Lai and Chen EURASIP Journal on Wireless Communications and Networking 2011, 2011:173

http://jwcn.eurasipjournals.com/content/2011/1/173

RESEARCH Open Access

Two-dimensional downlink burst construction in

IEEE 802.16 networks

*Yuan-Cheng Lai and Yen-Hung Chen

Abstract

Several burst construction algorithms for orthogonal frequency division multiple access were proposed. However,

these algorithms did not meet the downlink burst characteristics specified in the IEEE 802.16 standard. This article

therefore proposes the best corner-oriented algorithm (BCO). BCO not only complies with downlink burst

characteristics, but also considers the three issues to obtain high throughput, as follows: BCO maintains all free

slots as a continuous area by constructing each burst in the corner of the available bandwidth area for minimizing

external fragmentation; BCO shrinks the burst area to minimize internal fragmentation, if the requested bandwidth

has been satisfied; and for exploring the continuous subchannels with good channel quality, BCO ensures that the

burst adopts an optimal modulation coding scheme by selecting the excellent corner that can generate the

maximal throughput. The simulation results indicate that BCO achieves 2-9 times the throughput achieved by the

previous algorithms under a heavy load.

Keywords: burst construction, downlink, IEEE, 802.16, OFDMA

1. Introduction attempted to determine the optimal matches between

Because IEEE 802.16 uses the technique of orthogonal bursts and subchannels [3-8].

frequency division multiple access (OFDMA), the band- The IEEE 802.16 standard defines a number of specifi-

width resources are represented by a two-dimensional cations to alleviate the overhead of management mes-

area of slots, in which the two dimensions are time in sages and to concentrate the transmission power on

units of symbols and frequency in units of subchannels specific subchannels for battery-powered devices, as fol-

[1]. Therefore, the bandwidth allocation in IEEE 802.16 lows: (1) the burst must be a continuous bandwidth

must consider the construction of a two-dimensional area, (2) the shapes of the bursts used in downlink and

bandwidth area, called a burst, assigned to a connection. uplink transmissions should be rectangular and multi-

The subchannel diversity should be considered when rectangular, respectively, and (3) one burst should use

constructing bursts. Subchannel diversity means that a only one MCS based on the worst signal-to-noise ratio

connection uses a different modulation coding scheme (SNR) among the assigned subchannels [1,9].

(MCS) on various subchannels because the connection The previous researches that focused on the maxi-

encounters various channel qualities on various sub- mum matching problem violated the specifications in

channels [2]. Therefore, for each connection, each burst IEEE 802.16 standard, and are thus unpractical. There-

must be constructed in its corresponding best-quality fore, a number of researchers regarded the burst con-

struction problem as a variant of the bin packingsubchannels, i.e., the subchannels on which the connec-

problem. So-In et al. [10] designed the enhanced one-tion receives the optimal channel quality to maximize

bandwidth usage. Several algorithms for the IEEE 802.16 column striping with non-increasing area first mapping

burst construction problem were proposed to obtain the algorithm (eOCSA), which constructs each burst from

higher throughput. A number of researchers regarded bottom right to top left of the available bandwidth area.

this problem as a maximum matching problem and Wang et al. [11] developed the weighted less flexibility

first algorithm (WLFF), which constructs each burst on

athe best edge selected in the free bandwidth area. The

* Correspondence: pplong@gmail.com

best edge is the edge on which a constructed burstDepartment of Information Management, National Taiwan University of

Science and Technology, #43, Sec. 4, Keelung Rd., Taipei 106, Taiwan

© 2011 Lai and Chen; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons

Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in

any medium, provided the original work is properly cited.Lai and Chen EURASIP Journal on Wireless Communications and Networking 2011, 2011:173 Page 2 of 18

http://jwcn.eurasipjournals.com/content/2011/1/173

generates the minimal variance of the sub-blocks in the 2. Background

free bandwidth area. Thus, constructing the burst on 2.1. IEEE 802.16 network

this best edge provides the most flexibility for the fol- The IEEE 802.16 network consists of a base station (BS)

lowing burst construction. eOCSA and WLFF conform and a number of subscriber stations (SSs). The BS pro-

to the specifications (1) and (2); however, they comple- vides connectivity, radio resource management, and

tely neglect subchannel diversity and the specification control of SS, which supports the connectivity with the

(3). BS.

