用於WiMAX系統中以最大長方形為基礎之下行鏈路資料叢集分配演算法

國立交通大學

電機資訊國際學位學程

碩士論文

用於 WiMAX 系統中以最大長方形為基礎之下

行鏈路資料叢集分配演算法

Maximum Rectangle-Based Down-Link Burst

Allocation Algorithm for WiMAX Systems

研究生：Jimmy Yau Zhong, 邱家俊

指導教授：李程輝博士

用於 WiMAX 系統中以最大長方形為基礎之下

行鏈路資料叢集分配演算法

Maximum Rectangle-Based Down-Link Burst

Allocation Algorithm for WiMAX Systems

研究生：邱家俊

Student: Jimmy Yau Zhong

指導教授：李程輝

Advisor: Tsern-Huei Lee

國立交通大學

電機資訊國際學位學程

碩士論文

A Thesis

Submitted to EECS International Graduate Program

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master

June 2011

Hsinchu, Taiwan, Republic of China

中華民國一百年六月

用於 WiMAX 系統中以最大長方形為基礎之下

行鏈路資料叢集分配演算法

研究生：邱家俊

指導教授：李程輝

國立交通大學

電機資訊國際學位學程

摘要

著名的行動通訊規格

802.16e WIMAX 在近年來開始被採用。因為其有著

高資料傳輸量和佈建的低成本，漸漸變成無線寬頻服務的主流方法。

WiMAX 使用 OFDMA 當作主要的傳輸方法。而從規格書上可知，一個資

料突衝需要被規畫成一個矩形才能放在以時間和頻率所組成的下載訊框裡，

但是如何設定矩形的形狀來減少浪費的空間並沒有詳細的介紹。近幾年不

少人提出了許多突衝規畫的演算法。在本篇論文裡，我們介紹了一個高效

率包裝的演算法，其能夠達到高傳輸資料量、減少未使用的槽、並且最小

化其規畫時所需的額外資訊。我們將提出的演算法和

eOCSA 還有 OBBP

這些之前提出來的演算法做比較。由數據結果可知我們的演算法較佳。

Maximum Rectangle-Based Down-Link Burst

Allocation Algorithm for WiMAX Systems

Student: Jimmy Yau Zhong

Advisor: Dr. Tsern-Huei Lee

EECS International Graduate Program

National Chiao Tung University

ABSTRACT

The IEEE 802.16e standard known as Mobile WiMAX has recently been introduced.

Because of the high data throughput and low cost deployment; it becomes one of the leading solutions for wireless broadband services. Mobile WiMAX makes use of

Orthogonal Frequency-Division Multiple Access (OFDMA) digital modulation

scheme as the transmission method. It is known from the standard that data burst in the

downlink subframe needs to be mapped into a time and frequency domain with a

rectangular shape, but how the data burst are placed in to minimize the wasted space

so that more data can be sent is not detailed. Many burst mapping algorithms have been proposed by researchers in these late years. In this thesis, we present an efficient

packing algorithm which achieves high throughput, reduces the number of unused

slots, and minimizes the mapping information overhead. We compare the performance

of our proposed algorithm with that of eOCSA and OBBP, high-performance packing

algorithms recently presented by other researchers. Numerical results show that our

Acknowledgement

This dissertation would not have been possible without the guidance and the help of

several individuals who in one way or another contributed and extended their valuable

assistance in the preparation and completion of this study.

First and foremost I want thank my parents, two sisters, and girlfriend for all their love, encouragement, and faithful support.

I would like to express my deepest gratitude to my advisor, Dr. Tsern-Huei Lee, for his

guidance, caring, and patience. Besides my advisor, I would like to thank Dr.

Yaw-Wen Kuo and Chi-Hsien Liu, whom were involved in this research work.

Thanks to my fellow lab mates and team members I have had the pleasure to work with. Special thank goes to Arleth Garth, a friend who stood by me through the good

and bad times.

Last but not the least; I also would like to make reference to Ms. Angel Yu, secretary

of the office of EECS International Graduate Program, for all the help and student

Table of Contents

1.1 OFDMA slots and bursts……….. 2

1.2 OFDMA Frame………... 4

Chapter 2: Problem Statement……….. 6

Chapter 3: Related Works………. 10

3.1 One Column Striping with non-increasing Area first mapping (OCSA)… 11 3.2 Enhanced OCSA (eOCSA) ………. 12

3.3 Orientation-Based Burst Packing (OBBP) ……….. 15

Chapter 4: Proposed Resource Allocation Algorithm………... 17

4.1 Maximum Rectangle-Based DL Burst Allocation………... 17

4.2 Enhanced Maximum Rectangle-Based DL Burst Allocation………. 25

Chapter 5: Performance Evaluation……….. 27

5.1 Simulation……… 27

5.2 Performance Analysis……….. 30

Chapter 6: Conclusion………... 35

References………. 36

List of Tables

Table 1. Fixed Versus Mobile WiMAX……….. 1

Table 2. Definition of OFDMA slot……… 4

Table 3. Ten random resource allocations example………... 23

List of Figures

Figure 1. Example of an OFDMA slot……….. 2

Figure 2. Example of an OFDMA frame structure in TDD mode……….... 5

Figure 3. IEEE 802.16 QoS architecture……….. 6

Figure 4. OCSA downlink burst mapping example………. 12

Figure 5. Example of resources allocated by OCSA algorithm and eOCSA Algorithm………... 13

