Elsevier Editorial System(tm) for Computer Networks Manuscript Draft Manuscript Number: Title: Greedy Scheduling Algorithm (GSA) - Design and Evaluation of an Efficient and Flexible WiMAX OFDMA Scheduling Solution Article Type: Regular Paper Keywords: WiMAX; OFDMA; QoS; Scheduler; DL-MAP Corresponding Author: Mr. Anatolij Zubow, Ph.D Corresponding Author's Institution: Humboldt University Berlin First Author: Anatolij Zubow, Ph.D Order of Authors: Anatolij Zubow, Ph.D; Daniel Camps-Mur; Xavier Perez-Costa, Ph.D; Paolo Favaro Abstract: WiMAX is one of the most promising technologies to provide broadband wireless access in the near future. In this paper we focus on the study of the combined performance of a WiMAX Base Station MAC downlink scheduler and OFDMA packing algorithm which mainly determine the usage efficiency of the available radio resources. We design and analyze an efficient and flexible solution, Greedy Scheduling Algorithm (GSA), and evaluate its performance as compared to several relevant alternative solutions. Specifically, we analyze their performance differences with respect to efficiency, flexibility to provide per subscriber station burst shape preferences, interference mitigation and computational load. Our results show that GSA achieves a performance close, even slightly superior, to the competing approaches considered in terms of efficiency, but significantly outperforms them in flexibility to provide per subscriber station burst shape preferences, interference mitigation and computational load. As a conclusion, the proposed GSA solution is a promising candidate to maximize the utilization of the available WiMAX radio resources at a low computational cost while at the same time being able to fulfill a wide range of requirements based on operators' preferences and/or network environment specifics.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Greedy Scheduling Algorithm (GSA)Design and Evaluation of an Efficient and Flexible WiMAX OFDMA Scheduling Solution

Anatolij Zubowa, Daniel Camps Murb, Xavier Perez Costab, Paolo Favaroc,

aHumboldt Universitat zu Berlin, Unter den Linden 6, Berlin, GermanybNEC Laboratories Europe, Network Research Division, Kurfursten-Anlage 36, Heidelberg, Germany

cVodafone Omnitel N.V., Via Jervis 13, 10015 - Ivrea (TO) Italy

Abstract

WiMAX is one of the most promising technologies to provide broadband wireless access in the near future. In this paper we focuson the study of the combined performance of a WiMAX Base Station MAC downlink scheduler and OFDMA packing algorithmwhich mainly determine the usage efficiency of the available radio resources. We design and analyze an efficient and flexiblesolution, Greedy Scheduling Algorithm (GSA), and evaluate its performance as compared to several relevant alternative solutions.Specifically, we analyze their performance differences with respect to efficiency, flexibility to provide per subscriber station burstshape preferences, interference mitigation and computational load. Our results show that GSA achieves a performance close, evenslightly superior, to the competing approaches considered in terms of efficiency, but significantly outperforms them in flexibilityto provide per subscriber station burst shape preferences, interference mitigation and computational load. As a conclusion, theproposed GSA solution is a promising candidate to maximize the utilization of the available WiMAX radio resources at a lowcomputational cost while at the same time being able to fulfill a wide range of requirements based on operators’ preferences and/ornetwork environment specifics.

Key words: WiMAX, OFDMA, QoS, Scheduler, DL-MAP

1. Introduction

The essential role played nowadays by the Internet has setthe competition among operators off in order to deliver the de-sired always-connected mobile support with the largest band-width at the lowest possible cost. In this arena, a plethora ofwireless access technologies have been developed being Wi-MAX one of the latest and most promising ones. Indeed, theIEEE 802.16 family of standards, which started with 802.16-2004 [1] followed by 802.16e-2005 [2] and recently by 802.16-2009 [3], have been developed in order to provide enhancedthroughput rates and coverage with respect to the current per-formance offered by well-established technologies such as HighSpeed x Packet Access (HSxPA) and Wireless LAN. An inter-national organization, namely the WiMAX Forum [4], guaran-tees the interoperability between products of different vendorsby defining certification programs.

The main features advocating in favor of the adoption ofWiMAX are mainly coming from the the advanced PHY layertechnology it builds upon. Scalable OFDMA (S-OFDMA) andAdvanced Antenna System (AAS) techniques such as MIMO orBeamforming ensure a higher degree of flexibility and allow atthe same time to fully exploit the characteristics of the wirelesschannel.

Email addresses: zubow@informatik.hu-berlin.de (AnatolijZubow), Daniel.CampsMur@nw.neclab.eu (Daniel Camps Mur),Xavier.Perez-Costa@nw.neclab.eu (Xavier Perez Costa),Paolo.Favaro@vodafone.com (Paolo Favaro)

OFDMA is a multiple access scheme stemming from Or-thogonal Frequency Division Multiplexing (OFDM), which al-lows to assign different subcarriers to different users in orderto implement advanced radio resource management algorithms.In WiMAX though subcarriers are not directly assigned to in-dividual users but are first grouped into basic radio resourceunits called subchannels, which are then scheduled among thedifferent users by the Base Station. The permutation schemeis the algorithm that defines how the physical subcarriers aremapped to the logical subchannels. Two basic types of permu-tation schemes have been defined in WiMAX that are suitedfor different types of environments. Distributed permutationschemes, like PUSC or FUSC [3], spread the subcarriers con-tained in a logical subchannel across the available spectrum,hence reducing fading effects by exploiting frequency diversity.Distributed permutation schemes are suited for mobile environ-ments. Instead, adjacent permutation schemes, like Band-AMC[3], map physically adjacent subcarriers into the same subchan-nels, allowing to efficiently exploit multiuser diversity at thecost of an increased feedback from the stations. Adjacent per-mutation schemes are suited for environments with low mobil-ity. Our work in this paper focuses on the distributed permuta-tion schemes used in WiMAX and in general in OFDMA.

Additionally, a key element that further enables differen-tiation among vendors when competing for their share of theWiMAX market is their ability to handle QoS. In WiMAX, asillustrated in Figure 1, uplink and downlink transmissions are

Preprint submitted to Computer Networks August 10, 2009

Main Latex documentClick here to view linked References

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

multiplexed in a TDD manner. Thus, the problem of assign-ing resources between different connections in order to provideQoS guarantees should be seen as a two dimensional problem,because both frequency and time are resources that can be al-located. The QoS and radio management algorithms imple-mented in WiMAX Base Stations will determine up to whatextent the agreed QoS is honored and the underpinning radioresources are efficiently used.

Our work in this paper focuses on the intricacies relatedwith the design and the analysis of an efficient and flexibleWiMAX OFDMA downlink scheduler algorithm, hereafter re-ferred to as Greedy Scheduling Algorithm (GSA). A uniquecharacteristic of GSA in front of other solutions in the stateof the art is that GSA has been designed with the aim of si-multaneously addressing a wide range of performance aspectslike efficiency, flexibility to provide per subscriber station burstshape preferences, interference mitigation and reduced com-putational load. Instead, previous work in the state of the artdesigned OFDMA downlink scheduling algorithms tailored tosolve a particular performance aspect. For instance, [5] and [6]presented downlink scheduling algorithms designed to maxi-mize efficiency, however a price was paid in terms of otherperformance aspects like flexible burst orientations or reducedcomputational complexity. [7] proposed an OFDMA downlinkscheduling algorithm aiming specifically at reducing the powerconsumption of the associated subscriber stations. In our ownprevious work [8] we proposed an OFDMA downlink sched-uler that balanced efficiency with execution time. The workpresented in this paper goes one step further and proposes anOFDMA scheduling solution which is able to efficiently con-sider multiple requirements at the same time. To the best of theauthors’ knowledge there is no solution in the state of the artable to simultaneously address as many performance aspects asour proposal, GSA, does.

The rest of the paper is structured as follows. In Section 2the major challenges to face while designing a WiMAX OFDMAdownlink scheduling algorithm are introduced. Our GSA sched-uling algorithm is then presented and analyzed in Section 3. InSection 4 the main algorithms against which GSA is comparedare described and a thorough performance evaluation is carriedout. The main results of our comparison are then summarizedin Section 5 that concludes the paper.

2. Challenges in the design of a WiMAX OFDMA downlinkscheduler

Figure 1 depicts the structure of the WiMAX TDD frame,which is composed of a downlink and an uplink subframe. Thesesubframes contain an integer number of slots, which is the ba-sic radio resource unit1 that can be used by the scheduling en-tity. The downlink subframe appears in the first position andcontains in the Preamble and the FCH fields the informationrequired by new stations to discover and join the network. The

1In PUSC one slot is composed of one subchannel and two OFDMA sym-bols.

FCH

Pre

am

be

l DL

-MA

P

FCH

Pre

am

be

l

DL

-MA

P

DL

Bu

rst #

6

DL Burst #1

DL Burst #2

DL Burst #3

DL Burst #4

DL Burst #5

DL Burst #6

Ranging Subchannel

UL Burst #1

UL Burst #2

UL Burst #3

UL Burst #4

OFDMA Symbol Number

Lo

gic

al S

ub

ch

an

ne

l In

de

x

Downlink Uplink

OFDMA Frame

Only Horizontal Mapping2D Mapping

UL

-MA

P

UL

-MA

P

Figure 1: WiMAX OFDMA frame.

downlink (DL) and uplink (UL) MAP messages are then used inthe downlink subframe to signal the position in the frame of thedata transmitted to and from the different stations. A Downlink-MAP (DL-MAP) Scheduler is the algorithm or logical entity re-siding in a WiMAX Base Station (BS) that decides how dataaddressed to its associated Subscriber Stations (SS) has to bemapped in the two-dimensional (2D) downlink OFDMA sub-frame. In the rest of the paper we refer to the downlink OFDMAscheduler in WiMAX as the DL-MAP scheduler.

In order to foster product differentiation the IEEE 802.16standard [1, 2, 3] did not specify any particular algorithm toperform this functionality. However, the standard did spec-ify that downlink data addressed to a particular SS should beencoded in 2D containers known as bursts. Interestingly, this2D mapping of data in the downlink direction differs from themapping algorithm employed in the uplink direction, which isindeed specified by the standard, and consists of allocating up-link transmissions in a one-dimensional (1D) raster in the time(horizontal) direction. The reasons for the different treatmentbetween uplink and downlink transmissions will be further dis-cussed in this section and, as it will be shown, constitute thecore of the challenges associated with the design of an efficientDL-MAP scheduler.

Designing a DL-MAP scheduler is a challenging task in thefirst place because the functionality associated with it, as pre-viously stated, is not specified in the standard. Therefore, aDL-MAP scheduler could be a complex intelligence targetinga multi-dimensional optimization of different aspects like per-flow QoS guarantees, fairness, or global utility functions, orcould as well be a much simpler algorithm which only tries tofind available space in a 2D OFDMA frame when requested todo so. Therefore, before stating the challenges associated withthe design of a DL-MAP scheduler, we need to define the ar-chitecture where the DL-MAP scheduler will be sitting in.

2.1. Online and Offline architecturesIn this paper we will introduce two different architectures

for the design of a WiMAX BS which to the best of our know-

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

QoS Scheduler

DL-MAP SchedulerChannel

Monitor

MCSi

MPDUi

Online:

MPDUi

Offline:

MPDU1,...,MPDUN

u1,...,uN

WiMAX DL

Frame Layout

Figure 2: Online and Offline WiMAX BS architectures.

ledge are representative of the common practice in the state ofthe art. These two architectures are depicted in Figure 2 andwill be hereafter referred to as the Online and Offline architec-tures.

Both architectures share three major logical entities: i) theQoS scheduler, ii) the DL-MAP Scheduler and iii) the Chan-nel Monitor. A high-level description of the functionality as-sociated with each of these entities is reported in the following.The main task of the QoS Scheduler is to appropriately schedulethe MSDUs belonging to different flows in order to fulfill eachflow’s QoS requirements as well as maintaining a certain notionof fairness between flows. Scheduling algorithms like DRR [9]or Proportional Fair [10] have been traditionally proposed toimplement this function. Once the QoS scheduler decides onthe MSDUs to be transmitted in the next OFDMA frame, itappends the correspondent MAC headers to each MSDU andhands over the correspondent MPDUs to the DL-MAP sched-uler. The DL-MAP scheduler is the entity in charge of decidingthe shape and position of the bursts where the different MP-DUs delivered by the QoS scheduler will be encoded within theOFDMA frame. It is in the interface between the QoS sched-uler and the DL-MAP scheduler where the Online and Offlinearchitectures differ.