A number of issues must be addressed to conform to The two layers in the IEEE 802.16 protocol stack are

the specifications and maximize the throughput. First, the physical layer, which transfers raw data, and the

external fragmentation may occur because the burst MAC layer, which supports the physical layer by ensur-

must be a continuous bandwidth area, which means that ing that the radio resources are used efficiently. The

the total available slots are sufficient to satisfy a burst; three duplex modes in the physical layer with OFDMA

however, the lack of contiguity may prevent their use by are Time Division Duplex (TDD), Frequency Division

this burst. Thus, the external fragmentation should be Duplex (FDD), and Half-duplex Frequency Division

avoided. Second, because of the rectangular shape of a (H-FDD). The TDD is the most attractive

downlink burst or improper slot allocation, internal duplex mode because of its flexibility. In addition, the

fragmentation may occur, which results from a burst modulation methods, that is quadrature phase shift key-

with capacity exceeding the requested bandwidth. The ing (QPSK), 16 quadrature amplitude modulation

internal fragmentation must be minimized because the (16QAM), or 64 quadrature amplitude modulation

unused slots internal to a burst are wasted. Third, (64QAM), and the associated coding rate for data trans-

because one burst must use one MCS based on the mission are selected according to the channel quality,

worst SNR among the assigned subchannels, it must be that is, signal-to-noise ratio (SNR).

constructed in its corresponding optimal block, i.e., a An IEEE 802.16 frame for downlink and uplink trans-

block in which a number of continuous subchannels missions is divided into downlink (DL) and uplink (UL)

have good SNRs. subframes in the time domain of the TDD mode (the

Therefore, this article proposes a one downlink burst right part of Figure 1). A burst is an allocated band-

construction algorithm, called the best corner-oriented width assigned to one dedicated connection of one SS

algorithm (BCO), to maximize the throughput. BCO not and is formed by slots. A slot is the minimal bandwidth

only conforms to the constraints in IEEE 802.16 stan- allocation unit, and consists of one subchannel and one

dards, but also considers these issues. To avoid external to three symbols. A subchannel is the smallest allocation

fragmentation, BCO constructs each burst in a corner of unit in the frequency domain, and a symbol is the smal-

the free bandwidth area to ensure that all free slots are lest allocation unit in the time domain. A number of

within a continuous area. A corner is the intersection of other fields in a frame provide specific functions. For

the horizontal edge and left-hand vertical edge of the example, preamble synchronizes each SS, DL/UL-MAP

free bandwidth area. To minimize internal fragmenta- describes the position and measure of each downlink/

tion, BCO shrinks the area of the burst if the requested uplink burst, and frame control header specifies DL sub-

bandwidth is satisfied to enable unused slots internal to frame prefix and the length of DL-MAP message.

this burst to be used by other bursts. BCO evaluates the In the IEEE 802.16, the SS must acquire bandwidth

channel quality in each corner to explore an optimal from the BS before transmitting or receiving data. On

block, and subsequently constructs the optimal burst in downlink, the BS broadcasts to all SSs, and each SS

the corner in which the burst can provide the largest picks up its destined packets. On uplink, SSs must

throughput. inform the BS of the bandwidth they require for data

This article is organized as follows: Section 2 presents transmission by sending a bandwidth request (BWR).

a discussion of the literature on the IEEE 802.16 net- Upon receiving the BWRs, the BS allocates the bursts in

work, the burst construction in downlink transmission, an uplink subframe to each SS, and subsequently broad-

and related studies. In Section 3, the problem statement casts this information through UL-MAP. After receiving

of the downlink burst construction is formally intro- UL-MAP, each SS uses the allocated burst to transmit

duced, and the issues to solve this problem are pre- its data.

sented. Section 4 provides a description of the proposed Figure 1 demonstrates that, for efficient bandwidth

BCO algorithm in detail. In Section 5, the superior per- use, the BS must consider several factors, including the

formance of BCO in comparison with eOCSA and power saving policy, quality of services (QoS) require-