Figure 6. Example of an orientation factor matrix by OBBP algorithm…………... 15

Figure 7. OBBP packing set rearrangement……….. 16

Figure8.Process for finding maximum rectangle area………... 18

Figure 9. Process for updating elements in set L……….. 22

Figure 10. Example of resources allocated by the proposed algorithm……… 24

Figure 11. Example of resources allocated by eOCSA and OBBP……… 25

Figure 12. Average efficiency vs. number of MS……….. 28

Figure 13. Average unused slots vs. number of MS……….. 29

Figure 14. Average over allocation slots vs. number of MS……….. 30

Figure 15. Illustration of the wasted slots of eOCSA algorithm……….... 33 Figure 16.Illustration of wasted space produced by mapping burst column wise…. 34

Chapter 1:

Introduction

Wireless communication systems have widely evolved in the recent years, making

WiMAX (Worldwide Interoperability for Microwave Access) one of the most

promising and popular networking infrastructures Broadband Wireless Access (BWA).

Each of the two WiMAX classifications, Fixed and Mobile, are based on a subset of

the IEEE 802.16 standards and are defined by the WiMAX forum. More specifically, Fixed WiMAX is based on the orthogonal frequency division multiplexing (OFDM)

physical layer of the 802.16-2004 specifications. While, Mobile WiMAX is based on

the orthogonal frequency division multiple access (OFDMA) physical layer of the

802.16e-2005 standard, which is a revision of the original Fixed WiMAX standard.

Table 1 shows a side-by-side comparison of the Fixed and Mobile WiMAX standards.

Fixed WiMAX Mobile WiMAX Standard IEEE 802.16-2004 IEEE 802.16e-2005

Multiplexing OFDM OFDMA

FFT size 256 Scalable (512, 1024, and so on)

Duplexing mode TDD, FDD TDD

Modulation scheme BPSK, QPSK, 16-QAM, and 64-QAM

QPSK, 16-QAM, and 64-QAM

Subcarrier spacing 15.625, 31.25, 45 kHz 10.94 kHz

Signal bandwidths 3.5, 7, and 10 MHz 5, 7, 8.75, and 10 MHz

Spectrum 3.5 and 5.8 GHz 2.3, 2.5, and 3.5 GHz

Within air interfaces defined in the IEEE 802.16 standard, OFDMA has shown to

reduce multi-path fading, while offering simultaneous access to multiple users to the

wireless medium. Longer transmission distances, higher data rates and better mobility

support are the main advantages of using this multicarrier technology.

The basic idea behind OFDMA is to divide the available frequency spectrum into several orthogonal carriers (often called subcarriers or tones). Due to this, inter-symbol

and inter-carrier interference are mitigated. Also, these subcarriers can be dynamically

assigned to different users for transmission, both in frequency and time dimensions.

This flexibility provides a way of boosting system performance, at the cost of

increasing the challenge of resource allocation.

1.1 OFDMA Slots and Bursts

In the mobile WiMAX specification, an OFDMA slot is the minimum possible data

allocation unit. It is defined in two dimensions, one in time (symbol number) and the

other in frequency (subchannel number) as illustrated in figure 1.

The definition of an OFDMA slot depends on the subchannelization or subcarrier

permutation used:

 Distributed permutation, logical subchannels are built from physically

distributed subcarriers along the available frequency spectrum. This channel

diversity reduces the effect of fast fading on mobile environments. Also, the

averaged Signal-to-Noise-Ratio (SNR) is practically the same for all subchannels (from the point of view of one user), so all subchannels are equally

adequate for all transmitters. The subchannelizations based on distributed

permutation includes DL PUSC (Downlink partial usage of subchannelization),

DL FUSC (Downlink full usage of subchannelization), and UL PUSC (Uplink

partial usage of subchannelization).

 Adjacent permutation, subchannels are built from physically adjacent

subcarriers. This leads to user diversity, due to the fact that users will get very

different SNR measures for different subchannels. The subchannelization based

on adjacent permutation includes DL AMC (Downlink adaptive modulation and

coding) and UL AMC (Uplink adaptive modulation and coding) subchannels.