In the Online architecture the QoS scheduler conveys to theDL-MAP scheduler one MPDU at a time. Thus, every time theDL-MAP scheduler receives a new MPDU from the QoS sched-uler it tries to determine the most efficient shape and placementfor the burst containing the delivered MPDU. Whenever there isno more space available to pack the next MPDU, the DL-MAPscheduler signals back to the QoS scheduler that the frame hasbeen filled up.

In the Offline architecture the QoS scheduler does not de-liver a single MPDU at a time to the DL-MAP scheduler but in-stead it delivers a set P = pi, i = 1..N of MPDUs. It is thenup to the DL-MAP scheduler to generate the optimal OFDMAframe layout out of the MPDUs contained in P . In addition,in order to abstract the different QoS or fairness requirementsfrom the DL-MAP scheduler, the QoS scheduler assigns in theOffline architecture a utility value ui to each MPDU pi. No-tice that while an Offline architecture enables a higher degreeof flexibility and probably a better performance it also involvesa higher complexity.

Finally, in order to perform its task, the DL-MAP schedulerneeds to know the transmission parameters associated with the

SS each MPDU is addressed to. For instance, to compute thesize of the burst required to accommodate a certain MPDU, theDL-MAP scheduler needs to know the Modulation and CodingScheme (MCS) currently being used to transmit data to the SSreceiving this MPDU. In both the Online and the Offline ar-chitectures the Channel Monitor is the logical entity in chargeof providing this input to the DL-MAP scheduler. Among themain parameters that the Channel Monitor could convey to theDL-MAP scheduler are the MCS and power (or multiple com-binations of both) to be used for each SS.

2.2. Major design challenges

Based upon the described architectures, the following majorchallenges have to be taken into account when designing a DL-MAP scheduler:

2.2.1. EfficiencyThe ultimate goal of a DL-MAP scheduler is to utilize the

scarce radio resources as efficiently as possible. It has to benoted though that the mapping of data in 2D bursts makes thistask particularly challenging in the downlink direction. The rea-sons are twofold. First, mapping data inside 2D bursts can re-sult in unavoidable padding. Consider for instance packing anMPDU encoded in a burst of size equivalent to 37 slots in a typ-ical 10MHz PUSC downlink WiMAX frame consisting of 30subchannels and 17 slots per subchannel. No valid 2D shape ofsize 37 slots can be considered in this layout, therefore paddingis unavoidable2. Second, the problem of efficiently packingrectangular bins inside a rectangular container is a well stud-ied problem in computer science which has been proved to beNP-complete [11].

In addition, efficiency can be understood differently depend-ing on whether an Online or Offline architecture is considered.In an Online architecture the DL-MAP scheduler simply triesto fill the downlink subframe as much as possible. Thereforeefficiency could be understood in this context as the percentageof occupancy of the downlink subframe. Instead, in an Offlinearchitecture each MPDU has a different priority represented byits level of utility. Therefore, in this case efficiency could be un-derstood as the aggregate level of utility carried in the WiMAXframe.

It is fair to notice that a 1D mapping of data, like the oneused for the uplink subframe, could possibly outperform in termsof efficiency the best performing 2D DL-MAP schedulers. Thereason is that padding is simply not a problem in 1D packingand empty spaces can be much better minimized. However, a2D packing was chosen in the downlink direction because itoffers a greater flexibility for the BS to address the specific re-quirements of each associated SS 3. These specific requirementswill be discussed in the next subsections.

2For instance the mentioned MPDU could be packed with one slot of pad-ding using a shape of dimensions 19 × 2 (38 slots).

3In the uplink this flexibility was sacrificed by mapping data in the timedirection. The reason is that such a mapping minimizes the bandwidth overwhich battery-limited SSs spread their transmission power, hence increasingrange, which was considered to be the most critical aspect in uplink.

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

2.2.2. Minimize OverheadRelated to maximizing efficiency there is the challenge of

minimizing overhead. In WiMAX each burst in the OFDMAframe is signalled with an entry in the DL-MAP. This overheadcan be quite significant specially when many small packets aretransmitted in the same frame, e.g. a network deploying VoIPservices.

In order to reduce this overhead the 802.16e standard offersthe possibility of concatenating different physical bursts. Theidea is to reduce overhead by identifying bursts encoded withthe same MCS with a single entry in the DL-MAP.4 However,when concatenation is used, the SSs signalled by the corre-sponding DL-MAP entry have to decode the whole burst worthof data and filter their intended information at the MAC layer.Therefore, as it will be discussed in the following, the concate-nation mechanism may not be desirable in all situations.

2.2.3. Flexible burst orientationAs previously mentioned, the DL-MAP scheduler should

leverage the flexibility provided by a 2D packing in order toaddress the specific needs of each associated SS.

A first example of how this flexibility could be used is thepossibility of achieving battery savings in SSs. Some batteryoperated stations could use the information available in the DL-MAP, transmitted at the beginning of the WiMAX frame, in or-der to improve power consumption by entering into sleep-modeduring the transmission of bursts that are not addressed to them.Hence, from a DL-MAP scheduler perspective, it would be con-venient to shape the data addressed to such stations in verticalbursts in order to maximize their sleeping times. Additionally,concatenation would probably not be a desirable mechanism inthis case since SSs would then be forced to decode additionaldata resulting in an increased power consumption.

Another reason to consider per SS shape preferences is toreduce the error rate. Some studies [12] have shown that in dis-tributed permutation schemes, due to frequency diversity rea-sons, packet error rate can be decreased by shaping bursts in avertical way. On the other hand, fast moving stations may ex-perience a time diversity gain if their intended data is shaped inthe time (horizontal) dimension.

Finally, per SS shape preferences can also be used to in-crease downlink SINR, if smart power allocations are used inthe BS. For instance, consider a single vertically-shaped burstspanning all the subchannels of a downlink OFDMA frame.In this case, assuming a distributed permutation like PUSC orFUSC, the best that a BS can probably do is to equally di-vide its transmitted power budget, Ptx, across all subchannels,Nsch, allocating a power equal to Ptx

Nschto each subchannel,

which ultimately determines the downlink SINR. Instead, if thesame burst is shaped in the horizontal (time) direction spanningM < Nsch subchannels in the frequency direction, a BS couldallocate a power equal to Ptx/M to each of the transmitted sub-channels resulting in a power gain and a corresponding increase

4This entry is a regular DL-MAP entry that, rather than referring to a singleConnection Identifier (CID), contains a list of CIDs for each encoded MPDU.

in downlink SINR of Nsch

M with respect to the previous case5. Inaddition, per SS shape preferences should be supported withoutsacrifizing efficiency.

2.2.4. Reduced InterferenceWiMAX is a cellular technology and as such interference

plays a very important role in the overall performance of thesystem. Distributed permutations like PUSC or FUSC havebuilt in mechanisms for interference mitigation. For instance,different ID Cell6 parameters can be assigned to co-channelBSs in order to randomize (average) the interference betweenbursts transmitted during the same OFDMA symbols.

In this context, a very important tool in order to manageinterference between co-channel BSs in a WiMAX network isthe DL-MAP scheduler itself. Notice that for a certain amountof load present in the network it is the way that the DL-MAPschedulers sitting in co-channel BSs schedule the transmittedbursts within the OFDMA frame which ultimately determinesthe level of interference, i.e. subcarriers being reused during thesame OFDMA symbol in co-channel cells. It is hence importantto design a DL-MAP scheduler considering its potential impacton interference in the network.

2.2.5. Reduced Computational ComplexityLast but not least, another compelling constraint to be ad-

dressed while designing a DL-MAP scheduler is the tight timeof operation. This fact is related to the duration of the WiMAXframe, which is typically 5ms and limited by the maximumspeeds supported by the system7.

Complexity in the DL-MAP scheduler operation could betraded-off by planning several WiMAX frames in advance. How-ever, this approach reduces the flexibility of the DL-MAP sched-uler to adapt to the instantaneous incoming traffic and to chan-nel variations and requires a more tight coupling between theDL-MAP and the QoS schedulers. Therefore, an efficient DL-MAP scheduler should be able to perform its operation as fastas possible, ideally in less than one frame duration.

3. Greedy Scheduling Algorithm (GSA)

In this section we present our Greedy Scheduling Algorithm(GSA) which is designed to tackle the several DL-MAP sched-uler challenges introduced in the previous section. Given themany different aspects to be considered by a DL-MAP sched-uler, attempting to design a solution that simultaneously opti-mizes all those different aspects would probably result in toocomplex algorithms to be utilized in the context of WiMAX.Therefore, we apply a divide and conquer approach and de-sign GSA as an algorithm composed of different modules where

5In order to enable intelligent power allocations the 802.16e standard allowsthe BS to specify a power boost between -12dB and +9dBs with a granularityof 3dBs for each transmitted burst.

6The ID Cell parameter can be understood as the random seed that deter-mines the subchannel to physical subcarrier mapping in distributed permuta-tions like PUSC or FUSC.

7Currently the 802.16e standard targets terminals moving at speed of up to120 Km/h.

4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

GSA

DL-MAP Scheduler

QoS

Scheduler

Packing

Pack Headers (FCH,

MAPs)

Queue

empty ?

Increase MAP +

Handle blocking

bursts

Free-Space

Defragmentation

Burst Shaping + Free-

Space Search

Free-Space

found?

Pack Burst

Reduced

MAP?

Deconcatenate

Decrease DL-MAP

No

Yes

No

No

Yes

Reinsert in Packet

Queue

Frame Transmission

Yes

Utility Assignment

Virtual 2D Dequeuing

Concatenation

Initial DL-MAP

Estimation

Preparation

Concatena-

ted burst ?

Yes

No

Channel

Monitor

Figure 3: Flow-chart that illustrates the processes involved in the GreedyScheduling Algorithm (GSA). The dashed blocks represent functionalities thatwill be enabled during offline packing mode only.

each individual module effectively tries to optimize one of theperformance aspects introduced in Section 2.

Our GSA algorithm is divided in two major stages whichare executed sequentially for every WiMAX frame: i) the BurstPreparation stage and the ii) the Burst Packing stage. Figure 3illustrates the modular architecture of GSA. The Burst Prepa-ration stage is in charge of obtaining from the QoS schedulerthe set of MPDUs to be transmitted in the next WiMAX frameand, by obtaining from the Channel Monitor the MCS neededto transmit each MPDU, present to the Burst Packing stage thebursts to be transmitted in the next frame. We assume in the de-sign of GSA that the BS evenly distributes power between sub-carriers and that only one MCS is considered at a time for eachSS. For each candidate burst to be transmitted the Burst Pack-ing stage will find an appropriate allocation within the WiMAXframe. Our two stage design is similar to the concept of Macroand Micro OFDMA scheduling introduced in [6]. However, theactual algorithms that we have used for our design completelydiffer from those presented in [6]. Next, we introduce in detailthe Burst Preparation and Burst Packing stages of our GSA al-gorithm which has been designed to be able to operate in bothOnline and Offline modes.

3.1. Burst Preparation Stage

The operation of the Burst Preparation stage varies depend-ing on whether GSA operates in an Online or in an Offline ar-chitecture. In an Online architecture the Burst Preparation stagesimply retrieves one MPDU at a time from the QoS Scheduler,and after obtaining the required MCS from the Channel Moni-tor, presents a burst to be packed to the Burst Packing stage. Onthe other hand, in an Offline architecture the Burst Preparationstage applies a set of algorithms in order to obtain the optimal

set of MPDUs from the QoS Scheduler to be transmitted in thenext WiMAX frame. Specifically, in an Offline architecture theoperation of the Burst Preparation stage is subdivided into thefollowing modules: i) Utility Assignment, ii) Virtual 2D De-queuing and iii) Concatenation.