WLFF is demonstrated by simulation. Finally, conclu- ments, channel quality variation, DL/UL bandwidth

sions and future studies are given in Section 6. ratio, and burst structure. Bandwidth allocation isLai and Chen EURASIP Journal on Wireless Communications and Networking 2011, 2011:173 Page 3 of 18

http://jwcn.eurasipjournals.com/content/2011/1/173

Figure 1 Bandwidth allocation in IEEE 802.16 network.

generally performed in two phases, flow scheduling and to allow a more flexible construction, although the

burst construction, because it is difficult to consider all uplink burst must be constructed with a multi-rectangu-

of these factors in a single step [9]. The objective of lar shape for reducing power consumption of SSs [9].

flow scheduling is to estimate the appropriate number Third, the SS has various levels of SNR on various sub-

of slots to assign to each connection and to subse- channels because of the variable noises on each sub-

quently schedule these connections according to their channel. To minimize the overhead and the complexity

QoS requirements, power saving policy, DL/UL band- of MAC control messages, each burst uses only one

width ratio, and other related factors. Several algorithms MCS based on the worst SNR of all assigned

for flow scheduling were evaluated in the literature (e.g., subchannels.

[12]). In burst construction, however, the burst for each Figure 2 shows an example of the construction of a

connection must be constructed according to the num- downlink burst for a connection with 15 slots allocated

ber of the allocated slots, the burst structure, channel by the flow scheduler. For simplicity, the SNR of each

quality variation, and computational complexity. This subchannel is transformed into its corresponding MCS

study considered the burst construction in the downlink (bytes/slot). A downlink burst can be presented as a rec-

transmission, i.e., downlink burst construction. tangle with a height-width pair (h,w) placed on a start-

ing slot (y,x), which is represented by a row-column

2.2. Burst construction in downlink transmission manner, for example, [(y,x),(h,w)] = [(0,0),(3,5)], as

The downlink burst structure specified by the IEEE shown in Figure 2. The MCS used by this burst is 9

802.16 standard is based on the downlink-partial usage bytes/slot, which is the worst MCS of its occupied sub-

of subchannels (DL-PUSC) method [1], in which the channels, i.e., subchannels 0 to 2.

burst uses partial subchannels in the OFDMA frequency

range. The downlink bursts have three distinct require- 2.3. Related studies

ments. First, the burst must be a continuous area to Because the construction of bursts that can provide the

minimize DL-MAP overhead because DL-MAP is trans- optimal throughput is a NP-hard problem [9], several

mitted at the lowest data rate for robustness (e.g., QPSK algorithms were proposed to raise throughput and were

modulation) and to ensure that all SSs can decode their classified as the max matching solutions and bin packing

embedded contents even under poor channel conditions. solutions. The objective of max matching solutions for

Second, the shape of the downlink burst is a rectangle burst construction is to assign bursts to their best-Lai and Chen EURASIP Journal on Wireless Communications and Networking 2011, 2011:173 Page 4 of 18

http://jwcn.eurasipjournals.com/content/2011/1/173

Figure 2 An example of constructing a downlink burst.

quality subchannels. Therefore, the researchers [3-8] independently use different MCSs. Thus, these burst

transformed this problem into a max matching problem construction solutions make unreasonable assumptions

and attempted to determine the optimal matches and do not comply with the IEEE 802.16 specifications.

between bursts and subchannels to maximize the Burst construction can be regarded as a process of

throughput. Sheu et al. [3] utilized the Hungary algo- placing items of variable heights, widths, and values into

rithm, which is a commonly used combinatorial optimi- a two-dimensional area to maximize the total value of

zation algorithm for the assignment problem with m all items in the area. Thus, the burst construction pro-

connections and m subchannels. Their approach first blem can be regarded as a variant of the bin packing

forms a subchannel assignment matrix, in which each problem, the objective of which is to determine the opti-

row represents one connection and each column repre- mal shape and position of each burst in the bandwidth

area for maximizing the overall throughput of all con-sents one subchannel. The entry in the matrix indicates

structed bursts. However, the traditional studies inthe channel condition with regard to a connection, e.g.,