In the DL FUSC, each slot is composed of a subchannel and an OFDMA symbol. For

the DL PUSC, one slot is one subchannel by two OFDMA symbols. And for the UL PUSC each slot is composed of a subchannel by three OFDMA symbols. However, in

the case of the DL/UL AMC, one slot is one subchannel by three OFDMA symbols.

For all the cases, an OFDMA slot contains 48 data subcarriers. Table 2 lists a

Subchannelization scheme Number of subchannels per slot Number of symbols per slots DL FUSC 1 1 DL PUSC 1 2 UL PUSC 1 3 DL/UL AMC 1 3

Table2: Definition of OFDMA slot

The OFDMA symbols generated out of a user terminal are mapped into a

two-dimensional data block called burst. Burst, in general terms, is a data region that is a

two-dimensional data block consisting of a group of contiguous OFDMA slots.

1.2 OFDMA Frame

The OFDMA frame structure is composed of “DL/UL data bursts”, which carry user data traffic; “Preamble”, which is used by the PHY layer for synchronization and

equalization; “Frame control header (FCH)”, which describes the length of the

MAP message, the repetition encoding, and the channel coding applied to the DL-MAP; “DL/UL MAPs”, which contains the number of burst and their locations (offset

and lengths in time-frequency domain); “Channel quality information channel (CQICH)”, which is used for DL CINR report and MIMO mode selection feedback; “ACK channel”, which provides feedback for DL HARQ; and “Ranging subchannel”,

which is allocated to MSs (Mobile Stations) for initial ranging, periodic ranging,

handover ranging, and bandwidth request.

Mobile WiMAX employs Time Division Duplex (TDD) technology, where the same

period, with a gap (TTG & RTG) following each period. Frequency Division Duplex

(FDD) technology can also be supported. FDD use different frequencies for uplink and

downlink. Figure 2 illustrate an example of OFDMA frame structure for TDD.

Chapter 2:

Problem Statement

In Mobile WiMAX networks, the BS is responsible of performing most of the system

decisions. From the perspective of QoS, these decisions include: call admission control,

scheduling and resource allocation as shown in figure 3.

Figure 3: IEEE 802.16 QoS architecture

New connections are accepted or rejected by the Call Admission Control (CAC) based on the available system capacity, in order to assure that the presence of a new

connection will not impair the QoS requirements of the existing ones. After

connections are accepted, the network traffic must be properly prioritized according to

a certain scheduling policy. This scheduling policy should be able to guarantee that

process may be needed, which is resource allocation. Network traffic must be logically

mapped into a time-frequency matrix before it is actually transmitted through the

wireless medium. After this step is done, some resource request may remain unmapped,

so it must be returned to the scheduler for its transmission in a later frame.

Although the scheduler is the main factor affecting QoS fulfillment, the resource allocation step may impair it. This can happen if the prioritization established by the

scheduler is not fully respected. Also, a bad resource allocation policy may severely

reduce the utilization of the spectral resources.

A burst is an allocation for transmitting data aimed to one or more MSs. Some

restrictions are imposed to the way the resource allocator maps the scheduler traffic. First, data from different MSs being sent in a burst must be transmitted using the same

Modulation and Coding Scheme (MCS). Depending on the channel quality perceived

by a certain user, it may be needed or preferred to use an MCS or another. For

example, an SS which is far away from the BS may need to use a reliable scheme

(such as QPSK-½); while another one with good channel conditions would use a more

efficient scheme (such as 64-QAM-¾). Obviously, if two MSs will transmit using the

same MCS, they will also be able to use the same burst.

The shape of a single burst is mandatorily rectangular and cannot overlap each other.

Burst dimension is specified in the DL-MAP IE with its symbol offset, subchannel

In order to design an efficient DL mapping algorithm, there are some factors to take

into account, which are MAP overhead, unused slots, over-allocated bursts, and QoS

preservation.

When a new burst is inserted into the frame, the respective IE must be added to

DL-MAP. Obviously, the MAP uses slots which could be otherwise used to transmit data. This means that there is a certain MAP overhead directly proportional to the number of

bursts in a frame. Also note that to one MS, multiple bursts can be assigned. However,

such an allocation increases the DL-MAP overhead. It is also possible to put the data

of multiple MSs into a single burst. Thus, reducing the number of bursts may leave

more slots for sending data.

Due to the rectangular shaping restrictions, it is possible that some slots of the frame

may not get assigned to any burst. These unused slots are considered as a waste of

bandwidth, so the mapper should minimize its number. On the other hand, some slots known as “over allocated” may be also internally wasted in a burst because more slots

than required are given to a MS to fulfill with the rectangular shape of the burst. In this

work, we assume that only one burst is allocated to each MS and a burst is never

shared by multiple MSs.

