Eighth IEEE PerCom Workshop on Pervasive Wireless Networking 2012
Performance Evaluation of Scheduling Algorithms for Mobile WiMAX Networks Preethi Chandur∗‡ , Karthik R.M†‡ and Krishna M. Sivalingam∗‡ ∗ Indian Institute of Technology Madras, Chennai 682036, India † Centre of Excellence in Wireless Technology, IIT Madras-Research Park, Chennai 600113, India ‡ India UK Advanced Center of Excellence in Next Generation Networks, Systems and Services (IU-ATC)
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[email protected]} Abstract—In this paper, we study the performance of some of the key scheduling algorithms such as Proportional Fair Scheduler, Modified Longest Weighted Delay First and Exponential Rule. We also propose a new algorithm called EXPQW, a variant of the Exponential rule which assigns weights to the subscriber stations based on their queue length and waiting time thereby ensuring fairness and quality of service for non-real time applications. We also present three hierarchical schedulers which use a combination of the exponential rule for waiting time and queue-length and other scheduling rules. The algorithms have been implemented in OPNET Modeler 16. The scenarios studied are: the baseline configuration specified by the 802.16m Evaluation Methodology Document with up to 60 users in a service region and a saturated frame scenario with up to 120 users. The results indicate that EXPQW and the hierarchical schedulers have comparable throughput and fairness values with algorithms like Proportional Fair Scheduler and Modified Longest Weighted Delay First in moderately loaded and heavily loaded scenarios.
I. I NTRODUCTION This paper studies scheduling algorithms for wireless broadband systems and in particular, WiMAX based systems. The IEEE 802.16e standard based on which WiMAX is derived, is designed to provide aggregate data rates up to 50Mbps to mobile users over an area of 50km [1]. WiMAX utilizes MIMO (Multiple Input Multiple Output), Adaptive Antenna System techniques, Adaptive Modulation Coding and Scalable OFDMA techniques. To provide quality of service (QoS) required by the applications, a number of protocol components have to be designed and implemented for admission control, scheduling, traffic shaping and traffic policing. This paper concentrates on scheduling algorithms. The scheduling algorithms have to select from among the different users and different transmission channels depending on the channel conditions, mobility and bandwidth requirements of the user while ensuring fairness, stability and throughput optimality [2]. The scheduling algorithms have to consider the various system limitations including scarce bandwidth, relatively high bit error rates, interference and distortion. Scheduling for Orthogonal Frequency Division Multiple Access (OFDMA) based systems can be described as the problem of assigning users for a given sub-channel at a given time slot based on the users’ channel conditions and resource requirements. A number of scheduling algorithms have been
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proposed for wireless mesh, mobile and relay networks. These algorithms can be broadly divided into channel aware and channel unaware schedulers, depending on whether they take the channel conditions into consideration or not [3]. In this paper, a detailed performance study of channel aware schedulers such as, Proportional Fair Scheduler (PF) [4], Modified Longest Weighted Delay First (MLWDF) [5], Exponential Waiting time (EXPW) and Exponential Queue length (EXPQ) algorithms [2] has been conducted. The algorithms are compared with respect to parameters such as system throughput, application level delay of real-time applications and fairness index of all the service classes. A new variation of the exponential rule (called EXPQW) has been proposed which evaluates the waiting time and queue length of the users to decide the users’ priority. In addition, the performance of a few hierarchical schedulers [6] has been studied. These algorithms implemented in OPNET Modeler 16 [7], have been evaluated under a moderately loaded and under a heavily loaded network. The simulation results show that the MLWDF rule performs very well for rtPS service classes. The impact of combining this with PF and the proposed EXPQW for nrtPS and BE classes has been studied. The results of scheduling delay sensitive applications with EXPW and delay insensitive applications with EXPQ have been presented. This study provides results based on the WiMAX architecture; however, the trends would also be applicable to other 4G systems such as Long Term Evolution (LTE or LTE-A). II. E XISTING S CHEDULING A LGORITHMS In this