3.1.1. Utility AssignmentMPDUs delivered by the QoS scheduler to GSA are first as-

signed a utility value according to the utility metric chosen byan operator. Given that operators are in general interested inmaximizing the profit they obtain from users per resource uti-lized, a possible example of how operators could use the con-cept of utility functions would be to assign larger utility valuesto MPDUs for premium users as compared to MPDUs for reg-ular users. Another example of how operators might use util-ity functions would be to consider to give priority to MPDUsfrom clients currently experiencing better channel conditions,i.e. opportunistic scheduling, in order to maximize the systemthroughput. More complex utility policies, combining simul-taneously several performance aspects, could also be transpar-ently supported within this framework. The optimal way to de-fine a utility policy in order to fulfill a certain objective is outof the scope of the work presented in this paper. In Section 4though an example will be provided for illustrative purposes.

3.1.2. Virtual 2D DequeuingThe goal of the Virtual 2D Dequeuing module is to select

from the MPDUs available at the QoS scheduler the ones tobe transmitted in order to maximize the utility contained in theWiMAX frame being scheduled. The Virtual 2D Dequeuingstage performs the following operations which are detailed inAlgorithm 18. Firstly, it creates the list P 1 by retrieving fromthe QoS Scheduler an initial set of MPDUs, obtaining for eachof these MPDUs its utility value and its required MCS. Eachelement in this list is then a burst represented by its utility andthe number of OFDMA slots needed to accomodate it. No spe-cial ordering is assumed in list P 1. The initial list P 1 may havean aggregated size (in slots) above the capacity of the WiMAXframe. Secondarily, the Virtual 2D Dequeuing module createsthe list P 2 by selecting from the list P 1 a subset of bursts thatdo fit in the WiMAX frame and maximize the amount of car-ried utility carried. Thus, performance will heavily depend onthe actual algorithm used to create list P 2.

Although other algorithms could be used, in our design wehave simply considered a greedy knapsack algorithm over listP 1 in order to create list P 2. A greedy knapsack algorithm op-erates in the following way. For each burst in P 1 its utility perslot is computed, i.e. ui

bi, where ui is the utility of burst i in P 1

and bi its size in slots. Then a list P 1′is created containing the

elements of P 1 in decreasing order of utility per slot. Finally,list P 2 is created by considering the first χ elements of P 1, with

8In Algorithm 1, b(pi), h(pi) and p(pi) represent respectively the size inslots, the amount of overhead in slots and the number of padding slots needed topack burst pi. The amount of padding is computed by obtaining the smallest 2Dshape that fits in the WiMAX downlink subframe and has size equal or biggerthan b(pi).

5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

χ as big as possible, such that its aggregated size is below τC ′,where C ′ = C − FCH − ulmap is the capacity of the down-link WiMAX subframe after subtracting the FCH and UL-MAPoverheads and 0 < τ < 1 is a parameter that accounts for thetypical efficiency achieved by the Burst Packing stage9.

Algorithm 1 Virtual 2D Dequeuing. Arrangement of packetsdelivered by the QoS scheduler.

1: procedure VIRTUAL 2D DEQUEUING2: P 1 ← fromQoSScheduler3: P 1′ ← maxUtilPerSlotSort(P 1)4: P 2 ← (p1′

1, · · · , p1′χ), χ = arg max

χ∈[1,|P 1|]

(∑χi=1(b(p1′

i) +

h(p1′i) + p(p1′

i)) ≤ τ · C ′) . Virtual 2D dequeuing5: end procedure

Notice, that the applied greedy heuristic, i.e. giving prefer-ence to bursts carrying more utility per slot, is a very fast way toobtain a candidate set of bursts that result in a notable increasein the amount of utility carried in the WiMAX frame.

3.1.3. ConcatenationIn order to efficiently utilize the scarce radio resources the

Concatenation stage tries to reduce DL-MAP overhead by con-catenating together bursts transmitted with the same MCS. TheConcatenation stage operates over the list P 2 generated by theVirtual 2D Dequeuing stage. Like in the Virtual 2D Dequeuingstage we use for Concatenation a heuristic that tries to greedilyconcatenate bursts transmitted with the same MCS. Our pro-posed heuristic is detailed in Algorithm 2.

Algorithm 2 Packet concatenation algorithm.1: procedure GSAGREEDYCONCATENATE(P 2)2: P 3 ← ∅, v ← ∅3: for i = 1 to |P 2| do4: M ← pi . packets to be concatenated5: for j = i+ 1 to |P 2| do6: if mcs(pi) 6= mcs(pj) ∨ pi ∈ v ∨ pj ∈ v then7: continue8: end if9: if s(M ∪ pj) ≤ max bsz then

10: M ←M ∪ pj, v ← v ∪ pj11: end if12: end for13: P 3 ← [P 3 pconcat(M)] . append to list14: end for15: return P 3

16: end procedure

A key to understand the behavior of Algorithm 2 is the vari-able max bsz which represents the maximum burst size to beconsidered after concatenation. Thus, the proposed algorithm

9To obtain the results presented in this paper we have used τ = 0.95

circulates over the list P 2 in a greedy manner while trying toconcatenate together bursts transmitted with the same MCS aslong as their aggregated size is below max bsz. The parametermax bsz depends on the OFDMA frame size, and in our im-plementation we have empirically set it to 2

3C, which provideda good performance in our experiments, where C is the capac-ity of the WiMAX downlink subframe. After the concatena-tion procedure completes, the list P 3 contains the concatenatedbursts to be delivered to the Burst Packing stage, where a con-catenated burst is represented by a size in slots equal to the sumof the sizes of its component bursts, and a utility correspondingto the addition of the utility of its component bursts.

Notice that the presented concatenation procedure can becompletely or partially disabled by limiting its input to a subsetof the bursts in the list P 2.

3.2. Burst Packing StageAfter the Burst Preparation stage GSA contains a list of

bursts in P 3 that are candidates to be transmitted in the WiMAXframe. Notice that at this stage each burst is simply defined byits size (number of required OFDMA slots). This list of burstsis the input required by the Burst Packing stage to generate thefinal WiMAX downlink frame layout.

The Burst Packing stage operates over the bursts defined inthe list P 3 in a FIFO manner. For each burst in P 3 the BurstPacking stage selects a suitable 2D shape and a position for thisshape within the WiMAX downlink subframe. When there isnot enough space in the WiMAX frame to pack any burst inP 3 or the list P 3 is empty, the Burst Packing stage terminatesand the WiMAX frame layout construction is considered com-pleted. The Burst Packing stage tackles hence the challenge offinding for each candidate burst an appropriate 2D shape to con-tain this burst while filling up as much as possible the downlinkWiMAX subframe. Notice that since the Burst Packing stageoperates over the bursts in P 3 in a FIFO manner it can seam-lessly operate in an Online architecture, where the bursts wouldbe delivered one by one by the QoS Scheduler.

Next, we describe in detail our proposed implementationof the Burst Packing stage. For the sake of clarity we divideour explanation in three subsections: i) the Offline Optimiza-tion module, ii) the Burst Packing stage operation, and iii) anextended explanation of the Core Packing Engine.

3.2.1. Offline OptimizationSince in an Offline architecture the DL-MAP scheduler has

a priori information about all the candidate bursts to be packed,GSA applies two simple heuristics that increase the efficiencyof the Burst Packing engine. The first heuristic consists in re-ordering the bursts contained in the list P 3 by decreasing orderof size (in slots). The reason why this heuristic is effective issimple. Bigger bursts can be packed in an easier way when theWiMAX frame is empty, because when the occupancy in theWiMAX frame increases the fragmentation in the frame alsoincreases making it more difficult to find a feasible allocationfor big bursts. Packing bigger objects first is a commonly usedheuristic in many packing problems. The second heuristic con-sists in appending at the end of P 3 the bursts that were initially

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

discarded by the Burst preparation stage10. The idea behind thisheuristic is to try to use as much as possible empty spaces leftin the WiMAX frame in case some of the initial bursts in P 3

can not be packed. After these heuristics are applied over listP 3, the Burst Packing stage operates in a FIFO manner over thedifferent bursts.

3.2.2. Burst Packing Stage OperationThe Burst Packing stage operates in the following way. For

each burst in P 3 a set of 2D candidate shapes suitable to con-tain this burst is generated and evaluated according to a plural-ity of criteria. Then, from the generated set of shapes the bestperforming shape that can be packed in the WiMAX frame isselected. The Burst Packing stage continues to operate over thebursts in list P 3 until either list P 3 is empty or none of the re-maining bursts can be succesfully packed. Next, we illustratethe operations performed by the Burst Packing stage over eachburst in P 3 which are depicted in Algorithm 3.

Algorithm 3 Burst Packing Stage1: procedure FSSEARCH(p)2: b sp ← ∅, b shape ← ∅, b sz ← 03: for szi = s(p) to s(p) + max padding do4: shapes ← genShapes(szi ,max facts)5: for all shapei ∈ shapes do6: fsi ← search(shapei)7: a← largestBlock(fsi , shapei)8: padding ← (szi − s(p))/s(p)9: a′ ← a · (1− padding)

10: a′′ ← a′ · shapeMetric(shapei)11: if fsi 6= ∅ ∧ a′′ > b sz then12: b sp ← fsi , b shape ← shapei , b sz ← a13: end if14: end for15: end for16: return (b sp, b shape)17: end procedure

The operation of the Burst Packing stage can be easily un-derstood through an example. Consider that a burstB of size 11slots has to be packed in the WiMAX frame. Several 2D shapescould be considered to accomodate this burst, for instance 1×11or 11× 1. In addition if one extra slot of padding would be al-lowed, 3 × 4, 4 × 3, 2 × 6, or 6 × 2 would be other candidate2D shapes. As it can be seen in the example, increasing theamount of allowed padding increases the number of candidate2D shapes.

Once a burst in P 3 has been selected for packing, the BurstPacking stage generates the set shapes of candidate 2D shapesthat can contain this burst, line 4 in Algorithm 3. Each of theseshapes is then weighted according to several factors and thebest performing shape is selected. In Algorithm 3 two vari-ables are defined that bound the amount of cadidate 2D shapes

10Notice that to keep the packing time low, the maximum size of list P 3 canbe bounded to a suitable value.

to be considered for a given burst in P 3. These variables aremax padding and max facts which can be used to trade-offperformance with execution time. As its name indicates, thevariable max padding limits the amount of padding in each2D shape, and together with max facts is used to limit thetotal number of shapes under consideration11.

In order to obtain the most suitable 2D shape contained inthe set shapes each candidate 2D shape is ranked with respectto three different criteria:

i. The packing efficiency of the resulting WiMAX framelayout if the candidate shape would be packed.

ii. The amount of padding of the candidate shape.

iii. The degree of compliance of the candidate shape to theshape preferences defined by the SS that the current burstis addressed to.

In order to implement these different criteria it can be ob-served between lines 6 and 10 in Algorithm 3 that each shapein the set shapes is ranked with a value equal to:

rank(shapei) = a× (1− padding)× shapeMetric(shapei)

Where a is an integer value that measures how efficient wouldthe resulting WiMAX frame layout be if shapei would be pack-ed. Higher values of a indicate higher values of efficiency. TheCore Packing Engine is the module in charge of performing thisevaluation and obtaining the value of a, and will be describedin detail in the next section. The variable padding measuresthe percentage of padding slots in shapei, resulting higher per-centages of padding in lower rank values. Finally, the functionshapeMetric measures the degree of compliance of shapeito the shape preferences of the SS that the current burst is ad-dressed to.

Notice that different shapeMetric functions can be con-sidered for different stations. Hence, by properly defining ashapeMetric function any kind of shape preferences can besupported in GSA. An example of shapeMetric functions thatcould be used to accommodate vertical or horizontal shape pref-erences are the following:

shapeMetricV ert ←height(shapei)width(shapei)

shapeMetricHorz ←width(shapei)height(shapei)

Thus, vertical 2D shapes would be highly ranked in case ofshapeMetricV ert, and horizontal 2D shapes otherwise.

Next, we describe in detail the Core Packing Engine whichresides at the heart of GSA and it is key in order to generateefficient WiMAX frame layouts with minimal complexity.

11The maximum padding is calculated as max padding = min(s(p) ·1.1, s(p) + ψ), where ψ is empirically set to 10 and 20 for 10Mhz and 20Mhzchannels respectively; max facts is empirically set to 10 for 10Mhz channelsand 20 for 20Mhz channels.

7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

free capacity

list (sorted)

OFDMA

frame

allocated

capacity list

1

2

Figure 4: In GSA the free and the allocated capacities are represented by alinked list of rectangles.