SNR. The Hungary algorithm is subsequently applied to operational research are not applicable for the burst

determine the optimal connection-subchannel match. construction because they focus on packing objects with

Chen et al. [4] proposed the dynamic frequency selec- fixed shapes and values [13-15]. Thus, a number of algo-

tion approach, in which each connection selects its sub- rithms were proposed [10,11,16-21]. The eOCSA algo-

channel according to the probability distribution, where rithm proposed by So-In et al. [10] constructs the first

the selection probability is determined by channel qual- burst in the bottom right-hand corner of the available

ity. Toufik and Knopp [5] presented a max-min alloca- bandwidth area, and subsequently constructs another

tion policy, which first constructs a matching graph burst if the available bandwidth area above the previous

(from subchannels to connections) and subsequently burst is sufficient. Otherwise, eOCSA subsequently con-

iteratively removes the edge with minimal weight from structs the burst on the left-hand edge of the previous

the matching graph until a perfect match is obtained. If burst. The approaches [16-18] were designed in a

twoormoreconnectionsselectthesamesubchannel, method similar to eOCSA, but with minor modifica-

the probability of selecting this subchannel decreases. tions. Cicconetti et al. [19] further evaluated the internal

All connections subsequently repeat the selection based fragmentation of the burst constructed in different

on the modified probabilities. This process continues directions, that is, vertical direction or horizontal direc-

until each subchannel is only chosen by one connection tion, and subsequently selected the direction that experi-

or until the maximal number of iterations is reached. A enced less fragmentation to construct the burst. Eshanta

number of studies applied greedy methods to allocate et al. [20] also proposed two approaches. One method

the best subchannel to the connection with the highest constructs bursts with the fixed width in a vertical

transmission rate [6-8]. However, as shown in Table 1, direction and the other constructs bursts with the fixed

height in a horizontal direction.these studies assumed that a subchannel is occupied by

only one burst. They also assumed that the subchannels The WLFF [11] constructs the burst on the best edge

assigned to one burst are disjointed and can in the free bandwidth area. The best edge is the edge onLai and Chen EURASIP Journal on Wireless Communications and Networking 2011, 2011:173 Page 5 of 18

http://jwcn.eurasipjournals.com/content/2011/1/173

Table 1 Comparisons among related studies

Author Year Solution Complexity Requested Shape of DL Subchannel

bandwidth burst diversity

4Sheu et al. [3] 2007 Hungary algorithm O(M ) No No Yes

iChen et al. [4] 2006 DFS O(L) No No Yes

3Toufik and Knopp 2004 Max-min allocation O(M ) No No Yes

[5]

Najeh et al. [6] 2005 Greedy O(LM) No No Yes

Kivanc et al. [7] 2003 O(LM) No No Yes

Ergen et al. [8] 2003 O(LM) Yes No Yes

2

So-In et al. [10] 2009 Sequentially construct bursts from one side to O(L ) No Yes No

another

2

Sarigiannidis et al. 2010 O(L ) No Yes No

[16]

Erta et al. [17] 2007 O(LM) No Yes No

Ohseki et al. [18] 2007 O(LM) Yes Yes No

2

Cicconetti et al. 2010 O(L ) No Yes No

[19]

2

Eshanta et al. [20] 2011 O(L ) No Yes No

2

Wang et al. [11] 2008 WLFF O(L ) No Yes No

2

Zubow et al. [21] 2010 GSA O(L ) No Yes No

L, number of connections; M, number of subchannels; i, maximum number of repetition.

which a burst is constructed, and generates the minimal number of slots allocated by the flow scheduler and the

variance of the sub-blocks in the free bandwidth area. requested bandwidth for C, respectively. Although thei

Thus, constructing the burst on this best edge provides flow scheduler estimates A according to the requestedi

the most flexibility for the following burst construction. bandwidth W,italsoconsidersseveralotherfactorsi

The greedy scheduling algorithm [21] was designed in a when performing this estimation. Thus, the throughput

manner similar to WLFF. However, none of the bin provided by A may be lower than W because the flowi i

packing solutions considers subchannel diversity. scheduler does not allocate sufficient slots in the current