section, a brief introduction has been provided to some of the existing scheduling algorithms used in wireless networks. The network architecture consists of a centralized Base Station (BS) and a set of Subscriber Stations (SS) which constitutes a basic service region. The region may be an entire cell or a sector within a cell. The scheduling algorithm is executed at the BS based on the SSs’ bandwidth requests. The scheduling algorithm uses scheduling frames which contain transmission opportunities (TX-OP) for the different users [1]. There are five main classes of traffic: Unsolicited Grant Service (UGS); extended real-time polling service (ertPS); real-time polling service (rtPS); non real-time polling service (nrtPS) and best effort (BE). As per the 802.16e standards [1], UGS and ertPS applications are to be allotted
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as much bandwidth as the SS requests. The system would use admission control mechanisms to limit the number of such applications at a given time. Thus, scheduling decisions need to be made for the other three application classes. A recent survey of scheduling algorithms for mobile WiMAX networks is available in [3]. A brief description of a few scheduling algorithms are given below. A. Proportional Fair Scheduler (PF) The PF provides a good trade off between system utility and fairness by selecting the user with highest instantaneous data rate relative to its average data rate [8], [9]. The PF was designed specifically for the BE service class and hence does not guarantee any QoS requirement such as delay, jitter and latency. Let µi (t) be the data rate supported by the channel of user i where i = 1, 2, . . . , N at time instant t. Assume that the data rate remains constant over a time slot and µi be the average data rate supported by user i. Let j be the selected µi (t) user, the scheduling rule is given by: j = arg max . µi i
by:j = arg max γi µi (t) exp i
ai Qi (t) − aQ p 1 + aQ
!
1 X ai Qi (t) N i In [10], EXP is combined with sub carrier allocation algorithm for rtPS and nrtPS service classes. The priority given to the users with large queue lengths can compromise the fairness of users with slightly less values. In such a case, the credits for the offending users have to be reduced, by modifying EXPQ. D. Hierarchical Schedulers where aQ =
One way to improve system performance is to use different scheduling rules for different service classes since it has been observed that different algorithms tend to perform well for different classes. In [11], UGS was allocated fixed bandwidth, Earliest Deadline First was applied to rtPS, Weighted Fair Queuing to nrtPS and Round-robin to BE services. Other similar ideas are reported in [6], [12], [13]. In this paper, the performance of three hierarchical schedulers based on combinations of above scheduling algorithms has been studied.
B. Modified Longest Weighted Delay First (MLWDF) The MLWDF was designed to meet QoS requirements and has been proved to be throughput optimal in [5]. A scheduling rule is termed throughput optimal if it satisfies the rule that it renders the queues to be stable if any other rule can do so. Suppose γi > 0 is a constant and Wi (t) is the waiting time of the packet at the head of the line. Let j be the selected user, µi (t) MLWDF rule is given by: j = arg max γi Wi (t). µi i
(a) EXPQ
(b) EXPW
(c) Ideal allocation
C. Exponential Rule algorithms: EXPW and EXPQ It is important to ensure that a proper trade-off is maintained between fairness, throughput-optimality and delay requirements for a scheduling algorithm to be truly efficient. The Exponential Rule (EXP) algorithm is designed to achieve this goal [2]. The intuition is to increase the weight of the far away users on slight improvement in their channel conditions while keeping the weight of the users near by proportionately less since they would get very good data rates due to their proximity. It has been proved to be throughput optimal and to satisfy QoS requirements [2]. Let γi > 0 and ai > 0 be constants and Wi (t) be the waiting time of the packet at the head of the line of user i. Let j be the selected user, exponential rule for waiting time (EXPW) is given by: ai Wi (t) − aW √ j = arg max γi µi (t) exp i 1 + aW X 1 where aW = ai Wi (t) N i When the differences in the waiting times are very small, a small change in the waiting time is reflected as a large change in the exponent value due to which, the rule behaves like the PF algorithm. Users can also be prioritized based on their respective queue length values. Let Qi (t) be the queue-length of the user i. Let j be the selected user, exponential rule for queue length (EXPQ) is given
Fig. 1.