3.2.3. Core Packing EngineThe Core Packing Engine is queried by the Burst Packing

stage in order to find a suitable allocation for each candidate2D shape considered for a given burst in the list P 3. Formally,packing rectangles inside a rectangular bin (the WiMAX down-link subframe) is a generalization of the 1D bin-packing classi-cal problem already studied in [13] in 1974. Both the 1D and2D bin packing problems have been proved to be NP-complete.In the case of GSA, we want to design a 2D bin packing algo-rithm which fulfills the following requirements:

1. FIFO processing of bursts to be packed, because we wantto support both Offline and Online architectures.

2. Efficient filling up of the WiMAX downlink subframe.

3. Fast enough to be used within the context of WiMAX.

In order to obtain a fast 2D packer we propose to use, again,a greedy algorithm which operates based on the following prin-ciple. The 2D WiMAX OFDMA downlink packing problem canbe transformed into a 1D searching problem by modeling thefree space in the WiMAX frame as an ordered linked list of rect-angular elements. Figure 4 illustrates this idea. GSA actuallymaintains two linked lists. The free capacity list which is popu-lated by a set of non-overlapping rectangular shapes represent-ing the current unallocated free space in the WiMAX frame,ordered in increasing order of area of the contained rectangles,and an allocated capacity list which is populated by a set ofnon-overlapping rectangular shapes which represent the databursts that have been already allocated in the WiMAX frame,no special ordering is needed on this list. As mentioned before,representing the free space in the frame as a set of non overlap-ping rectangular shapes transforms the packing problem into asimple search. When the Core Packing Engine wants to find asuitable position to allocate a given 2D shape it simply performsa search in the free capacity list until it finds a free space rect-angle where the 2D shape can fit. Notice that maintaining thefree capacity list sorted in increasing area of rectangular shapesminimizes the searching time, because a 2D shape is greedilyassigned to the first free space rectangle that can contain it. In

A

free capacity

list (sorted)

OFDMA

frame

A

allocated

capacity list

1

2

A

free capacity

list (sorted)

OFDMA

frame

A

allocated

capacity list

1'

2'

B

new burst

B

B

placement

Before placement After placement

Figure 5: Free capacity list and allocated capacity list before and after a packingoperation.

addition, notice that a matching rectangle from the free capacitylist will have in general a bigger area than the 2D shape beingconsidered. In this case we pack the 2D shape in the upperright corner of the free space rectangle. The reasons for pre-ferring placements in the upper right corner are detailed laterin this section. Finally, after a 2D shape has been assigned theallocated space is transferred to the allocated capacity list andthe free capacity list is updated to represent the new situation inthe frame. Figure 5 illustrates the procedure of packing a burstin the WiMAX frame.

Figure 5 also illustrates an inherent challenge of our ap-proach. Since the 2D shape to be packed does not have, ingeneral, dimensions that perfectly match those of the free spacerectangles represented in the free capacity list, small rectanglesappear in the free capacity list when a burst is inserted. Noticethat having the free space in the WiMAX frame represented bymany small rectangular shapes is a problematic situation, be-cause the Core Packing Engine can only succesfully pack a 2Dshape when it finds a free rectangle of dimensions bigger thanthe ones of the considered 2D shape. The Core Packing Enginewill function in a better way if the free space available in theWiMAX frame is represented with few bigger non overlappingrectagular shapes. To solve this problem the Core Packing En-gine makes use of a Defragmentation algorithm which is runevery time a 2D shape is allocated.

Next, we describe the Defragmentation algorithm used byGSA. Afterwards, we will focus our attention on other spe-cific WiMAX aspects which are important to the overall per-formance of the Core Packing Engine: i) managing the spaceallocated for the DL-MAP overhead, and ii) several heuristicefficiency optimizations.

DefragmentationAs mentioned before the representation of free space is key

to the efficient operation of the Core Packing Engine. Repre-senting free space with few big rectangular shapes is better thanrepresenting it with many small rectangular shapes. In order tomaintain always a proper representation of the free space in theWiMAX frame, defragmentation over the free capacity list isneeded.

8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Algorithm 4 GSA free space defragmentation algorithm.1: procedure FREESPACEDEFRAG2: repeat3: toBreak ← true4: for all i = 1 to |FS | do5: for all j = 1 to |FS | do6: (fs ′i, fs

′j)← defrag(fsi, fsj)

7: a← max(s(fsi), s(fsj))8: a′ ← max(s(fs ′i), s(fs

′j))

9: if a′ > a then10: fsi ← fs ′i, fsj ← fs ′j , toBreak ← false11: end if12: end for13: end for14: until toBreak15: FS ← sort(FS )16: end procedure

layout 1

layout 2

new burst

BA

B’

A’

If (max(A’,B’) > max(A,B))

→ replace A,B by A’,B’

defragmentation:

Figure 6: Free space defragmentation process in GSA.

Algorithm 4 depicts the defragmentation algorithm that wehave used for GSA. The basic idea is to convert two old freespace rectangles into two newer rectangles if they overlap in atleast one corner and the size of the biggest rectangle increases.Figure 6 illustrates the operation of the defragmentation algo-rithm.

Finally, when the defragmentation process completes, thefree space is again sorted in increasing order of the area of thecontained non overlapping free space rectangles.

DL-MAP ManagementAn aspect that makes the DL-MAP Scheduling problem in

WiMAX different from traditional 2D bin packing problems, isthe fact that the size of the DL-MAP overhead is unknown a-priori. Indeed, the size of the DL-MAP overhead depends onthe number of bursts transmitted in the WiMAX frame, but thenumber of bursts that can be transmitted also depends on thespace occupied by the DL-MAP overhead.

In WiMAX the DL-MAP overhead is not constrained to bea 2D shape, instead it is packed in the downlink subframe inthe vertical direction starting from the end of the FCH field.Notice that DL-MAP scheduler algorithms can enter a deadlocksituation if there is enough space to pack a certain burst butthere is not enough space to increase the DL-MAP in order to

B1

OFDMA frame

new burst

FC

HD

L-M

AP

DL-M

AP

B2

B3B4B5

B6

blocking

rectangle

free space

DL-MAP

cannot be

increasedB1

OFDMA frame

FC

HD

L-M

AP

DL-M

AP

B2

repacked blocking

rectangle

free space

increased

DL-MAP

B6

B7

B3B4B5B7

Before reallocation After reallocation + placement of new burst

Figure 7: Illustration of the burst reallocation operation in GSA.

signal this next burst. Figure 7 (left) illustrates this problem 12.DL-MAP schedulers in the state of the art solve this prob-

lem by pre-allocating an integer number of columns (OFDMAsymbols) in the WiMAX downlink subframe for the DL-MAPoverhead, so as to be reasonably sure that the previous dead-lock situation will never occur. The problem with this approachis that the space pre-allocated for the DL-MAP can in generalnot be used to transmit data bursts and hence the scarce radioresources become underutilized. In GSA we have tried to solvethis problem by using an a priori estimation of the DL-MAPsize, while allowing the Core Packing Engine to pack belowthe estimated DL-MAP area if needed. If the DL-MAP over-head grows above the initial estimation, the Core Packing En-gine tries to reallocate any data burst blocking the growth of theDL-MAP. The reallocation procedure is illustrated in Figure 7and is simply performed through a new search in the free capac-ity list. If reallocation is not possible the Core Packing Engineterminates indicating that the 2D shape could not be success-fully packed.

The heuristic used to estimate the required DL-MAP sizevaries depending on whether GSA operates in an Offline or inan Online architecture. In an Offline architecture we estimatethe original DL-MAP overhead in the following way:

ˆdlmap =⌈F0 + |P 3| · F5 + |P 2| · F6

SC

⌉·R

Where the constants Fi and SC are defined in Table 1, R isthe number of repetitions used for the DL-MAP, and QPSK 1

2is the MCS employed to transmit the DL-MAP. Our estimationhence basically considers that all concatenated bursts in the listP 3 are going to be successfully packed. In case of an Onlinearchitecture GSA does not have a priori information about thenumber of data bursts to be packed in the frame, therefore GSAsimply estimates the initial DL-MAP overhead as an integernumber of columns based on the number of bursts in the list P 3

and the number of DL-MAP repetitions.It is also possible that the original estimation of the DL-

MAP size turns out to be bigger than the actual size needed.

12Although there is enough free space for burst B7, this can not be packedbecause the DL-MAP can not be increased due to the blocking burst B6.

9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Variable ValueF0 fixed dl map overhead (72 bits)F1 harq dl map ie (fixed part) (18 bits)F2 max harq subbursts per ie (16)F3 harq dl map ie (per 16 subbursts) (56 bits)F4 dl map ie harq chase subburst (36 bits)F5 dl map ie fixed overhead (44 bits)F6 dl map ie cid size (16 bits)R map repetitions (2)SC num subcarriers slot (48)

Table 1: Parameter used for the calculation of the DL-MAP size.

In that case GSA incorporates the possibility of shrinking thespace initially allocated to the DL-MAP overhead in order toincrease the space for data bursts. If none of the rectangles inthe free capacity list is large enough to fit a specific 2D shape,the Core Packing Engine algorithm reduces the space reservedfor the DL-MAP overhead to the minimum size needed to sig-nal the bursts already packed in the frame plus the new burst.Then, GSA tries to repack the mentioned burst. If after the DL-MAP has been readjusted the burst still cannot be packed, thealgorithm terminates.

Efficiency Optimization HeuristicsTo finalize the description of the Core Packing Engine in

GSA we introduce three heuristic optimizations that increasethe efficiency of the presented algorithm:

1. Start Packing from the upper right corner of the WiMAXframe. In order to reduce the possibility of blocking thegrowth of the DL-MAP we configure the Core PackingEngine to start packing bursts as far from the DL-MAPas possible, i.e. the upper right corner of the downlinksubframe. Notice that this heuristic is not mandatory, andas we will discuss in the next section GSA can be config-ured to start packing in any part of the WiMAX frame ifthat would result benefitial.

2. Maximize the largest free rectangle. To maximize ef-ficiency it is important to achieve a dense packing ofbursts inside the WiMAX frame. In addition, as previ-ously mentioned, it is important for the performance ofthe Core Packing Engine to maintain in the free capacitylist rectangular shapes as big as possible. Hence a heuris-tic that we use to select an appropriate 2D shape for a cer-tain burst in P 3 is to preferentially select 2D shapes thatresult in big free rectangles in the free capacity list. Thisheuristic is applied by the Burst Packing stage in Algo-rithm 3, where a represents the size of the biggest emptyrectangle in the free capacity list when a given 2D shapeis packed in the WiMAX frame. Hence, bigger values ofa increase the rank of the considered 2D shape.

3. Deconcatenation. If, even after the DL-MAP size hasbeen re-adjusted, the largest free rectangle in the free ca-pacity list is not enough to fit a specific concatenatedburst, the burst is fully deconcatenated and the result-ing deconcatenated bursts are reinserted in the packing

queue. Notice that this heuristic increases efficiency, be-cause decontenated bursts can potentially fill up emptyspaces in the WiMAX frame, but it also results in a higherexecution time. Therefore, if execution time is critical,this heuristic can simply be disabled without affecting theoverall behavior of GSA.

In order to finalize our description, Algorithm 5 depicts thedetailed operation of the Core Packing Engine.

Algorithm 5 GSA 2D packing algorithm.1: procedure GSAPACK(P, dlmap)2: packHeaders(FCH,ULMAP, dlmap)3: reducedDLMap ← false . Marker4: for i = 1 to |P | do5: if ∼ incrDLMap() then . Increase MAP?6: if ∼ repackBlocking() then7: break . Cannot increase MAP; terminate8: end if9: incrDLMap()

10: end if11: freeSpaceDefrag()12: (fs, shape)← fsSearch(pi)13: if fs 6= ∅ then . Free space block found14: pack(fs, shape) . Pack the burst15: reducedDLMap ← false16: else . Not sufficient free space found17: if reducedDLMap then . Already reduced18: deconcatenateAndReeinsert(pi)19: else . Decrease MAP to min. required size20: decrDLMap(), reducedDLMap ← true21: i = i− 1 . Retry packing pi22: end if23: end if24: end for25: decrDLMap() . Adjust MAP to final size26: end procedure

3.3. Interference Mitigation

As explained in Section 2 a major challenge to be tackledby the DL-MAP scheduler is to be able to reduce interference.In this section we describe an extension of GSA which signif-icantly improves its performance in interference limited envi-ronments. We refer to this extension as interference aware GSAor iGSA.