Table 1 shows the summary of these methods. The downlink subframe. Conversely, the throughput provided

complexity refers to the time complexity consumed by by A may exceed W because the burst allocator con-i i

the burst construction algorithm. The required band- structs the burst in an excellent block.

width implies that the algorithm not only considers the A two-dimensional matrix R represents the used

allocated slots, but also considers the requested band- MCSs on different subchannels for each connection in

width during burst construction. This is because the order to investigate the effects of subchannel diversity,

bandwidth provided by the allocated slots may exceed where R(i, j) specifies the MCS used by C on the jthi

the required bandwidth of the connection when the subchannel. A downlink subframe is composed of M×N

burst is constructed on good-quality subchannels. slots, where M is the number of subchannels and N is

Therefore, these unused slots can be further utilized by the number of slots within one subchannel.

the other bursts if the algorithm extra considers the A downlink burst can be represented as a rectangle

requested bandwidth. with a height-width pair placed on a starting slot; i.e., a

downlink burst B=[(y, x),(h, w)], where (y, x)and(h,

3. Problem statement w) represent the starting slot and the height-width pair,

This section first defines a number of used notations respectively. Let B be the downlink burst constructedi

and formally states the problem of the two-dimensional for C. In addition, let NOS and MCS denote the num-i i i

downlink burst construction. ber of occupied slots and the MCS adopted by B ,i

respectively. Th is the throughput achieved by connec-i

3.1. Notations tion C,anditsvalueismin(NOS ×MCS ,W ), wherei i i i

A two-phase bandwidth allocation is used, as described in NOS×MCS is the bandwidth that can be supported byi i

Section 2.1. Let C be the set of all downlink connec- B.Whenthe valueof NOS ×MCS exceeds theall i i i

tions, and let L be the number of all downlink connec- requested bandwidth W , connection C only requiresi i

tions, i.e., L=|C |. In addition, let C represent the ith W to transmit its data; therefore, the effective through-all i i

connection after flow scheduling. A and W denote the put is W. All used notations are listed in Table 2.i i iLai and Chen EURASIP Journal on Wireless Communications and Networking 2011, 2011:173 Page 6 of 18

http://jwcn.eurasipjournals.com/content/2011/1/173

Table 2 Used notations

Notation Definition

C The set of all downlink connectionsall

L The number of all downlink connections, i.e., L=|C |all

C The ith connection after the flow scheduling phasei

W The requested bandwidth for C, in terms of bytesi i

A The number of allocated slots for C in the flow scheduling phasei i

M The of subchannels in a downlink subframe

N The number of slots within one subchannel

R The MCS matrix for different connections on different subchannels, where R(i,j) specifies the MCS used by C on the jth subchanneli

B The constructed downlink burst for Ci i

NOS The number of occupied slots by Bi i

MCS The MCS adopted by Bi i

Th Throughput achieved by Ci i

3.2 Problem and Issues constructer constructs each burst on its corresponding

Problem statement: Given a downlink subframe of M×N inferior-quality subchannels and uses a low MCS; the

slots, the set of C (all C,W,and A), and the MCS bandwidth is inefficiently used. An example of optimalall i i i

matrix R, construct all B to maximize the overall block exploration is shown in Figure 3c, in which thei

throughput of C is low when B is constructed in anThi 1 1throughput .

0≤i≤L−1 inferior block (i.e., subchannels 1, 2, and 3), whereas the

Inefficient bandwidth usage must be eliminated to throughput is high when B is constructed in an optimal1

solve this problem. The following issues must be care- block (i.e., subchannels 5 and 6).

fully considered when designing a downlink burst con- 4. Best corner-oriented algorithm

struction algorithm. BCO not only complies with the downlink burst struc-

1. External fragmentation ture specified in IEEE 802.16 standards, but also consid-

A downlink burst with a rectangular shape may cause ers the issues discussed in Section 3.2. To avoid external

external fragmentation. External fragmentation refers to fragmentation, BCO maintains all free slots as a contin-

the division of available slots into small pieces that can- uous area by constructing each burst in the corner. To

not meet burst requirements. Figure 3a shows an exam- minimize internal fragmentation, BCO expands the

ple of a connection C with A = 12 slots. The burst B1 1 1 burst by one slot height in steps. At any step, if the

cannot be constructed because the free bandwidth was throughput of the constructed burst exceeds the

divided into pieces that were too small to accommodate requested bandwidth, the burst is large enough and is