Frame allocation example.
III. P ROPOSED S CHEDULING RULES This section presents motivation for designing new scheduling rules and the details of the proposed scheduling rules. Consider a mobile WiMAX network with one BS and four SSs. Suppose that the available spectrum is subdivided into two OFDM sub-channels, C1 and C2 and there are 5 time slots in a scheduling frame. Let r(SSj ) be the rate between SS j and BS over a given sub-channel. Let the varying channel conditions for each user be represented by the following different rates in (bytes/time-slot) r(SS1 ) = 35, r(SS2 ) = 40, r(SS3 ) = 28 and r(SS4 ) = 28. Let node SS1 run a UGS connection which requires 210 bytes to be sent in each frame. Node SS2 runs a FTP session which can send up to 160 bytes of data per frame with a minimum throughput requirement of at least one slot per frame. Nodes SS3 and SS4 each have rtPS connections which require 28 bytes to be sent in every frame. Assume that each user has data to transmit in every frame. When we apply EXPQ algorithm to this scenario, priority is given to every user on the basis of the amount of data present in the user queue and thus, the user allocation is done as shown in Fig. 1(a).
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Observe that the users SS3 and SS4 are not scheduled though they are running real-time applications with strict delay requirements. Suppose EXPW algorithm is applied to the same scenario, since waiting time for every user is same, each user gets an equal share of the resources and the users are scheduled as shown in Fig. 1(b). Nodes SS3 and SS4 are assigned one tile each and the remaining 8 slots are divided equally between SS1 and SS2 . We see that the UGS connection which has strict delay and minimum throughput requirement has not been allocated adequate resources. The ideal allocation for the above scenario would involve a certain level of trade off between queue length and waiting time as shown in Fig. 1(c). The UGS connection is given the highest priority and 6 slots are assigned to it. Next, the rtPS connections with strict delay and bandwidth requirements are assigned one slot each. The remaining slots are allocated to the nrtPS application. As seen above, there is a need to assign priorities to the users based on their queue length and waiting time. A variant of the EXP rule which tries to achieve this objective has been proposed in the paper. A. Proposed EXPQW rule In EXPW algorithm, the difference between the instantaneous waiting time of the head-of-line packet of every user and the average head-of-line packet waiting times of all the users are measured and normalized by some constant value. As long as the differences are small, the EXPW works like PF and assigns similar weights to all the users. If the difference for one user is slightly more, the exponential function magnifies this value and assigns a higher priority to this user. Similarly, in EXPQ algorithm, the differences between the queue-lengths of the users are highlighted. For a given traffic type with fixed throughput and delay values, it has to be decided for every user whether the head-of-line packet has been waiting for a long time or the user has been generating more packets when compared to the other users. In order to compare these two metrics with different dimensions, they have to be mapped onto a common scale using an appropriate mapping function. Let xi (t), represent the waitingtime or queue-length of the user i at time t; xi be the average waiting-time or queue-length of user i calculated during the creation of every UL-MAP. Consider the mapping function xi (t) . For every user having connections f (x) where f (x) = xi µi (t) of the same type, calculate : Wi = γi exp(f (wi ) − 1) µi µi (t) and Qi = γi exp(f (qi ) − 1). Here, f (wi ) and f (qi ) µi µi (t) represent the f (xi ) values for user i. The ratio ensures µi that the channel conditions do not drastically affect the priority given to the user. For every i, Pi = max(Wi , Qi ). Once all the Pi values are computed, the user to be scheduled, j is given by: j = arg max Pi . i