The overall performance of the system will depend on the combined efficiency of the

QoS scheduler and DL-MAP packing algorithm which try to maximize the radio

In a WiMAX system, QoS provisioning depends not only on the QoS scheduler but

also on the DL-MAP packing algorithm. Because of the rectangular shape of the bursts

in the WiMAX downlink subframe, it could happen that no solution is found that can

pack all the data in rectangular bursts with the available capacity. Moreover, a packing

algorithm should be able to adapt to packing constraints and still be reasonably

efficient. Because of design considerations some kind of rectangular shapes or positions in the WiMAX frame might be preferred, e.g., allocations in the subchannel

dimension due to power consumption reasons.

A common length of a WiMAX frame is 5ms considering both downlink and uplink

subframes. In this case, 5ms is the maximum time that a WiMAX base station has for

taking decisions on a per frame basis, in order to maximize performance. Thus, the computational load required by both the QoS scheduler and DL-MAP packing

algorithm should be kept as low as possible to meet this tight deadline without

Chapter 3:

Related Works

In this section, we briefly review some of the other burst mapping algorithms for

Mobile WiMAX. Basically, the downlink scheduler generates a set of requests to be

allocated, which is determined based on current backlog, QoS parameters (delay and

throughput), and available capacity. Then, a burst mapping algorithm allocates these

requests in a rectangle form.

Several algorithms have been proposed recently to address the burst mapping problem

[2]-[13]. For example, in the algorithm proposed by Ohsekiet. al. [2], data of multiple MSs having same physical mode are placed into “bucket”, a defined resource unit in

the vertical direction (subchannels). Multiples buckets are assigned in case a single

bucket is not sufficient to contain the data. Buckets with same length and physical

mode are put together to form a burst.

In the algorithm presented in [6], separate queues are provided for delay tolerant users

and delay sensitive users. Delay tolerant users are not served until they have sufficient

data to fill an entire row of slots in the time domain. If a delay sensitive user is picked

by the scheduler, as many slots as needed are allocated to serve all its queued data.

the user with larger request that can fit into this space is mapped. This algorithm may

end up needing more rows than available and some signaling overhead.

Work in [7], provides a study of the impact of the performance of a WiMAX DL-MAP

packing algorithm on the overall system performance and set an upper bound on the

maximum achievable resource usage by a DL-MAP packing algorithm. The OCSA, eOCSA and OBBP algorithms proposed in [7]-[10], respectively, are related to our

work and, therefore, will be reviewed in depth.

3.1 One Column Striping with non-increasing Area first mapping (OCSA) [7]

The One Column Striping with non-increasing Area first mapping (OCSA) algorithm

consists of three steps. First, a set of resources to be allocated is obtained from the downlink scheduler, based on its demand, QoS parameters (delay and throughput) and

available capacity. Then, the algorithm will sort this set of resources in the descending

order.

In the second step, the resources to be allocated are mapped into the DL subframe

from right to left and from bottom to top as shown in figure 4. The algorithm selects

the largest resource to map and checks if it can fit into the available space in the subframe. Because a burst can have many possible width-height combinations [7], the

algorithm selects the combination with the smallest width or in a second case, the

Figure 4: OCSA downlink burst mapping example

In the third step, the OCSA algorithm will search for the largest resource to be

allocated that can fit into the space (or strip) left on top or above the burst mapped in

the previous step 2. In this step, the combination with the least height is selected. This

process is repeated until no further allocation can be done in the strip or if there is no

resource that can fit. After this, the algorithm moves leftward to fill the remaining

empty columns in the DL subframe and repeats from step 2. Remaining resources after the algorithm terminates are sent back to the scheduler to be mapped in a future frame.

Figure 5 (a) shows an example of resources allocated by the OCSA algorithm. In this

example, the set of resources to be allocated are {92, 76, 65, 48, 25, 20, 17, 10, 5, 2}.

3.2 Enhanced OCSA (eOCSA) [7]

This algorithm is very similar to its predecessor (OCSA) [7]. The resources to be

allocated are also mapped into the DL subframe from right to left and from bottom to top, but this time, a few enhancements are added to make it more efficient.

Figure 5: Example of resources allocated by (a) the OCSA algorithm and (b) the eOCSA algorithm

Same as in the OCSA algorithm, the first step is to sort in the descending order the set

of resources to be allocated that were obtained from the downlink scheduler. In the

second step, the eOCSA algorithm selects the biggest resource to be allocated and

calculates W*A H/  and H*A W/ * , where A is the area of the resource to be

allocated, H is the maximum height that can be used for allocation, and W*andH* are,

In the third step, the algorithm looks for the next largest resource to be allocated that

can fit into the space left on top of the burst mapped in the previous step 2. But here,

the widths of all following bursts allocated in the strip are fixed to that of the burst

allocated in the previous step. In other words, the height of each burst allocated in the

strip is computed as the smallest integer greater than or equal to the area of the burst

divided by the fixed width. This process is repeated until no space can be allocated or if there is no resource that can fit into this space. After this, the algorithm moves

leftward to fill the remaining empty columns in the DL subframe and repeats from step

2. Figure 5(b) shows an example of the eOCSA algorithm handling the same resources

allocated by the OCSA algorithm illustrated in Figure 5(a). One can see that eOCSA

has a better performance than OCSA for this example. The OCSA algorithm leaves 38

empty slots and over allocates 10 slots. The numbers for the eOCSA algorithm are 6 and 11, respectively.