B , although the total free bandwidth was sufficient for1 not further expanded, even when the number of occu-

A .1 pied slots is smaller than the number of allocated slots,

2. Internal fragmentation i.e., NOS <A. To explore an optimal block, BCO con-i i

The number of occupied slots, NOS,mustequalthei structs a virtual burst in various corners, and subse-

allocated number of slots, A , for any connection C .i i quently selects the best corner in which the burst

However, the throughput provided by A may exceed Wi i provides the largest throughput.

when the burst B isconstructedinanoptimalblocki

and thus, has an excellent MCSi.Thiscausesinternal 4.1. Definition of corners

fragmentation, which means that only some slots within BCO avoids external fragmentation by constructing a

a burst are used to transmit data, and the remaining are burst starting from the corner and limiting it by the

wasted. Figure 3b shows an example of internal frag- bounded width and height. The corner, bounded width,

mentationinthat C only uses ten slots to transmit1 and bounded height are formally defined as follows:

data, and the remaining two slots are wasted. given the available bandwidth area before constructing

3. Optimal block exploration the ith burst, the edge set, E, surrounding this area in ai

The SS experiences various levels of SNR on different counterclockwise order is defined by

subchannels resulting from variable noises on each sub- j j jJ J0 0 1 1E = {H ,V ,H ,V ,...,H,V ,...,H ,V },where Hi i i i i i i i i ichannel. The burst must be constructed in its corre-

j

and are the jth horizontal and vertical edges, respec-sponding optimal block, i.e., a block in which a number Vi

of continuous subchannels have excellent SNRs, and jtively. The corner, is defined as an available slot,CRithus,itcanuseasatisfactoryMCS.Thus,iftheburstLai and Chen EURASIP Journal on Wireless Communications and Networking 2011, 2011:173 Page 7 of 18

http://jwcn.eurasipjournals.com/content/2011/1/173

Figure 3 Examples of issues by constructing B with A =12 slots and W =270 bytes: (a) External fragmentation; (b) Internal1 1 1

fragmentation; (c) Optimal block exploration.Lai and Chen EURASIP Journal on Wireless Communications and Networking 2011, 2011:173 Page 8 of 18

http://jwcn.eurasipjournals.com/content/2011/1/173

j temporary burst, B , with a possible height-width pairtmpwhich is the intersection of H and left-hand verticali

and calculates the throughput that this burst can pro-jkedge V of . The corresponding bounded width andHi i vide. The steps are listed as follows:

j jk Initialization: h = 1// set initial heightheight are defined as H and V ,where H andii i

Step 1: Determine the width w for h by considering k j k jV denote the lengths of and V , respectively.Hi ii Ai, Wi, and the width H . i

Therefore, constructing a burst in the corner indicates jStep 2: B =[(y, x)(h, w)], where . In addi-(y,x)= CRtmpj ithat one of the vertices of the burst lies in ,andtheCRi tion, calculate the throughput of B . tmp

j bestwidth and height of this burst are restricted by H and Bi Step 3: Record the optimal burst with the opti-tmp

k mal height-width pair obtained thus far.V , respectively. Figure 4a demonstrates that three cor-i

Step 4: h = h+1;ners are located on slot(0,4), slot(3,0) and slot(7,0) at

kconstructing the ith burst, and their corresponding If h ≤V , go to step 1.i

(height, width) pairs are (3,4), (5,4), and (5,8), respec-

bestWhen the loop ends, B provides the optimaltmptively. Figure 4b presents an example of constructing

j1 throughput among all B virtually constructed in .burst B in the CR . CRtmpi i i