B. Hierarchical schedulers In this paper, the performance of three hierarchical schedulers has been studied. UGS and ertPS traffic types are allocated fixed bandwidth. EXPW / MLWDF algorithms are applied to rtPS connections since they prioritize users based on their waiting times. EXPQ / PF / EXPQW are applied to non-real time applications such as nrtPS and BE. PF favours users with good channel conditions where as EXPQ favours users with heavily loaded queues. EXPQW assigns users based on queue-length and waiting time. The combination of EXPW for rtPS, EXPQ for nrtPS and BE has been referred to as, ‘H1’; MLWDF for rtPS, EXPQW for nrtPS and BE as ‘H2’; MLWDF for rtPS, PF for nrtPS and BE as ‘H3’. IV. P ERFORMANCE E VALUATION This section presents the performance study, based on implementation of the algorithms described in OPNET Modeler 16. The same scheduler is used for both uplink (UL) and downlink (DL) scheduling. The frame has been divided equally into the UL and DL subframes. Two different traffic scenarios are considered: the baseline configuration specified in [14] and a saturated scenario. [14] specifies fixed operational values for the BS and SS, the nature of application traffic, mobility of the mobile stations and their channel conditions. The metrics studied are: system load, system throughput, Jain’s fairness index, delay of real-time applications and delay variation. System Load is defined as the total traffic sent to the WiMAX MAC layer from the higher layers of the BS and SS. System throughput refers to the total traffic sent to the higher layers from the WiMAX MAC layers. End-to-end application level delay is measured at every SS and the average delay of all the SS are compared. The common simulation parameters for baseline and saturated frame scenarios as specified by [14] are : OFDMA profile: 20 MHz; Number of sub-carriers: 2048; Site-to-site Distance: 1.5 Km; BS Tx Power: 46 dBm; MS Tx Power: 23 dBm; Frame duration: 5 ms; Mobility: 3 km/hr - 60% of users, 30 km/hr - 30%, 120 km/hr - 10%; and PDU size: 1500 bytes. The baseline and saturated frame scenarios differ only in the number of users and the traffic mix. In the saturated frame scenario, the same set up has been run for both free space environment as well as ITU Vehicular A channel model. The purpose of running the experiment in a free space environment is to study the system performance without interference and path loss. The following two sections describe the two scenarios in more detail. In several cases, the obtained values have been mentioned without any graphical representation mainly due to space constraints. A. Baseline Scenario The baseline configuration as specified by [14], provides a standard topology and configuration for the comparison of new scheduling rules. The WiMAX cell is divided into three sectors and SSs are randomly distributed in each of these sectors. The traffic mix for baseline scenarios is as mentioned in Table I.
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TABLE I T RAFFIC M IX FOR BASELINE AND S ATURATED F RAME S CENARIOS Appln.
Type of service
VoIP FTP HTTP NRTV Gaming
UGS nrtPS BE rtPS rtPS 2.1
% of users of users Base. Sat. 30% 21.7% 10% 48.3% 20% 48.3% 20% 25.0% 20% 11.7%
Max. Allowed Traffic(Kbps) Base. Sat. 32 64 256 384 256 384 128 128 128 128
Min. Reserved Traffic(Kbps) Base. Sat. 8 8 32 32 32 32 32 32 32 32
Load Throughput
System Throughput (in Mbps)
2.05
2
B. Saturated Frame Scenario
1.95
1.9
1.85 EXPQ EXPQW EXPW
H1
H2
H3
MLWDF
PF
H2
H3
MLWDF
PF
(a) 30 users 4.85
Load Throughput
4.8 4.75 System Throughput (in Mbps)
users is seen to be around 4.8 Mbps. The difference in offered loads among the algorithms is due to application level data generation variability. The maximum tolerated delay of rtPS applications such as NRTV, Gaming and video conferencing is 150 to 200 ms, beyond which the quality of the application suffers heavily. For the baseline scenario with 60 users, the rtPS delay values range from 27.4 ms to 28 ms for all the algorithms. The mean opinion score (MOS) values of the VoIP call for the algorithms calculated by OPNET using R-values were measured to be approximately 3.49 and 3.46 for 30 and 60 user scenarios respectively, for all the algorithms. Fairness was also measured but not reported here since the algorithms had approximately the same fairness measure. In summary, there was no clear distinction among the algorithms with respect to all the metrics in this scenario since it is moderately loaded.