Note that most of the bursts allocated by the OCSA and eOCSA algorithms have the

least widths, which implies that the active time and consequently, the energy

consumption, of each MS is minimized. Besides, by considering mapping the larger

resources first, utilization of downlink resources can be maximized. Our proposed

algorithm presented in the next chapter also adopts these ideas. However, we do not restrict allocation of resources in a strip, after it is created. Instead, whenever a burst

is to be allocated, our algorithm finds the largest rectangle in the remaining

3.3 Orientation-Based Burst Packing (OBBP) [10]

The principle of OBBP algorithm is to group up the resources based on the orientation

factor (OF) and pack the bursts of each group in column-wise or row-wise in the DL

subframe. The algorithm divided into three stages. First stage is a pre-packing stage to

prepare and classify the requests. A process called OF calculation will give the

possible dimension for each rectangle. For example, given a set of resource request B = {10, 6, 8, 5}, the orientation factors for this set will be {10x1, 1x10, 2x5,5x2}, {6x1,

1x6, 3x2, 2x3}, {8x1, 1x8, 4x2, 2x4}, and {5x1, 1x5}. If a request has one or more

OFs out of the DL subframe range, those out of range OFs are removed. And if the

request has no OF within the DL subframe range, the size of this request is increased

by 1 and OF calculation is performed again until at least one within range OF is found.

In the second stage, the resources are classified according to their common number of

symbols (vertically) or subchannels (horizontally) in a matrix generated from the OFs

obtained on the first stage. Figure 6shows an example of the OF matrix produced using

Then, packing sets are selected by calculating the maximum sum of the elements in

each columns or rows of the OF matrix. From this set, bursts are going to be placed in

a descending order. Later, the algorithm rearranges the packing sets by sorting them in

descending order so that burst packing starts from the bottom-right corner and grow in

the two dimensions, leaving the unallocated slots at the top-left corner, as shown in

Figure 7.

Figure 7: OBBP packing set rearrangement (a) Before and (b) After

In the third stage, overlapped rectangles with its maximum possible dimensions are

made with the unallocated slots. The remaining set of resources is sort in descending

order and an optimum rectangle for the first resource that has the minimum difference between the rectangle area and burst size is selected for packing remaining burst.

Chapter 4:

Proposed Resource Allocation

Algorithm

4.1 Maximum Rectangle-Based DL Burst Allocation

This section presents the proposed two-dimensional rectangular burst mapping algorithm. For convenience, define the coordinate of the bottom-right corner of the

downlink subframe as (1,1). Assume that there are s slots and c subchannels in the downlink subframe. Let Ωbe the set of all slots in the subframe and Ω = {(x,y)|1 ≤ x

≤ s, 1 ≤ y ≤ c }. The algorithm allocates slots to a request iteratively, and U denotes the set of current unallocated slots. In each iteration, a rectangle r in U is allocated to a selected request. r is denoted by [(x0,y0); w, h] where (x0,y0), w, and h are its bottom-right corner, the width , and the height, respectively. Initially, U = Ω and the request set with N elements is denoted by Ai= {𝐴𝑖}𝑖−1𝑁 .The proposed algorithm consists of

three phases as follows.

First phase: finding max rectangle

In this first phase, we will basically look for the rectangle which gives the maximum

area in U that will be used to allocate the burst and will be denoted by R= [(x0,y0); W,

In order to find the largest rectangle in U, a set L ⊂ Ω is maintained during the mapping process, which is used to record the bottom-right corners of candidate

rectangles. It is clear that, initially, we will only have L = {(1,1)}and it is obvious that after mapping the first burst, there could be many possible rectangles.

Now, the question is: how can these bottom-right corners will help to find the largest

rectangle? From these bottom-right corners, we will extend to the left and up all the

possible until there is no more unallocated space that can be used, and among these

created rectangles, the one with the maximum total area is selected to allocate the burst.

Figure 8, illustrates an example of the process for finding the maximum rectangle area for burst allocation, where we can see the resulting two candidate rectangles after the

first burst is allocated, a red one and a blue one, denoted as R1 and R2 respectively.

Notice that the bottom-right corners of each rectangle, which are the elements in L, are

also shown. And that the blue rectangle gives the maximum area for allocation

between these two rectangles.