The basic idea behind iGSA is illustrated in Figure 8. Gen-erally WiMAX BSs in a cellular deployment do not operate atfull capacity, i.e. the WiMAX frames are not full all the time. Inthese scenarios there is the possibility of reducing interferencein the system by scheduling transmissions in co-channel BSsin non-overlapping time-frequency regions within the WiMAXframe. The challenge is to implement this scheduling in a dis-tributed way without assuming any communication between co-

10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I II

III

I

III III

II

III

I I

III III

II

I

II

I

II

II

III

II

I

II

(a) 1x3x3 Reuse Pattern (b) 3x1x1 Reuse PatterniGSA packing regions

1 3 2

3 2 1

2 1 3

Figure 8: iGSA configuration for reuse patterns 1x3x3 and 3x1x1. In 1x3x3 we have 3 interferers in the first tier, whereas in 3x1x1 there are 6 interferers in thesecond tier. The roman numerals depict the used frequencies whereas the arabic numerals represent the priority of the packing region within the frame.

channel BSs 13.In order to minimize interference between co-channel BSs

iGSA defines a set of N non-overlapping regions within theWiMAX frame, assigns a preferred iGSA region to each co-channel BS, and configures these BSs to pack with more pri-ority within their assigned iGSA region. This concept is il-lustrated in Figure 8, where N = 3 non-overlapping regionshave been considered. To describe the behavior of iGSA, Fig-ure 8 also illustrates two typical cellular deployments, a three-sectorial deployment and a frequency reuse three deployment.The roman numbers in each sector/cell represent different fre-quencies and the different colours the preferred iGSA regionsof operation. Notice how in the mentioned deployments the in-terfering cell/sectors in the first tier always operate in differentiGSA regions, and hence interference between them can be re-duced. Interferers placed further away can reuse iGSA regionsbut due to the increased distance their mutual interference issmaller. The iGSA region to be assigned to each sector/cellcould be statically configured by an operator in the same waythat frequencies of operation are configured.

From Figure 8 it is clear that iGSA can minimize interfer-ence when the load to be served in each sector/cell completelyfits within one iGSA region, i.e. the offered load is smaller than1N of the nominal sector/cell capacity, whereN is the number ofnon-overlapping iGSA regions. However, when more load hasto be served, iGSA needs to make use of other iGSA regionswithin the WiMAX frame. In this case the different co-channelBSs expand their regions of operation in a way that the numberof co-channel BSs in the first tier simultaneously operating inthe same iGSA region is minimized. This mechanism is alsoillustrated in Figure 8 for N = 3.

iGSA can be easily enabled as an optional module within

13Such communication should be per WiMAX frame hence consuming nonnegligible amounts of bandwidth.

the GSA framework by performing the following modificationsin the Burst Packing Stage:

1. In order to implement the partition of the WiMAX framein iGSA regions, the free capacity list is initialized with anumber of free rectangles corresponding to the differentiGSA regions.

2. When the Burst Packing Stage looks for suitable 2D sha-pes in Algorithm 3, shapes belonging to the preferrediGSA region of operation should be preferred. This isimplemented by multiplying the rank assigned to a given2D shape in Algorithm 3 by a priority value prio(fsi).For instance if N = 3 iGSA regions are considered, apriority equal to 3 is given to the preferred region of op-eration, a priority equal to 2 is given to the iGSA regionused when the load to be served is above 1

3 , and a priorityof 1 is given to the remaining iGSA region.

3. The free space sorting algorithm is adapted so that freespace rectangles are ordered according to their priority indecreasing order. Free space rectangles having the samepriority are then ordered according to their burst size.

4. Finally, the defragmentation algorithm is also modifiedso that only bursts belonging to the same iGSA regioncan be defragmented14.

3.4. Complexity

In order to bound the computational complexity of our GSAalgorithm we provide in this section a worst case analysis of itsexecution time. In this analysis we consider the WiMAX frameto be a two dimensional matrix of T time slots and S OFDMA

14Only when a burst can not be successfully packed the free capacity list iscompletely defragmented without considering the notion of iGSA regions.

11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

subchannels, and we compute separately the worst case excu-tion time of the Burst Preparation and the Burst Packing stagesof GSA:Burst Preparation Stage:

1. Let |P 1| = N1 , where N1 is unbounded2. Sort the elements in P 1 according to their utility per slot

metric O(N21 )

3. Virtual 2D dequeuing O(N1)4. Greedy packet concatenation, where |P 2| = N2 O(N2

2 )5. Sort the elements inP 2 according to their slot sizeO(N2

2 )

Notice that N2, number of bursts to be concatenated, can bebounded by the maximum number of slots in the WiMAX frame,i.e. T · S15. Hence, the complexity of the Burst PreparationStage can be bounded by O(N2

1 +N1 +(T ·S)2 +(T ·S)2) =O(N2

1 ).Burst Packing Stage:

1. Let |P 3| = N3 and X be the number of items in the freecapacity list.

2. For-each element in list P 3 O(N3)(a) Adapt the size of the DL-MAP O(X)(b) De-fragment the free capacity list O(X3)(c) For-each candidate shape size O(C1)

i. For-each 2D factorization O(C2)A. Search in free capacity list O(logX)B. Rank each shape O(X)

(d) Append the selected shape to the allocated capacitylist and update/sort the free capacity list O(X2)

Constants C1 and C2 are bounded by max padding andmax facts respectively. In addition X , the number of free rect-angles in the free capacity list, can be bounded in the followingway. ConsiderN3 input bursts to the Burst Packing Stage, in theworst case every time a packing operation is performed the freecapacity list increases by one element, as illustrated Figure 5,and the defragmentation algorithm can not reduce the numberof rectangles in the list. Hence, if N3 bursts are finally packed,and assuming that initially there were two rectangles in the freecapacity list, the packing process completes with X = N3 + 2free space rectangles in the free capacity list. Furthermore, N3,the number of successfully packed bursts, is bounded by N1,the total number of input bursts. So the size of the free capacitylist is bounded by X < N1 + 2. Summarizing, the complexityof the Burst Packing Stage can be bounded by:

O(N3 · (X +X3 + C1 · C2 · (logX +X) +X2))

= O(N1 · (N1 +N31 + C1 · C2 · (logN1 +N1) +N2

1 ))

= O(N41 +N3

1 + C1 · C2 ·N21 + C1 · C2 ·N1 · log(N1))

= O(N41 )

Finally, considering together the Burst Preparation and BurstPacking stages, the overall worst case execution time of GSAcan be estimated as O(N2

1 ) + O(N41 ) = O(N4

1 ), i.e. polyno-mial time.

15The minimum size of a burst is one slot.

4. Performance Evaluation

The goal of this section is to evaluate whether our designedGSA DL-MAP scheduler is able to fulfill the DL-MAP sched-uler challenges introduced in Section 2. In addition we willcompare GSA with three relevant related approaches in orderto evaluate the potential improvements achieved.

Instead of focusing on a single aspect of the DL-MAP sched-uling problem, we target a global evaluation of the different al-gorithms under study considering those that, in our understand-ing, are the most important goals to be achieved by a DL-MAPscheduler. For this purpose we divide our evaluation in fourdifferent aspects: i) Efficiency, ii) Flexibility, iii) InterferenceManagement and iv) Computational Load.

This evaluation is performed by means of simulations usinga customized WiMAX simulator that implements the WiMAXTDD frame structure and the different DL-MAP algorithms un-der study. In each simulation we considered a WiMAX frameof duration 5ms containing a single PUSC zone and a 35/12DL/UL ratio. Two MAP repetitions and QPSK 1/2 were usedfor the DL-MAP. If not otherwise indicated a 10Mhz channelbandwdith is considered, which results in a WiMAX down-link subframe composed of 30 logical subchannels. For eachWiMAX frame we randomly generated a set of MPDUs, repre-senting the output of the QoS Scheduler, and assigned to eachMPDU a MCS level obtained from a certain MCS distribution,representing the output of the Channel Monitor. Thus, given theset of MPDUs delivered by the QoS scheduler and the corre-sponding MCS level needed to transmit each MPDU assignedby the Channel Monitor, we evaluated how the different DL-MAP scheduling algorithms under study perform on each ofthe previously mentioned performance aspects. In order to gainstatistical confidence we considered for each algorithm understudy a total of 104 WiMAX frames and averaged the obtainedresults. The corresponding confidence intervals have not beenincluded in the figures because they were too small to add sig-nificant information.

Specifically, for our MPDU generation process we used theMPDU size distribution depicted in Table 2 which was obtainedby SPRINT in a data collection campaign16. This packet sizedistribution is dominated by TCP flows, the 1500B packet sizecorresponds to TCP segments and the 40B size to TCP ACKs,and could therefore represent a typical DSL access network,which is a relevant scenario because one of the target use casesfor WiMAX is the wireless DSL service. In addition the MCSdistribution used is depicted in Table 3 and corresponds to atypical urban scenario where most of the users are located closeto the BS.

4.1. DL-MAP Scheduling AlgorithmsOne of the questions that we want to investigate is which are

the advantages and drawbacks of using a 2D DL-MAP sched-uler versus a 1D one. Hence, besides GSA, we consider for ourevaluation the following algorithms: i) The MaSP/MiSP5 algo-rithm proposed in [6] which is a representative example in the

16https://research.sprintlabs.com/packstat/

12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Percentage MPDU Size (bytes)18.89% 4012.09% 15006.14% 624.67% 14204.60% 5253.61% uniform(40,1500)

Table 2: MPDU Size Distribution

Percentage MCS Level20% 64QAM 3/420% 64QAM 2/320% 64QAM 1/215% 16QAM 3/415% 16QAM 1/25% QPSK 3/45% QPSK 1/2

Table 3: MCS Distribution

literature of a 2D DL-MAP scheduler, ii) a 1D-Vertical and a1D-Horizontal schedulers, which are examples of 1D DL-MAPschedulers mapping data in the WiMAX frame in the verticaland horizontal directions respectively, and finally iii) an IdealDL-MAP Scheduler which we use to obtain an upper bound tothe maximum efficiency achievable by the different algorithms.Next, we provide a detailed description of these algorithms.

MaSP/MiSP5With their MaSP/MiSP5 Cohen at al. proposed an efficient

solution to the DL-MAP scheduling problem in OFDMA [6].Similar to GSA they divided the scheduling task in two steps.First, the Macro Scheduling (MaSP) algorithm decides whichsubset of the MPDUs delivered by the QoS scheduler will betransmitted, and how these MPDUs will be concatenated, inorder to maximize the overall utility carried in each WiMAXframe. In order to reduce MAP overhead, unlike GSA, theMaSP algorithm considers that all MPDUs transmitted with thesame MCS have to be concatenated. The MaSP algorithm has aparameter 0 ≤ β ≤ 1 used to trade off optimality (β = 0) witha decrease in complexity; in our simulations we used β = 0.1,which showed to be a good trade-off between efficiency andexecution time. Thereafter, the Micro Scheduling (MiSP) algo-rithm decides the actual shape and position for each burst. Al-though the authors proposed three different MiSP algorithms,in their paper it was shown that MiSP5 was the best performingone. Therefore, in this paper we consider only MiSP5. Finally,the MaSP/MiSP5 algorithm can only be used in an Offline ar-chitecture17.

1D-Vertical and 1D-HorizontalIn order to evaluate the potential gains and drawbacks of a

1D DL-MAP scheduler, we consider a 1D-Vertical and a 1D-

17In addition we introduced a slight modification to the MiSP5 algorithmin order to increase its efficiency by allowing it to pack below the DL-MAPoverhead which was not possible in the original proposal.