4.7 4.65 4.6 4.55 4.5 4.45 4.4 EXPQ EXPQW EXPW
H1
(b) 60 users Fig. 2. Mean Offered Load and Mean System Throughput for the Baseline Scenario.
The document also specifies the application parameters for FTP, HTTP, VoIP, gaming and NRTV (Near Real Time Video). The multipath channel model applied is ITU Vehicular A and vehicular environment is the path loss model. The maximum achievable load for the given setup are 4.224 Mbps and 9.08 Mbps for 30 and 60 users scenarios respectively. The minimum achievable load for the same scenarios are 0.74 Mbps and 1.532 Mbps respectively. Fig. 2 presents the results for the baseline scenario with 30 and 60 users in a cell. In all cases, the system throughput closely matches the offered load expected, since the system is not saturated. The maximum load (and throughput) with 60
The performance of the algorithms is next studied for a saturated frame scenario. In this scenario, the UL frames are loaded to 95 - 98% and DL data frames are loaded to 70 73% of their total capacities. The simulation parameters for the saturated frame scenario are as earlier, with the following changes: Site-to-site distance is 2 Km; MS Tx power is 30 dBm; 10% of the users have speeds of 80 Km/hr instead of 120 Km/hr. The delay and the fairness graphs have been plotted by calculating the 95% confidence interval for 10 different random seed values. The traffic mix of the saturated frame applications is presented in Table I. Note that 6% of users in both the scenarios run FTP, HTTP and UGS connections simultaneously. 1) Saturated Frame Scenario with free space channel model: The saturated scenario with free space channel model and no mobility has been run for 60 users and 120 users. System load and throughput results are presented in Fig. 3(a)(b). It is seen that there is a noticeable difference between the load and throughput with 120 users, due to buffer overflow. H2 and H3 algorithms provide much higher priority to video conferencing application than the other algorithms. Since the UL and DL subframes are equal, the DL frame gets filled up mostly by real-time applications such as UGS, ertPS, rtPS and MAP leaving very less number of slots for nonreal time bidirectional applications such as FTP. Due to lack of resources, less FTP traffic gets sent and received which accounts for the drop in load and throughput. From Fig. 3(c) that presents the 95th percentile of delay, it is seen that H1, H2, H3, PF and MLWDF algorithms satisfy the delay constraint with 120 users. The mean rtPS delay values with 120 users were 1,180 ms for EXPQ, 592 ms (EXPQW), 161 ms (EXPW), 114 ms (H1), 92-ms (H2), 52 ms (H3), 145 ms (MLWDF) and 159 ms (PF). The delay with H1, H2 and H3 are thus observed to be lower than that of MLWDF and PF. The fairness values in both the scenarios is nearly 1 for all the service classes. 2) Saturated Frame Scenario with vehicular channel model: The results for saturated frame scenario with 60 mobile users,
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16.7
8.2 Load Throughput
8.1 8 System Throughput (in Mbps)
System Throughput (in Mbps)
16.6
Load Throughput
16.5
16.4
16.3
7.9 7.8 7.7 7.6 7.5 7.4
16.2 7.3 7.2
16.1 EXPQ EXPQW EXPW
H1
H2
H3
MLWDF
EXPQ EXPQW EXPW
PF
H2
H3
MLWDF
PF
(a) Mean Offered Load and System Throughput
(a) 60 users 900
Load Throughput
22
H1
800 700 Delay (in milliseconds)
System Throughput (in Mbps)
20
18
16
14
600 500 400 300
12 200 10
100
8
0 EXPQ EXPQW EXPW
H1
H2
H3
MLWDF
PF
EXPQ EXPQW EXPW
(b)
1400
70
1200
60 Packet Delay Variation (in milliseconds)
Delay (in milliseconds)
(b) 120 users
1000
800
600
400
200
95th
H1
H2
H3
MLWDF
PF
Percentile of Delay of RTPS users
50
40
30
20
10
0 0 EXPQ EXPQW EXPW
(c)
95th
H1
H2
H3
MLWDF
PF
EXPQ EXPQW EXPW
Percentile of Delay of RTPS users (120 users)
H1
H2
H3
MLWDF
PF
(c) Packet Delay Variation of RTPS users
Fig. 3. Performance under Free Space Saturated Frame Scenario with 120 users and no user mobility.