The following pseudo code is the procedure used on the first phase to find the largest

rectangle in U.

1 1 2 2

Find the maximum rectangle in

Given {( , ), ( , ), , ( , )} [ 1, 1] 0 [ 1] 0 [ 1] 0; [ 1] 0 For 1 to do{ For to 1 do{ If ( , ) then [ , n n w h U L x y x y x y height s c area n R n R n i s j c i j U height i               ] [ , 1] 1 else [ , ] 0 } } For 1 to do{ ( , ) ( , ) 0 While ( , ) do{ min( , [ , ]) ( , ) ( 1, ) If [ ] then{ [ ] [ ] temp temp k k temp temp temp temp temp temp temp temp

temp k w j height i j height i j k n x y x y w h c x y U h h height x y x y x y w x x area k w h area k w h R k                   [ ] } } If [ ] [ 1] then _ e [( , ); [ ], [ ]]

/ /Record the maximum rectangular } return _ e h k k w h w R k h area k area k Max r c x y R k R k Max r c     

Second phase: burst allocation

In this second phase, after finding the maximum rectangle, the selection of the largest

element in A for mapping, say Ak, such that Ak ≤ W∙H is the next step, or in other words, the largest resource request that can fit into this maximum rectangle is selected

for mapping. Remember that W and H stand for the maximum rectangle‟s width and height respectively.

Once the selected the resource request for mapping, the dimension of the burst is

defined. Set its width equal to w, where w has the minimum possible width and thus, minimizing MS power consumption. According to the following equation (H stands for the maximum rectangle‟s height):

w = 𝐴𝑘/ 𝐻

Set its height, denoted as h as follows:

h = 𝐴_𝑘/ 𝑤

After the dimension for the burst is defined, allocation of the burst or rectangle r = [(x0,y0);w, h]for request Ak to the downlink subframe begins. The location of the burst inside the downlink subframe will depends on the maximum rectangle. The bursts are

The algorithm terminates if there is no resource request that can fit into the available

space of the selected rectangle or we can say that, when no such Ak exists, the algorithm ends.

Third phase: update L

Now that the burst has been placed into the downlink subframe, in this third phase,

resource request Ak is removed from the set of resource request A and the set of

bottom-right corners L needs to be updated, so that the maximum rectangle can be found properly once the algorithm goes back to the first phase.

In order to update L, assume that r = [(x0,y0); w, h]is allocated in the second phase.

The set of bottom-right corners L is updated as follows. First, any element (𝑥, 𝑦) ∈ (𝐿 ∩ 𝑟)is removed out ofL. In simple words, the bottom-right corner of the just allocated burst is removed from L. Second, search for possible candidates from (x0

+w,y0+h)to the right. Any element (x,y0+h) is added to set L if there existsℎ′ > 0

such that [ 𝑥, 𝑦₀ + ℎ); 𝑥₀ + 𝑤 − 𝑥, ℎ′_{] ⊂ 𝑈and[ 𝑥 − 1, 𝑦}

0 + ℎ); 𝑥0 + 𝑤 − 𝑥 + 1, ℎ′] ⊄

𝑈.Then the algorithm, again search for possible candidates, but this time, from (x0

+w,y0+h) to bottom. Any element(x0+w,y)is added to L if there exists𝑤′ > 0 such

that[ 𝑥₀ + 𝑤, 𝑦); 𝑤′, 𝑦₀ + ℎ − 𝑦] ⊂ 𝑈 and [ 𝑥₀ + 𝑤, 𝑦 − 1); 𝑤′, 𝑦₀ + ℎ − 𝑦 + 1] ⊄ 𝑈.

Continuing the previous example in figure 8, figure 9 illustrates how this third phase works. Notice that a new burst was allocated. The algorithm start from point (x0 +w,y0

+h) following each arrow directions, looking for the maximum length that can be reached and are this points were the bottom-right corner (the others two red points on

figure 9) are defined as elements in L. The figure also shows that any element already in L is skipped to avoid duplication.

Because the maximum rectangle is used for allocation in each iteration, we name the

proposed algorithm the maximum rectangle-based DL burst allocation algorithm.

Figure 9: Process for updating elements in set L

Before leaving this subsection, we present an example to explain the operation of the

There are ten MSs with request set A as shown in Table 3 and the sum of all requests is equal to 360. A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 Resource (slots) 38 8 86 32 41 3 115 7 17 13 Mapping: (Width, Height) 2, 19 8, 1 3, 29 4, 8 2, 21 3, 1 4, 29 1, 7 1, 17 1, 13 Over Allocation 0 0 1 0 1 0 1 0 0 0

Table 3: Ten random resource allocations example

Initially, we have U = Ω and L = {(1,1)}. Consider the first iteration. The largest rectangle in U is [(1,1);12,30] and the largest request smaller than 12x30=360 isA7 =

115. According to the second phase, the rectangle [(1,1);4,29] is allocated toA₇. The

over allocation (shown in black) would be of only 1 slot. And the set L is updated as L = {(1,30),(5,1)}.