Horizontal DL-MAP schedulers. These schedulers function ina similar way than GSA in order to select the MPDUs deliv-ered by the QoS scheduler 18. However, unlike GSA whichmaps bursts in two-dimensional regions, these algorithms per-form one-dimensional packing in the vertical or horizontal di-rection. Actually, the 802.16e standard defines for the down-link subframe a special packing algorithm when the BS em-ploys Hybrid-ARQ (H-ARQ) [2]. This H-ARQ packing modecan be basically understood as a 1D vertical packing where theBS starts to allocate the different bursts in the frequency direc-tion. Hence, the BS simply states in the DL-MAP the size ofeach different burst and a SS can discover the position of itsintended data simply by adding up the sizes of the bursts pre-viously signaled in the DL-MAP. Formally, as defined in the802.16e standard, the size of the DL-MAP overhead in bitswhen using H-ARQ mode19 is different than when using reg-ular PUSC. Specifically, the size of the DL-MAP in bits whenn bursts are packed using H-ARQ is computed as:

map(n) = F0 +⌈F1 + d nF2

e · F3 + F4 · n8

⌉· 8

Where the different constants Fi are defined in Table 1.The 1D-Vertical scheduler that we have used for our evalu-

ation closely resembles the H-ARQ packing mode as defined inthe 802.16e standard. In addition, we have extended it to alsoconsider packing in the horizontal (time) direction, thus obtain-ing our 1D-Horizontal scheduler.

Ideal SchedulerIn order to obtain an upper bound to the optimal efficiency

that can be obtained by GSA and the previous considered DL-MAP schedulers we define an Ideal Scheduler that tries to max-imize efficiency without considering the required complexity20.The main properties of this Ideal Scheduler are:

• Given a set of MPDUs delivered by the QoS schedulerwith an utility value, ui, attached to each MPDU, theIdeal Scheduler solves a collapsing knapsack problem,according to the algorithm defined in [14], in order tofind the subset of MPDUs that fit in the WiMAX down-link subframe carrying the highest total utility.

• The ideal scheduler assumes that bursts can be packedwithout any shape limitations and therefore no slot pad-ding occurs. This is similar to the 1D packing algorithms.

• In order to minimize the required DL-MAP overhead theIdeal scheduler considers that all MPDUs transmitted us-ing the same MCS can be concatenated together.

The reason for using a collapsing knapsack instead of aclassical knapsack in order to obtain the subset of MPDUs thatmaximize the utility carried in the WiMAX frame in the Ideal

18The only difference is that for the 1D schedulers we use a knapsack algor-tihm to create list P 2 instead of the greedy heuristic used in GSA.

19We considered Chase-Combining H-ARQ.20Notice that this scheduler is not feasible in practice.

13

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

scheduler, is that the actual capacity available in the WiMAXframe for data transmissions reduces with the number of packedbursts due to the DL-MAP overhead. Indeed, this reduction isnot linear, i.e. it is not correct just considering a fixed over-head per burst, because the DL-MAP overhead associated witha burst varies depending on whether this burst can be concate-nated with other bursts or not. Formally the collapsing knap-sack problem is defined as follows:

Instance: A set S of items s1, s2, ..., sm and a capacityfunction c() which depends on the packed items: c(S) : ℘(S)→N. Each item si has utility p(si) and a weight w(si).

Objective: Find a subset S′ ⊆ S of items such that thissubset has a feasible packing, namely,

∑sj∈S′ w(sj) ≤ c(S′),

and the aggregated utility∑sj∈S′ p(sj) is maximized.

In our case we defined the function c() in order to considerburst concatenation as follows:

c(S) = C ′ −R ·⌈map(S)

SC

⌉Where the size of the DL-MAP in bits is:

map(S) = F0 +MCS∑m=1

δ(|Sm|) · F5 + |Sm| · F6

Sm ⊆ S ∧ ∀x ∈ Sm : mcs(x) = m

Where δ(x) is 1 when x > 0 and 0 otherwise. The rest of pa-rameters are defined in Table 1, C ′ is the available size in theDL subframe after considering the FCH and UL-MAP over-heads and R is the number of MAP repetitions.

4.2. EfficiencyIn this section we evaluate how the different DL-MAP sched-

uling algorithms under study are able to utilize the scarce radioresources. Since some of the algorithms can only function in anOffline architecture, e.g. MaSP/MiSP5, we will start comparingthe performance of the different algorithms under this architec-ture. Afterwards we will study specifically for GSA which arethe effects of considering an Online architecture instead of anOffline one.

As introduced in Section 2, efficiency can be understooddifferently according to the particular objectives of a serviceprovider. For instance a certain service provider may want tooptimize cell throughput, while another one may prefer to opti-mize a certain notion of fairness. In order to allow for this flex-ibility, in the Offline architecture the QoS scheduler conveysfor each MPDU a utility value that can be defined according toan arbitrary policy. For the purpose of this evaluation we willconsider two different utility policies:

• Throughput Utility. To implement this policy the QoSscheduler simply defines as the utility of each MPDU thenumber of bits of this MPDU. Thus, when the DL-MAPscheduler maximizes the utility carried in the WiMAXframe it maximizes the number of carried bits, and hencethe observed throughput. Notice that under this policyMPDUs transmitted with a higher MCS should be treatedpreferentially by the DL-MAP scheduler.

• Random Utility. With this policy we model an arbitrarypolicy set by a service provider. Each MPDU is simplyassigned by the QoS scheduler a random amount of util-ity, where utility is defined as a random integer numberbetween 0 and the size of the MPDU.

Figures 9(a) and 9(b) illustrate the performance of the dif-ferent algorithms under study for the two utility policies con-sidered when we increase the amount of load offered by theQoS scheduler. Both the 1D vertical and horizontal schedulersare depicted in Figure 9(a) and 9(b) using a single line becausethe two algorithms provide the same performance in terms ofefficiency, since in 1D scheduling only the total capacity ofthe frame is important. As observed in Figures 9(a) and 9(b)the higher the load offered by the QoS scheduler the more thepossibilities for the DL-MAP scheduler to build an optimizedframe layout, but also the higher the complexity. We vary theamount of offered load from 0.8 (the aggregated size in slotsof the MPDUs delivered by the QoS scheduler equals 80% ofthe capacity of the WiMAX frame) to 3 (the aggregated sizein slots of the MPDUs delivered by the QoS scheduler equalsthree times the capacity of the WiMAX frame). Logically allalgorithms perform equally when the load offered by the QoSscheduler is below 1 (all the delivered MPDUs can be packed)and the performance differs as the amount of offered load in-creases.

All algorithms though perform significantly close for bothutility policies under study, with GSA slightly outperformingMaSP/MiSP5 and being close to the 1D and Ideal schedul-ers. The differences between the best performing algorithm, theIdeal scheduler, and the worst one, in this case the MaSP/MiSP5algorithm, are always below 8.3% and 7% respectively. Thisfact is significant because it indicates that 2D schedulers likeGSA or MaSP/MiSP5 can perform in terms of efficiency al-most as well as 1D or even Ideal schedulers do. Thus, the goalwill be for the 2D schedulers to be able to provide the addi-tional benefits of two-dimensional packing without decreasingtheir performance in terms of efficiency.

In order to gain a deeper understanding on the performancedifferences between the different DL-MAP schedulers, Figure9(c) illustrates the packing efficiency of the different algorithmsfor the case of the random utility policy. Similar results wereobtained in the case of the throughput utility policy. Packingefficiency is defined as follows:

pack efficiency =total packed − total padding

C − FCH− ulmapBeing C the capacity of the downlink subframe, total packedthe total number of packed slots in the WiMAX frame andtotal padding the total padding in slots.

Figure 9(c) shows that the packing efficiency of GSA is onaverage 2.5% higher than the one of MaSP/MiSP5, which ex-plains the slightly higher level of utility observed in these ex-periments. In addition the packing efficiency of the 1D andthe Ideal schedulers is on average 6.5% and 7.6% points higherthan in GSA respectively.

So far we have always considered an Offline architecture inthe evaluation of the different algorithms. However, GSA has

14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

1 1.5 2 2.5 39

10

11

12

13

14

15

16

17

18

Offered Load

Tot

al T

hrou

ghpu

t (M

bps)

Throughput Utility

GSA

MaSP/MiSP5

1D−vert/horiz

1D−Ideal

2.5 2.6 2.7 2.8 2.9 315.5

16

16.5

17

17.5

18

GSA

MaSP/MiSP5

1D−vert/horiz

1D−Ideal

(a) Performance under a Throughput Efficiency

1 1.5 2 2.5 320

25

30

35

40

45

50

55

Offered Load

Tot

al U

tility

(×10

3 )

Random Utility

GSA

MaSP/MiSP5

1D−vert/horiz

1D−Ideal

2.5 2.6 2.7 2.8 2.9 3

48

50

52

54

56

(b) Performance under a Random Efficiency

1 1.5 2 2.5 365

70

75

80

85

90

95

100

Offered Load

Effi

cien

cy (

%)

Random Utility

GSAMaSP/MiSP51D−vert/horiz1D−Ideal

(c) Packing Efficiency

1 1.5 2 2.5 39

10

11

12

13

14

15

16

17

Offered Load

Tot

al T

hrou

ghpu

t (M

bps)

Throughput Utility

GSA−offlineGSA−online1GSA−online2

(d) Offline vs Online performance of GSA

Figure 9: Efficiency comparison of the different algorithms

been designed to be able to operate in an Online architecture aswell. As explained in Section 2 an Online architecture sacrificesefficiency but promises a reduced complexity.

Figure 9(d) illustrates the efficiency trade off, when consid-ering an Online architecture in the case of GSA. For the sakeof space we only include the results for the throughput utilitypolicy. In this experiment we considered two different OnlineGSA algorithms: i) online-1 is a simple online algorithm whereGSA stops its operation whenever a MPDU delivered by theQoS scheduler can not be packed in the WiMAX frame, andii) online-2, which represents an improvement over the previ-ous algorithm where upon a MPDU not being able to be packedGSA retries with another one, until all the MPDUs available inthe QoS scheduler have been explored.

Looking at Figure 9(d) it is clear that performance is sacri-ficed when using an Online architecture. The reason is that in anOnline architecture the DL-MAP scheduler can not make use of

the extra offered load delivered by the QoS scheduler in order tooptimize efficiency, e.g. pre-selecting from the set of deliveredbursts the ones carrying higher utility. A better performancewhile respecting the constraints imposed by an Online architec-ture is obtained in the case of the online-2 algorithm, because inthis case GSA keeps requesting data to the QoS scheduler untilno more data is available in the QoS scheduler queues.

Based on the presented results we conclude that GSA, al-though using simplified greedy heuristics, can provide in a typ-ical scenario a performance in terms of efficiency similar to theone of more complex approaches existing in the state of the art,and even close to the performance of 1D and Ideal DL-MAPschedulers. In addition, GSA can operate in both Offline andOnline architectures. In the next sections we will show howGSA can provide additional benefits with respect to the other al-gorithms under study without sacrificing performance in termsof efficiency.

15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

4.3. FlexibilityAs mentioned in Section 2 an important requirement for a

DL-MAP scheduler is its ability to accommodate per SS shapepreferences. Hence, we analyze in this section the WiMAXframe layouts obtained with the different algorithms under study,in order to see how this requirement is fulfilled.

In order to compare the burst shapes obtained with the dif-ferent DL-MAP schedulers, we need an objective way to mea-sure how vertical or horizontal a packed burst is. For this pur-pose we define the vertical and horizontal indexes which areused to compare the shapes obtained with the different DL-MAP schedulers. The vertical and horizontal indexes are ob-tained in the following way. Consider that a burst of B slotshas to be packed in the WiMAX frame. In this case the max-imum vertical dimension that could be used to pack this burstis:

max height =

B ,B ≤ Sd BdB

S ee , B > S

Where S is the number of subchannels building up the WiMAXdownlink subframe. Similarly, the maximum horizontal dimen-sion that could be used to pack this burst is:

max width =

B ,B ≤ Td BdB

T ee , B > T

Where T is the number of time slots the WiMAX downlinksubframe lasts for.

Given that a certain DL-MAP scheduler algorithm packsthe mentioned burst of B slots in a 2D shape of dimensionsheight(B) and width(B), the horizontal and vertical indexesof such a burst can be computed in the following way:

IdxV ert =height(B)max height

, IdxHorz =width(B)max width

Where 0 < IdxV ert,Horz ≤ 1, and a value of one indicates thatthe burst is packed in a completely vertical or horizontal way.