Fig. 4. Performance under Saturated Frame Scenario with 60 users and vehicular mobility.
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ITU Vehicular A multipath channel model and vehicular environment path loss model has been presented in this section. There is a single service pool maintained for all polling service applications, rtPS and nrtPS, in OPNET. In case of EXPW and MLWDF, when the delay of nrtPS queues become larger than that of the rtPS queues, the delay of rtPS applications increases. If the channel conditions of users running nrtPS applications are better than that of the users running rtPS applications, they will have a higher priority in PF. The queue length of nrtPS applications is generally much larger than that of the rtPS applications. Hence, nrtPS queues are given a higher priority through EXPQ. H1, H2 and H3 assign metrics specifically to rtPS queues first and then to nrtPS queues; thereby reducing the delay and delay variation in rtPS applications. The reduced rtPS delays are reflected in the increased nrtPS upload and download response delays. In the 60 user saturated frame scenario with mobility, the FTP download and upload response times were observed to be, 16.35s and 15.74s for PF, 17.69s and 16.73s for MLWDF, 17.89s and 15.98s for EXPQW, 18.34s and 16.02s for EXPW, 17.14s and 15.28s for EXPQ, 19.16s and 17.34s for H1, 29.19s and 24.96s for H2, 22.15s and 25.54s for H3 respectively. H2 and H3 have high FTP upload and download times when compared to all the others, due to the priority given to the rtPS requests. Fig. 4 presents the performance under saturated scenario with 60 users and vehicular mobility. EXPQ, EXPW and EXPQW provide much better throughput than the PF algorithm, but their delay exceeds the 200 ms upper bound of video conferencing applications. For VoIP packets, the delay of 100 ms is met for all the users in all the scheduling schemes. H1, EXPW, MLWDF and PF algorithms on the other hand, provide comparable throughput while strictly meeting the delay requirement as seen in the Fig. 4(b). The average delay for EXPQ is 605 ms, EXPQW is 433ms, EXPW is 219ms, H1 is 177ms, H2 is 39.6ms, H3 is 38ms, MLWDF is 183ms and PF is 125ms. Thus, H2 and H3 are seen to have the least overall delay. At the same time, the delay variation of the hierarchical algorithms are much less than MLWDF as seen in Fig. 4(c). The Jain’s Fairness index for BE,rtPS and nrtPS service classes are presented in Table II. The MOS values were recorded to be 3.346 for the 60-user saturated frame scenario with mobility and varying channel conditions and 3.347 for the saturated frame scenarios in free space environment across all the algorithms. In summary, it is difficult to state that one algorithm outperforms all others. However, H1, H2 and H3 can achieve lower delay variation with a slight trade-off in throughput. V. C ONCLUSIONS In this paper, some common scheduling algorithms used in mobile broadband wireless networks have been evaluated and analyzed in the context of IEEE 802.16e standard, using OPNET Modeler 16. A new scheduling rule (EXPQW) has been proposed, that weighs queue-length and waiting-time
TABLE II FAIRNESS I NDEX FOR S ATURATED F RAME S CENARIO WITH 60 USERS AND MOBILITY