Lemma: Provided with a downlink subframe of M×N In Step 1, A and W were used to calculate the widthi i

slots and number of connections, L, the available band- when the height was given, to alleviate internal fragmen-

width area is continuous if each downlink burst is con- tation. BCO first calculated the width w where (w ×h)1, 1

structed in the corner. was equal to the allocated slots A. BCO calculated thei

width wProof: Mathematical induction is applied to prove the that the throughput provided by the burst2

(w ×h) to satisfy the requested bandwidth W.Subse-claim. For L = 1, which indicates that only one burst is 2 i

jrequired to be constructed, the free slots are maintained quently, BCO used the minimum of w , w,and H as1 2 i

as a continuous area after this burst is constructed in

the width. This is because if w is the minimum, con-2 j j kand limited by H and V .CR 0 00 structing a burst with a larger width w will exceed the1

requested bandwidth, resulting in internal fragmenta-Suppose that all free slots are maintained as a contin-

juous area when L = s.When L = s+1,the(s+ 1)th tion. In addition, H, as the minimum, indicates thatijburst is constructed in one of the corners (i.e., )CRs+1 the available bandwidth area located in this corner with

j k the height h is insufficient to accommodate a burst withand limited by the corresponding H and V .s+1 s+1

A slots. Therefore, the burst should be shrunk by usingij Constructing burst in maintains this burst adja-CRs+1 j

H as its width. The exact calculations of w and w1 2icent to other constructed bursts. In addition, limiting

are described in the following section.j

the burst by H prevents the horizontal division ofs+1 Furthermore, examining each possible height of a

the continuous free bandwidth area. Conversely, con- burst can avoid the phenomenon of throughput anom-

j k aly. The throughput anomaly indicates that a burst withstructing burst in and limiting it by V pre-CR s+1s+1 a large height may anomaly cause lower throughput

vent the vertical division of the continuous free than a burst with a small height when the burst with a

bandwidth area. Consequently, the free slots, after con- large height uses an inferior MCS. Figure 5 shows an

structing the (s+ 1)th bursts, are not divided and are, example in which the throughput provided by the burst

therefore, maintained as a continuous area. Thus, by the B(h = 3), referring to the burst with height 3, is consid-

mathematical induction, the available bandwidth area is erably lower than that provided by the burst B(h=2)

always a continuous area. because B(h = 3) used an inferior MCS, although B(h =

3) is larger than B(h = 2). In this case, a burst with a

4.2. Burst construction small height that provides large throughput should be

BCO minimizes the internal fragmentation by exploring constructed to avoid slot waste.

the optimal height-width pair of the burst constructed

j

in the selected . The optimal height-width pair indi-CR 4.3. Pseudo code of the BCO algorithmi

Figure 6 shows the pseudo code of BCO. To constructcates that the burst with thispairprovidestheoptimal

burst B for each connection C,BCOfirstusesthethroughput or the smallest area. To obtain the optimal i i

FindCorner function to obtain CRList, which containsheight-width pair, BCO repeatedly constructs aLai and Chen EURASIP Journal on Wireless Communications and Networking 2011, 2011:173 Page 9 of 18

http://jwcn.eurasipjournals.com/content/2011/1/173

j j kFigure 4 An example of constructing a burst in the corner. (a) An example for explaining , and ; (b) Construct the burst B inCR H V iii i1 with eight slots.CRiLai and Chen EURASIP Journal on Wireless Communications and Networking 2011, 2011:173 Page 10 of 18

http://jwcn.eurasipjournals.com/content/2011/1/173

Figure 5 An example of throughput anomaly. (a) Construct B(h = 2); (b) Construct B(h = 3).

j bestthe corners from the available bandwidth area. The subsequently compares with B to determineB ii

FindCorner function returns the CRList by examining

which is superior, i.e., which has higher throughput or

the horizontal and the vertical edges of the available

which occupies the fewer slots under the same obtained

bandwidth area. BCO subsequently explores the optimal j jbestthroughput. If is superior, BCO sets B to .B Bicorner by virtually constructing the burst in each corner i i

jto address the optimal block exploration (line 6-13), i.e., After virtually constructing all and obtaining the bestBi

BCO repeatedly invokes the ConstructBurst function to best bestburst B , BCO constructs B as B .ii ij jvirtually construct a burst in the corner .BCOB CRi i