In the second iteration, the maximum rectangle in U is [(5,1);8,30] and the largest one in the updated request set smaller than 8x30=240 isA₃= 86. According to the second

phase, the rectangle [(5,1);3,29] is allocated toA₃ with 1 slot of over allocation. The

set L is updated again, giving L = {(1,30),(8,1)}.

In the third iteration, [(8,1);5,30] is the maximum rectangle in U and is used for allocation. The rectangle [(8,1);2,21] is allocated to A₅ with 1 over allocated slot. After

The algorithm is executed for six more iterations to sequentially allocate rectangles to

requests A₁, A₄, A₉, A₁₀, A₂, and A₆.

The result is shown in Figure 10. For this example, there is no allocation to request A8.

The total number of over allocated slots is 4 and the total number of unused slots is 3.

The packing efficiency (percentage of slots used) of the proposed algorithm is 98.06% with over allocated slots and unused slots being counted as wasted. Figure 11

illustrates the packing results for eOCSA and OBBP using the same ten random

resource allocations used in the previous example of the proposed algorithm.

Figure 11: Example of resources allocated by eOCSA and OBBP

4.2 Enhanced Maximum Rectangle-Based DL Burst Allocation

Slight changes have been made to the maximum rectangle-based burst allocation

algorithm while remaining quite similar. Furthermore, we no longer select only the

maximum rectangle, but consider all possible rectangles to map the request in order to

reduce over allocated slots.

rectangles within the unallocated spaces of the downlink subframe. But, the selection

of the rectangle with the highest total area is omitted in this phase of the enhanced

version of the algorithm.

Moving into the second phase, once selected the largest request from A, say Ak. The

dimension of the burst for request Ak is computed for each of the candidate rectangles

found in the previous first phase. After this has done, the algorithm will proceed to

evaluate the amount of over allocation slots that the burst will have in each of these

candidate rectangles. So, the one with the least over allocation slots will be selected for

the mapping. Then, the mapping of the burst and the rest of the algorithm will be done

in the same way as the original version of the algorithm.

It is worth to mention that while considering all possible rectangles of the first phase,

in some cases, the request may not fit into a rectangle and so, this rectangle is

discarded. And because for each rectangle, the burst will have different dimensions,

allowing the algorithm to select the rectangle that will result in less over allocation

slots for the burst. In case there are two candidate rectangles, which gives the same

amount of over allocation slots, the candidate rectangle with the largest total area will

be selected.

The enhanced maximum rectangle-based burst allocation algorithm results, not only in

Chapter 5:

Performance Evaluation

In this chapter, the performance of the proposed algorithm and its enhanced version

will be discussed and from now on, the two algorithms will be referenced as

MaxRectangle and eMaxRectangle respectively. In order to study the performance of

MaxRectangle and eMaxRectangle algorithms, simulations for these two algorithms,

eOCSA and OBBP has been performed and compared. Notice that OCSA was not

included into the comparison due to the fact that its enhanced version, the eOCSA algorithm performs better.

5.1 Simulation

For performance evaluation, the total number of unused slots, over allocated slots, and

packing efficiency is studied. The over allocations and unused slots are averaged and

normalized over 10,000 trials.

Simulation parameters used for each algorithm where the same and are resumed on

Table 4, which are from the suggestions of the WiMAX forum. Resource requests

were generated randomly with the sum of all requests equal to 1230 slots or 360 slots, as a constraint. We assume each MS needs only one burst.

Parameter Value

Frame length 5 ms

Channel BW 10 MHz

Permutation scheme PUSC

Number of subchannels 30

DL subframe 12 slot columns

Total number of slots per DL subframe 30 x 12 slots

Simulation time 10,000 frames

Table 4: System simulation parameters

Figure 12 shows the average packing efficiency, which is defined as allocated slots

divided by total slots per frame, of all schemes to different number of mobile stations

per frame. We can see that the proposed algorithms achieve higher packing efficiency.

Figure 12: Average efficiency vs. number of MS

6 8 10 12 14 16 18 20 22 24

0.85 0.9 0.95 1

Number of Mobile Stations

E ff ic ie n c y ( a llo c a te d s lo ts / t o ta l s lo ts ) eOCSA MaxRectangle eMaxRectangle OBBP

Figure 13 illustrates the impact of the number of mobile stations on average unused

slots per downlink subframe. The average number of unused slots for the proposed

algorithms is smaller than that for eOCSA and OBBP. eOCSA wastes more slots

because of its third step, and OBBP algorithm allocate the resource into free rectangle

spaces in their third stage, resulting in fewer shapes that can be choose.

