real-time, 1 nrtPS, and 1 BE connection with CID , , , respectively. The connections from SS1 to SS5 apply with QPSK modulation, and connections from SS6 to SS7 apply with 16-QAM modulation. All the other connections initially adopt 64-QAM modulation. This is for simulating the different channel conditions.
4.3. Performance Evaluation for Fairness and QoS Guarantees
4.3.1. Scenario 1: Intraclass Fairness and QoS Guarantees of Real Time Connections
Intraclass fairness index.
4.3.2. Scenario 2: Intraclass Fairness and QoS Guarantees of Real-Time and nrtPS Connections
4.3.3. Scenario 3: Intra-and Interclass Fairness and QoS Guarantees of All Classes
For intraclass fairness evaluation of the BE class, we compare the FQFC with the priority-only scheduler regarding average throughput. Figure 18 shows that the average throughputs of all BE connections under the FQFC control are nearly the same. The priority-only scheduler provides more resource to the last four BE connections since they employ higher rate modulation. Table 2 shows that the intraclass fairness index of the FQFC is close to zero, which is much lower than the one of the priority-only scheduler. For every real-time connection, FQFC sets the goal delay below the deadline for a certain distance in terms of the tolerable jitter. Since the goal is for priority and TXOP controllers to follow, intraclass fairness is achieved when real-time connections have almost the same loss rate and jitter performances based on the intraclass fairness criteria. For nrtPS connections, the FQFC control algorithm maintains their ratios of throughput achievement over minimum reserved rate as close to 1 as possible. Again, as long as BE connections can evenly share the remained resources from real-time and nrtPS connections, intraclass fairness of BE connections is achieved. Hence, the FQFC guarantees the intraclass fairness for the BE classes.
Interclass fairness index.
4.3.4. Scenario 4: Link Degradation
when the link degradation occurs, the FQFC adjusts the goal delay and the tolerable range according to the updated modulation. The FQFC continues to make the delay of real-time connection 3 fall in the tolerable range as shown in Figure 19. Hence, the FQFC can efficiently control the delay according to the goal delay and the tolerable range;
the service curves of nrtPS connections 11 and 13 in Figure 20 distinguish a throughput drop from the 4.0th second to the 6.0th second, whereas FQFC still maintains the throughput to meet the QoS requirements. The service curves of BE connections 23 and 25 in Figure 21 also distinguish a throughput drop from the 4.0th second to the 6.0th second. The resources are released to guarantee the QoS of real-time connection 7 as shown in Figure 19. For intraclass fairness in nrtPS connections and BE connections, all nrtPS connections keep almost the same resource usage ratio. For interclass fairness, nrtPS and BE connections release resources to guarantee the QoS of real-time connections. Hence, the FQFC guarantees both QoS and fairness even in case that wireless link degrades.
A fairness and QoS guaranteed scheduling approach with fuzzy controls FQFC algorithm is proposed for WiMAX OFDMA systems. Different from the utility-fairness, new fairness and QoS evaluation criteria in terms of loss rate, jitter, and throughput are proposed for different classes. The proposed FQFC scheme controls the delay, jitter, and throughput QoS parameters efficiently providing both fairness and QoS guarantees. Rather than using hard computation approaches such as utility-based optimizations, we use fuzzy controller to perform scheduling and resource allocations to resolve mapping among priority, transmission opportunity, and QoS requirements. The proposed FQFC scheme provides both intra- and interclass fairness guarantees in addition to QoS guarantees while implementation is with low complexity.