Figure 10(a) depicts the obtained average vertical and hor-izontal indexes in the WiMAX frame layouts generated by thedifferent algorithms under study. For a WiMAX frame gen-erated by a given DL-MAP scheduler each point in the graphrepresents the average vertical and horizontal indexes of all thebursts present in the frame. Different points represent differentsimulation runs.

As illustrated in Figure 10(a) the frame layouts generatedby the different algorithms are significantly different. As ex-pected, the 1D-Vertical and 1D-Horizontal DL-MAP schedul-ers generate bursts which have vertical and horizontal shapesrespectively. MaSP/MiSP5 exhibits a certain preference to gen-erate horizontal bursts, which is due to the way the algorithmhas been designed, although more squared shapes are also gen-erated21. In addition, GSA, when no shape preferences are con-sidered, generates bursts which are more or less square.

21In our experiments we have observed that MaSP/MiSP5 tends to generatemostly horizontal bursts when the offered load is below the frame capacity andgenerates more squared bursts when the offered load exceeds the frame capac-ity.

Unlike all the other algorithms though, GSA has the uniquefeature of being able to consider per SS burst shape preferences.This is easily achieved by means of the shapeMetric functionused to weight the selection of rectangular shapes to assign toa given burst as detailed in Algorithm 3. For instance, in orderto accommodate vertical or horizontal shape preferences, theshapeMetric function can be defined in the following way:

shapeMetricV ert(B)← height(B)width(B)

shapeMetricHorz(B)← width(B)height(B)

Where height(B) andwidth(B) are the vertical and horizontaldimensions of the 2D shape selected to accommodate a burst ofB slots.

In order to evaluate the ability of GSA to accommodate perSS shape preferences we perform the following experiment. Weconsider a WiMAX BS serving users belonging to two differentclasses, where each class of users has a certain shape prefer-ence. We analyze three different cases:

• Horz/Horz. In this experiment both classes of users pre-fer to shape their bursts as horizontally as possible.

• Vert/Vert. In this experiment both classes of users preferto shape their bursts as vertically as possible.

• Horz/Vert. In this experiment one class of users prefersto shape their bursts as horizontally as possible and theother class of users as vertically as possible.

In our experiment whenever a MPDU is delivered by theQoS Scheduler to the DL-MAP scheduler we randomly selectthe class of the user receiving this MPDU, and we assign oneof the previously defined shapeMetric functions according tothe shape preference of each class. In addition, when shapepreferences are considered GSA disables concatenation. Thereason is that this feature could result troublesome if users withdifferent shape preferences had to be concatenated together.

Figure 10(b) depicts the vertical and horizontal indexes ob-tained in the previous experiments where GSA considers shapepreferences. It is clearly observed in the figure how in theHorz/Horz and Vert/Vert experiments GSA achieves horizontaland vertical shapes which are reasonably close to those achievedby the corresponding 1D schedulers. Indeed, even when hori-zontal and vertical users have to be scheduled simultaneouslyin the same frame, in the Horz/Vert experiment, GSA is ableto clearly accommodate the shape preferences of each user, al-though the obtained shapes slightly degrade because shapingone user vertically limits the ability of shaping successive usershorizontally and viceversa. An example of the WiMAX framelayouts generated by GSA in the previous experiments is shownin Figure 11.

Since GSA is indeed able to accommodate per user shapepreferences, a question to be answered is which is the priceto pay in terms of efficiency when GSA considers flexibilityconstraints. Figure 10(c) depicts the performance of GSA with

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Horizontal Index

Ver

tical

Inde

x

GSAMaSP/MiSP51D−vert1D−horiz

(a) All algorithms

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Horizontal Index

Ver

tical

Inde

x

Horiz/Horiz (user 1/2)Vert/Vert (user 1/2)Horiz/Vert (user 1)Horiz/Vert (user 2)

(b) GSA with shape preferences

1 1.5 2 2.5 320

25

30

35

40

45

50

55

Offered Load

Tot

al U

tility

(×10

3 )

Random Utility

GSA

GSA (no concatenation)

GSA (horizontal preference)

GSA (vertical preference)

GSA (horizontal+vertical preference)

2.5 2.6 2.7 2.8 2.9 347

48

49

50

51

52

53

(c) Performance under a Random Utility Policy

1 1.5 2 2.5 370

72

74

76

78

80

82

84

86

88

90

Offered Load

Effi

cien

cy (

%)

Random Utility

GSAGSA (no concatenation)GSA (horizontal preference)GSA (vertical preference)GSA (horizontal+vertical preference)

(d) Packing Efficiency under a Random Utility Policy

Figure 10: Shape scatter plot for the different algorithms and the achieved performance.

and without shape preferences when considering a random util-ity policy. In addition, since concatenation is disabled whenshape preferences are considered, we also report the perfor-mance of GSA without shape preferences and without concate-nation, which allows us to understand how much performanceloss is due to the concatenation capability and how much is dueto the consideration of shape constraints. The results in Figure10(c) though, show that the performance loss when consideringshape constraints is indeed minimal in our scenario. This can beunderstood by looking at Figure 10(d) which depicts the pack-ing efficiency achieved in the previous experiments, and showsthat less than 4% of efficiency is lost by GSA in our experimentwhen considering shape constraints. The results obtained whenconsidering the throughput utility policy closely resembled theones with the random utility, and hence have not been included.

Thus, we conclude that our proposed GSA algorithm is ableto accommodate per SS shape preferences without a significant

performance degradation, which makes it a suitable candidateto fulfill the flexibility requirements needed to implement opti-mal resource management strategies in WiMAX networks.

4.4. Interference Management

In this section we will evaluate how the different DL-MAPscheduling algorithms studied in this paper perform in terms ofinterference avoidance. For this purpose, in addition to GSAwithout shape preferences, MaSP/MiSP5 and the 1D verticaland horizontal schedulers, we will also consider GSA with hor-izontal shape preferences, as defined in Section 4.3 and theinterference-aware GSA (iGSA) extension proposed in Section3.3 with three iGSA regions.

In our evaluation we consider two different WiMAX de-ployments, a deployment based on tri-sectorial BSs with Fre-quency Reuse 3 (1x3x3), and a deployment based on Omnidi-rectional BSs with Frequency Reuse 3 (3x1x1). The two de-

17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0 2 4 6 8 10 12 14 160

5

10

15

20

25

30

FCH

DL-MAP

slot

subchannel

UL-MAP

(a) Horz/Horz

0 2 4 6 8 10 12 14 160

5

10

15

20

25

30

FCH

slot

subchannel

UL-MAP

DL-MAP

(b) Vert/Vert

0 2 4 6 8 10 12 14 160

5

10

15

20

25

30

FCH

DL-MAP

slot

subchannel UL-MAP

DL-MAP

Horizontal

Vertical

(c) Horz/Vert

Figure 11: GSA frame layout of an arbitrary run for the 3 scenarios, Horz/Horz, Vert/Vert, and Horz/Vert.

Figure 12: Radiation pattern considered by the tri-sectorial BSs used in the1x3x3 deployment

ployments under study are depicted in Figure 8. As illustratedin the figure, we have considered for this evaluation the threeinterferers in the first tier in the case of the 1x3x3 scenario, andthe six interferers in the first tier in the case of the 3x1x1 sce-nario. This number of interferers represents 81% of the totalinterference power in a typical22 1x3x3 scenario and 100% ofthe total interference power in a typical 3x1x1 scenario.

In order to evaluate how the different DL-MAP schedulersmanage the level of interference in the network we have per-formed the following experiment. We increase the amount ofdownlink load to be served by each BS in each sector/cell inthe two deployments under study from 0 (the WiMAX frame isempty) to 1 (the WiMAX frame is full), and evaluate the perfor-mance of each DL-MAP scheduler, by computing the averagesubcarrier SINR perceived in a target station, placed for eachdeployment under study in the position indicated in Figure 8.

The average subcarrier SINR in the target station is obtainedby averaging the SINR of each used subcarrier in each OFDMAsymbol in the WiMAX frame23. Individual SINRs for each sub-

22Typically cellular studies are performed considering a 19 cell layout [15],however in order to reduce the required simulation time we have only consid-ered the effect of the strongest interferers.

23All the traffic in the target cell is considered to be directed to the targetstation.

Parameter ValueDeployment (reuse pattern) 3x1x1 and 1x3x3Channel Bandwidth 10MhzBS-BS distance 900mPath-Loss Model Erceg (terrain type B) as

defined in [15]SS noise figure 7 dBAntenna gains 0 and 17 dBiPermutation Scheme PUSC

Table 4: Parameters in Interference Experiment.

carrier are computed as SINR(k) = S(k)

N(k)+PN

i=1 Ii(k), where

S(k) is the received signal power in subcarrier k, N(k) is thepower of Noise in this subcarrier, Ii(k) is the amount of inter-ference power in subcarrier k coming from interferer i, and Nis the maximum number of interferers considered to be three inthe 1x3x3 deployment and six in the 3x1x1 deployment. Table4 depicts all the parameters used in our simulations, and Figure12 illustrates the radiation pattern considered by the tri-sectorialBSs used in the 1x3x3 deployment.

Figures 13(a) and 13(b) depict the average subcarrier SINRobtained in the target cell for the different DL-MAP schedulersunder study in the 1x3x3 and 3x1x1 deployments respectively.As observed in the figures the trends exhibited by the differ-ent DL-MAP scheduler are quite similar in both deployments,however the actual SINR values are different since interferenceis inherently higher in the 3x1x1 deployment.

As clearly seen in Figures 13(a) and 13(b) the 1D-VerticalDL-MAP scheduler is the worst performing approach in termsof interference management. The reason for such a bad perfor-mance is the following. Even though different cells or sectorsuse different ID Cell parameters that result in randomized sub-carrier to subchannel mappings, this randomization gain is min-imal if the DL-MAP scheduler performs a column wise packing(in the frequency direction), because even at low loads columns(OFDMA symbols) in the WiMAX frame are fully used by dif-ferent cells at the same time, which results in full interference.These results clearly show that in an interference limited sce-nario 1D vertical packing is not an optimal choice.

18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

15

20

25

30

35

40

Offered Load

Ave

rage

Sub

carr

ier

SN

IR (

dB)

Reuse Pattern 1x3x3

iGSAGSAGSA (horiz. pref)MaSP/MiSP51D−vert1D−horiz

(a) Interference management in reuse pattern 1x3x3 with different DL-MAPschedulers

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

12

14

16

18

20

22

24

26

Offered Load

Ave

rage

Sub

carr

ier

SN

IR (

dB)

Reuse Pattern 3x1x1

iGSAGSAGSA (horiz. pref)MaSP/MiSP51D−vert1D−horiz

(b) Interference management in reuse pattern 3x1x1 with different DL-MAPschedulers

Figure 13: Interference Management for the different DL-MAP Schedulers

On the other hand, the DL-MAP schedulers packing pref-erentially in the time direction (row wise), i.e. 1D-Horizontal,MaSP/MiSP5 and GSA with horizontal preferences, exhibit amuch better performance. These schemes can fully benefit fromthe PUSC randomization gain because a full column in the WiMAXframe will only be used when the amount of offered load in eachcell equals to 1 (the WiMAX frame is full). These schemes re-sult in a maximum gain with respect to the 1D-Vertical sched-uler of 17dBs in the 1x3x3 deployment and of 8dB in the 3x1x1deployment. In addition, GSA, when no shape preferences areconsidered, has a performance between the 1D-Vertical sched-uler and the horizontal scheduler, since as previously illustratedin Section 4.3, it constructs bursts which are more or less squar-ed. Finally, special attention is deserved by iGSA which signif-icantly outperforms any of the other approaches.

In our evaluation, iGSA provides a maximum gain of 12dBwith respect to the horizontal DL-MAP schedulers and of 25dBswith respect to the 1D-Vertical scheduler in the 1x3x3 deploy-ment, and a maximum gain of 6dB with respect to the horizon-tal DL-MAP schedulers and of 13dBs with respect ot the 1D-Vertical scheduler in the 3x1x1 deployment. The reason whyiGSA can provide an improved performance is that it placesthe interferers in the first tier in each deployment in differentpacking regions than that of the target cell. This concept is il-lustrated in Figure 8. Hence, given that iGSA considers threedifferent packing regions, when the load in each cell is below1/3 of the cell capacity, interference can be completely avoided.However, when the load offered in each co-channel cell growsabove 1/3 of the cell capacity, the performance of iGSA de-grades faster than that of horizontal DL-MAP schedulers. Thereason is that the target cell starts packing in a packing regionwhich has already been fully utilized by other interferer cells,and the randomization gain obtained in each iGSA packing re-gion is reduced with respect to the one obtained with horizon-

tal DL-MAP schedulers, simply because iGSA packing regionsspan only one third of the WiMAX frame. Notice though thatonly at considerably high amounts of load in each sector/cell(offered load bigger than 0.7) the performance of iGSA falls tothe level of the horizontal DL-MAP schedulers.