Type rtPS nrtPS BE
PF 0.97 0.97 0.98
(a) MLWDF 0.96 0.97 0.98
EXPQ 0.95 0.97 0.98
EXPW 0.96 0.98 0.98
(b) Type rtPS nrtPS BE
EXPQW 0.95 0.97 0.98
H1 0.96 0.97 0.97
H2 0.98 0.93 0.97
H3 0.98 0.94 0.99
of the users before assigning a metric and three hierarchical schedulers which are combinations of the exponential rules EXPW, EXPQ, MLWDF, EXPQW and PF. The proposed EXPQW rule provides system throughput comparable with PF in moderately loaded and heavily loaded scenarios. The hierarchical schedulers perform well for delay sensitive applications like video streaming, while ensuring fairness. They provide lower delay and jitter values when compared to the other algorithms by compromising slightly on system throughput. R EFERENCES [1] “IEEE 802.16e-2005 - IEEE Standard for Local and metropolitan area networks,” Dec. 2005. [2] S. Shakkottai and A. L. Stolyar, “Scheduling for multiple flows sharing a time-varying channel: the exponential rule,” Analytic Methods in Applied Probability, vol. 2, p. 2002, 2000. [3] C. So-In, R. Jain, and A.-K. Tamimi, “Scheduling in IEEE 802.16e Mobile WiMAX networks: Key issues and a survey,” IEEE Journal on Special Areas in Communications, vol. 27, 2009. [4] A. Jalali, R. Padovani, and R. Pankaj, “Data throughput of CDMAHDR a high efficiency-high data rate personal communication wireless system,” in Vehicular Technology Conference Proceedings, vol. 3, 2000, pp. 1854–1858. [5] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, R. Vijayakumar, and P. Whiting, “CDMA QoS scheduling on the forward link with variable channel conditions,” Bell Labs Technical Memo, Apr 2000. [6] C. Msadaa, Ikbal, Camara, Daniel, Filali, and Fethi, “Scheduling and CAC in IEEE 802.16 fixed BWNs: a comprehensive survey and taxonomy,” in IEEE Communications Surveys and Tutorials, 2010, pp. 1991– 1996. [7] OPNET Modeler 16. [Online]. Available: http://www.opnet.com [8] F. P. Kelly, A. K. Maulloo, and D. K. H. Tan, “Rate control for communication networks: Shadow prices, proportional fairness and stability,” The Journal of the Operational Research Society, vol. 49, no. 3, pp. 237–252, Mar. 1998. [9] H. Kushner and P. Whiting, “Convergence of proportional-fair sharing algorithms under general conditions,” IEEE Transactions on Wireless Communications, vol. 3, pp. 1250–1259, 2004. [10] H. Wang and V. B. Iversen, “Hierarchical Downlink Resource Management Framework for OFDMA Based WiMAX Systems,” in IEEE WCNC Proceedings, 2008, pp. 1709–1715. [11] K. Wongthavarawat and A. Ganz, “Packet scheduling for QoS support in IEEE 802.16 broadband wireless access systems,” International Journal of Communication Systems, vol. 16, pp. 81–96, 2003. [12] L.-F. Chan, H.-L. Chao, and Z.-T. Chou, “Two-tier scheduling algorithm for uplink transmissions in IEEE 802.16 Broadband Wireless Access systems,” in International WiCOM Proceedings, 2006, pp. 1–4. [13] J.-Y. Hwang and Y. Han, “An adaptive traffic allocation scheduling for mobile wimax,” in IEEE PIMRC Proceedings, 2007, pp. 1–5. [14] “IEEE 802.16m Evaluation Methodology Document: Evaluation Methodology for P802.16m-Advanced Air Interface,” Nov. 2008.
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