- IEEE Std. 802.16-2004 (Revision of IEEE Std. 802.16-2001) : IEEE standard for local and metropolitan area networks—part 16: air interface for fixed broadband wireless access systems. Revision of IEEE Std. 802.16-2001, October 2004Google Scholar
- IEEE 802.16e-2005 : IEEE Standard for Local and metropolitan area networks—part 16: air interface for fixed and mobile broadband wireless access systems amendment 2: physical and medium access control layers for combined fixed and mobile operation in licensed bands and corrigendum. February 2006Google Scholar
- Fattah H, Leung C: An overview of scheduling algorithms in wireless multimedia networks. IEEE Wireless Communications 2002, 9(5):76-83. 10.1109/MWC.2002.1043857View ArticleGoogle Scholar
- Liu Q, Wang X, Giannakis GB: A cross-layer scheduling algorithm with QoS support in wireless networks. IEEE Transactions on Vehicular Technology 2006, 55(3):839-847. 10.1109/TVT.2006.873832View ArticleGoogle Scholar
- Song G, Li Y: Cross-layer optimization for OFDM wireless networks—part I: theoretical framework. IEEE Transactions on Wireless Communications 2005, 4(2):614-624.MathSciNetView ArticleGoogle Scholar
- Ergen M, Coleri S, Varaiya P: QoS aware adaptive resource allocation techniques for fair scheduling in OFDMA based broadband wireless access systems. IEEE Transactions on Broadcasting 2003, 49(4):362-370. 10.1109/TBC.2003.819051View ArticleGoogle Scholar
- Chen C-L, Lee J-W, Chen S-Y, Kuo Y-H: Hierarchical cross-layer fuzzy control for compromise of multiple objectives in wireless mobile networks. Proceedings of the International Conference on Mobile Technology, Applications, and Systems (Mobility '08), September 2008, Yilan, Taiwan 1-7.Google Scholar
- Zheng D, Zhang J: A two-phase utility maximization framework for wireless medium access control. IEEE Transactions on Wireless Communications 2007, 6(12):4299-4307.View ArticleGoogle Scholar
- Liu K-H, Cai L, Shen X: Multiclass utility-based scheduling for UWB networks. IEEE Transactions on Vehicular Technology 2008, 57(2):1176-1187.View ArticleGoogle Scholar
- Lu N, Bigham J, Nasser N: An intra-class and inter-class utility-fair bandwidth adaptation algorithm for multi-class traffic in wireless networks. Proceedings of the Asia-Pacific Conference on Communications (APCC '06), August-September 2006, Busan, Korea 1-5.Google Scholar
- Lu N, Bigham J: On utility-fair bandwidth adaptation for multi-class traffic QoS provisioning in wireless networks. Computer Networks 2007, 51(10):2554-2564. 10.1016/j.comnet.2006.11.012MATHView ArticleGoogle Scholar
- Chen C-L: IEEE 802.11e EDCA QoS provisioning with dynamic fuzzy control and cross-layer interface. Proceedings of the 16th International Conference on Computer Communications and Networks (ICCCN '07), August 2007, Honolulu, Hawaii, USA 766-771.Google Scholar
- Chen C-L: Morphisms from IEEE 802.11 DCF specifications to its EDCA QoS practice with cross-layer interface. Proceedings of the 13th International Conference on Parallel and Distributed Systems (ICPADS '07), December 2007, Hsinchu, Taiwan 2: 1-8.Google Scholar
- Douligeris C, Develekos G: Neuro-fuzzy control in ATM networks. IEEE Communications Magazine 1997, 35(5):154-162. 10.1109/35.592110View ArticleGoogle Scholar
- Habib IW: Applications of neurocomputing in traffic management of ATM networks. Proceedings of the IEEE 1996, 84(10):1430-1441. 10.1109/5.537109View ArticleGoogle Scholar
- Yang K, Zhang J, Chen H-H: A flexible QoS-aware service gateway for heterogeneous wireless networks. IEEE Network 2007, 21(2):6-12.View ArticleGoogle Scholar
- Kazemian HB, Meng L: Neuro-fuzzy control for MPEG video transmission over bluetooth. IEEE Transactions on Systems, Man and Cybernetics, Part C 2006, 36(6):761-771.View ArticleGoogle Scholar
- Ben-Shimol Y, Kitroser I, Dinitz Y: Two-dimensional mapping for wireless OFDMA systems. IEEE Transactions on Broadcasting 2006, 52(3):388-396. 10.1109/TBC.2006.879937View ArticleGoogle Scholar
- Wengerter C, Ohlhorst J, von Elbwart AGE: Fairness and throughput analysis for generalized proportional fair frequency scheduling in OFDMA. Proceedings of the 61st IEEE Vehicular Technology Conference (VTC '05), May-June 2005, Stockholm, Sweden 3: 1903-1907.Google Scholar
- Shreedhar M, Varghese G: Efficient fair queuing using deficit round-robin. IEEE/ACM Transactions on Networking 1996, 4(3):375-385. 10.1109/90.502236View ArticleGoogle Scholar
- Jain R, Chiu D, Hawe W: A quantitative measure of fairness and discrimination for resource allocation in shared computer systems. In DEC Research Report. Digital Equipment, Littleton, Mass, USA; September 1984.Google Scholar
- Freitag J, da Fonseca NLS: Uplink scheduling with quality of service in IEEE 802.16 networks. Proceedings of the 50th Annual IEEE Global Telecommunications Conference (GLOBECOM '07), November 2007, Washington, DC, USA 2503-2508.Google Scholar
- Fong M-H, Novak R, McBeath S, Srinivasan R: Improved VoIP capacity in mobile WiMAX systems using persistent resource allocation. IEEE Communications Magazine 2008, 46(10):50-57.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.