Given the presented results we consider GSA and speciallyiGSA an excellent candidate to perform efficient interferencemanagement in future WiMAX networks.

4.5. Computational Load

Finally, we complete our evaluation by looking at the aver-age computational time required by the different algorithms inorder to build a WiMAX frame. Although the reported absolutetimes may not be significant as such, specialized hardware tai-lored to each algorithm could be built to optimize its executiontime, we believe that the reported results clearly illustrate thescaling properties of each algorithm which are relevant whenconsidering the development of a cost limited BS.

Figure 14(a) depicts the computational load for the differ-ent DL-MAP schedulers under study in terms of average frameconstruction time. In order to evaluate the scalability of the dif-ferent algorithms with respect to the size of the WiMAX framewe consider 10Mhz (30 subchannels in the WiMAX frame) and20Mhz (60 subchannels in the WiMAX frame) channel band-widths. For the sake of clarity, we only report the frame con-struction times obtained when considering a throughput utilitypolicy. No significant differences were observed when consid-ering a random utility policy.

The greedy principle used to design GSA shows its majorbenefits in terms of complexity. When considering a bandwidthof 10Mhz and an offered load delivered by the QoS sched-uler equal to three times the capacity of the WiMAX frame,the MaSP/MiSP5 algorithms takes on average around 200ms to

19

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Offered Load

Fra

me

pack

ing

time

(s)

Throughput Utility

GSA(30)

MaSP/MiSP5(30)

1D−v/h(30)

GSA(60)

MaSP/MiSP5(60)

1D−v/h(60)

0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

0.06

(a) Comparison of different algorithms

0.5 1 1.5 2 2.5 30

0.02

0.04

0.06

0.08

0.1

0.12

Offered Load

Fra

me

pack

ing

time

(s)

Throughput Utility

GSA(30)GSA (no concatenation, 30)GSA−online1(30)GSA−online2(30)GSA(60)GSA (no concatenation, 60)GSA−online1(60)GSA−online2(60)

(b) Onlline vs Offline in GSA (60 subchannels)

Figure 14: Computational Time required with the different algorithms

build a WiMAX frame, while for the same settings GSA re-quires only around 20ms24, resulting in an order of magnitudeimprovement for a similar level of efficiency. Indeed, whena 20Mhz bandwidth is considered GSA results in a 25 foldspeed up with respect to MaSP/MiSP5. Notice that faster ex-ecution times for MaSP/MiSP5 could be achieved by consid-ering a higher β value, but a price on performance would bepaid. In addition the greedy heuristics and the linear searchused to build a WiMAX frame in GSA result in frame construc-tion times very close to those of 1D schedulers, which can beconsidered to be a lower bound to the complexity required by2D DL-MAP schedulers.

Figure 14(b) compares the computational times achievedin GSA when considering Online or Offline architectures andagain 10Mhz and 20Mhz channel bandwidths. The same on-line schedulers for GSA, online-1 and online-2, that were in-troduced in Section 4.2 are considered now. In addition, weconsider GSA in the Offline architecture with and without theconcatenation feature.

The results depicted in 14(b) are at a first sight surprisingin the sense that in some cases GSA achieves faster frame con-struction times under an Offline architecture that under an On-line architecture. The main reason behind the obtained resultsis the fact that the highest computational effort is allocated byGSA in the Core Packing Engine. Notice, that regardless ofthe size of the burst to be packed, GSA spends a similar effortper burst on finding a suitable rectangular shape, updating thefree and allocated capacity lists and possibly performing de-fragmentation on the free capacity list. Hence, the number ofbursts to be considered by the Core Packing Engine in GSAbecomes a critical factor in terms of computation time.

24These execution times are the ones obtained in our simulator using com-modity hardware with a general purpose processor. We believe though, thatGSA running on an FPGA used in commercial WiMAX BSs can easily achieverunning times below 5ms (typical WiMAX frame time).

The previous fact indirectly turns the concatenation featureinto an excellent way to reduce complexity. Although, somecomputational effort is spent by the concatenation heuristic, Al-gorithm 2, in order to find suitable concatenations, this effort ispaid off by the reduced number of bursts to be later consideredby the Core Packing Engine. The previous argument explainswhy Offline GSA together with concatenation turns out to bethe fastest algorithm. Notice though, that if concatenation cannot be employed, e.g. when users have certain shape prefer-ences, the online-1 algorithm results as expected in a significantspeed up. Finally, notice that the proposed online-2 algorithmresults in frame construction times even above those of offlineGSA without concatenation. The reason is that even withoutconcatenation offline GSA reduces the number of bursts to beconsidered by the Core Packing Engine by means of the smartordering performed in the Burst Packing Stage (packing big-ger objects first). Although optimizations could be divised thatreduce the complexity of the online-2 algorithm, we have notconsidered them in this paper.

Finally, notice that in practice, thanks to features like theconcatenation or defragmentation modules which reduce thenumber of elements to be considered by the Core Packing En-gine, the complexity of GSA appears to be much better thanthe estimated O(N4

1 ) worst case bound. We conclude thus thatGSA is a feasible DL-MAP scheduler to be deployed in currentWiMAX equipment employing both Offline or Online architec-tures.

5. Conclusions and Future Work

In this paper, we have presented an efficient and flexibleWiMAX OFDMA downlink scheduling solution, Greedy Sched-uling Algorithm (GSA), where the 2D WiMAX OFDMA down-link packing problem was transformed into an efficient 1D search-ing problem. We identified the major challenges in the design of

20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

a DL-MAP scheduler, namely efficiency, allowing for flexibleburst orientations, ability to reduce interference, and minimalcomputational complexity. To the best of our knowledge GSAis the first DL-MAP scheduling algorithm to address all theseperformance aspects at the same time.

Through a thorough simulation study we were able to provethat GSA achieves in terms of efficiency a performance close,or even slightly superior, to the one of the considered alter-native approaches, but significantly outperforms any other ap-proach in terms of flexible burst orientations, interference mit-igation and reduced computational load, indeed addressing allthese requirements without sacrificing efficiency. In additionwe proved that GSA is highly independent from the architec-ture used in a WiMAX Base Station, being able to be deployedin both Online and Offline architectures. For all the previousreasons, we believe that GSA is an excellent candidate to ful-fill the stringent requirements of the next generation wirelessbroadband access networks based on OFDMA.

Finally, we have identified as promising future work severalareas where GSA could be extended to provide even furtherperformance improvements, for instance the support of SpaceDivision Multiple Access (SDMA) in downlink or the use ofsmart power allocations in order to improve channel capacity.

6. Acknowledgements

The authors would like to thank their colleagues from thedevelopment team of NEC’s WiMAX Base Stations for theircollaboration on this research work.

References

[1] IEEE Standard for Local and Metropolitan Area Networks. Part 16: AirInterface for Fixed Broadband Wireless Access Systems, IEEE 802.16-2004 Standard (October 2004).

[2] IEEE Standard for Local and Metropolitan Area Networks. Part 16: AirInterface for Fixed Broadband Wireless Access Systems - Amendmentfor Physical and Medium Access Control Layers for Combined Fixedand Mobile Operation in Licensed Band., IEEE 802.16e-2005 Standard(February 2006).

[3] IEEE Standard for Local and Metropolitan Area Networks. Part 16: AirInterface for Fixed Broadband Wireless Access Systems, IEEE 802.16-2009 Standard (May 2009).

[4] WiMAX Forum, http://www.wimaxforum.org.[5] Y. Ben-Shimol, I. Kitroser, Y. Dinitz, Two-dimensional Mapping for

Wireless OFDMA Systems, IEEE Transactions on Broadcasting.[6] R.Cohen, L.Katzir, Computational Analysis and Efficient Algorithms for

Micro and Macro OFDMA Scheduling, in: In proceedings of the IEEEInternational Conference on Computer Communications (INFOCOM),Phoenix, USA, 2008.

[7] C. Desset, E. de Lima Filho, G. Lenoir, Wimax Downlink OFDMABurst Placement for Optimized Receiver Duty-Cycling, IEEE Interna-tional Conference on Communications (ICC).

[8] X. Perez-Costa, P. Favaro, A. Zubow, D. Camps, J. Arauz, On theChallenges for the Maximization of Radio Resources Usage in WiMAXNetworks, Consumer Communications and Networking Conference(CCNC).

[9] C. Cicconetti, L. Lenzini, E. Mingozzi, C. Eklund, Quality of servicesupport in IEEE 802.16 networks, IEEE Network, pp. 50-55.

[10] C. Wengerter, J. Ohlhorst, A. von Elbwart, Fairness and throughput anal-ysis for generalized proportional fair frequency scheduling in OFDMA,in: In proceedings of Vehicular Technology Conference, pp. 1903- 1907Vol. 3, 2005.

[11] H. Kellerer, U. Pferschy, D. Pisinger, Knapsack Problems, Springer, 2004.[12] K. Balachandran, Design and Analysis of an IEEE 802.16e-Based

OFDMA Communication System, Bell Labs Technical Journal.[13] D. S. Johnson, A. Demers, J. D. Ullman, M. R. Garey, R. L. Graham,

Worst-case performance bounds for simple one-dimensional packing al-gorithms, SIAM J. COMPUT. (1974) 299–325.

[14] U. Pferschy, D. Pisinger, G. J. Woeginger, Simple but efficient ap-proaches for the collapsing knapsack problem, Discrete Applied Math-ematics 77(3) (1997) 271–280.

[15] A. Koc, Wimax system evaluation methodology, WiMAX Forum (2008)209.

Anatolij Zubow is a researcherin the Wireless Mesh NetworkingGroup at the department of ComputerScience at Humboldt-Universitat zuBerlin, Germany. He received his PhDdegree from Humboldt-Universitat in2009. His main research inter-est is cooperative mesh network-ing in wireless ad hoc networks.In addition he worked on severalWLAN and WiMAX related projects.

Daniel Camps Mur is a researcherin the Wireless Access Group at NECNetwork Laboratories in Heidelberg,Germany. He is currently pursu-ing a PhD at the telematics depart-ment of the Polytechnical Univer-sity of Catalonia (UPC), where heobtained an M.Sc degree in 2004.His current research interests includeMAC QoS and power saving proto-cols for WLAN and WiMAX net-

works. His master thesis work received the ’Mobile In-ternet and 3G Mobile Solutions’ award from the Span-ish Association of Telecommunication Engineers (COIT).In the field of network simulations he also received theOPNET’s Significant Technical Challenge Solved Award.

Xavier Perez Costa is a Chief Re-searcher at NEC Laboratories Europein Heidelberg, Germany, where he iscurrently the responsible of leadingseveral projects related with QoS pro-visioning and power saving in wire-less networks. In the Wireless LANarea he is managing a team responsibleof designing, configuring and evaluat-ing the QoS and power saving mecha-nisms of NEC’s dual-mode mobile ter-minals (3G/WLAN). In the WiMAX

area Xavier is leading a project focusing on the design of abase station MAC QoS scheduler for NEC WiMAX products.In the wireless multi-hop area he contributes to the EU FP7Carrier-Grade Mesh Neworks (CARMEN) project. Addition-ally, he regularly participates in IEEE 802.11 and Wi-Fi Al-liance meetings. Xavier received NEC’s R&D Award for theQoS configuration of the N900iL 3G/WLAN dual-mode ter-

21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

minal and OPNET’s Significant Technical Challenge SolvedAward. He holds both a telecommunications engineering de-gree and a Telematics Engineering PhD degree from the Poly-technic University of Catalonia (UPC), Barcelona, and receivedfor his PhD the award from the Spanish Official Associa-tion of Telecommunication Engineers (COIT) to the best PhDthesis on ’Multimedia Convergence in Telecommunications’.

22