Figure 13: Average unused slots vs. number of MS

Figure 14 indicates the average over allocated slots of all schemes to different number

of mobile stations per frame. The eMaxRectangle select the appropriate waste less rectangle to allocate. Thus the average number of over allocation slots for

eMaxRectangle is lower than that for MaxRectangle and eOCSA. Besides that, is

acceptable that OBBP results in less average over allocation slots than eMaxRectangle,

6 8 10 12 14 16 18 20 22 24 0 10 20 30 40 50 60

Number of Mobile Stations

N u m b e r o f u n u s e d s lo ts eOCSA MaxRectangle eMaxRectangle OBBP

because this last one has the lowest average number of unused slots, but still the

proposed algorithm has the best packing efficiency.

Figure 14: Average over allocation slots vs. number of MS

In this chapter, the simulation results shows that the proposed algorithm indeed

performs better than its competitors, but the reasons of this efficiency will be

explained in details through some examples in the following section.

5.2 Performance Analysis

In this section, more than presenting the reasons of the better performance of the

proposed algorithm against the ones compared with, an analysis of the things that

6 8 10 12 14 16 18 20 22 24 1 2 3 4 5 6 7 8

Number of Mobile Stations

N u m b e r o f o v e r a llo c a ti o n s lo ts eOCSA MaxRectangle eMaxRectangle OBBP

allows a downlink packing algorithm to achieve a high utilization of the resources is

presented.

Sorting the MSs resource request in decreasing order before packing will definitely

impact on how efficient the resource in the downlink subframe is going to be used by

the packing algorithm because is more easier to fit small burst into the remaining spaces on the subframe, specially when this is the last remaining burst that need to be

packed and on the contrary, a larger burst not packed will lead to more unused slots.

But, special care has to be taken when designing a packing algorithm since operations

like ordering the data to be packed by decreasing size might result in a larger

utilization of the WiMAX downlink subframe resources but not necessarily in a larger throughput. MSs with better channel conditions will have a better MCS and thus, one

slot can transmit more data than one MS with lower MCS.

So, in order to achieve higher throughput and an efficient use of the resources in the

downlink subframe, it is advised to pack based on the amount of slots the MS require

and the total data that would be packed. For example, pack the burst with the highest

amount of data to transmit first, in case of a tie, refer to the amount of slots assigned, selecting the highest. Consider the following scenario, a small burst with higher MCS

and a large burst with lower MCS and less data to transmit, in this case is better to

pack the smaller in order to achieve higher throughput but in some cases this might

Obviously, the reason why the proposed algorithm performs better than eOCSA and

OBBP is the way of allocating the burst into the downlink subframe. Because bursts

are required to have a rectangular shape, the proper dimension of it is another

procedure that needs our attention while designing an efficient packing algorithm.

A burst can have different dimensions with different resulting over allocation slots or no over allocation at all. Then, selecting the dimension with none or less over

allocation should be the best in order to maximize the utilization of resources in the downlink subframe. It‟s believed that packing efficiency of the proposed algorithm

could increase a little more if applied, but was not considered to lower the complexity

of the algorithm. Instead, like in the eOCSA algorithm, most of the bursts allocated by

the algorithm have the least widths, which imply that the active time and consequently, the energy consumption of each MS is minimized.

While allocating bursts vertically or column wise simplifies things, it may results in

some disadvantages. The problem of packing burst column wise or row wise is that,

for example, in eOCSA all bursts in a common column will have the same width, and

so, the burst will only adopt one dimension. For this reason, some burst may result in

more over allocated slots than others. Besides, the whole column can possibly not be filled completely with burst, given the reason that there‟s no resource request that can

fill the spaces left. And this will apply for all the columns of burst, resulting into more

unused spaces, as shown on figure 15. Notice that, only the first column of burst,

Figure 15: Illustration of the wasted slots of eOCSA algorithm

If we forget about how eOCSA compute its burst dimension, burst A₅ dimension (on

figure 15) could have been set with height equal to 4 and width equal to 2, resulting in only one over allocation slot, and if we didn‟t have this column width limits, the space

left could have been used to place other burst more flexibly. Figure 16 illustrates the

above mentioned; where the white area is the unallocated slots and the over allocated slots are shown in black.

Figure 16: Illustration of wasted space produced by mapping burst column wise

Like in eOCSA algorithm, OBBP can also have unused slots on the top of each

columns of burst. OBBP sort their columns in descending order, so that all unallocated

spaces will be located to the left and top of the downlink subframe, please refer back

to figure 7 on chapter 3. By doing this, all unused slots will be together, making it

easier to place a burst that could possibly not fit without this. Then, they try to allocate the burst in a selected rectangle that result with the least over allocated slots, but

because they still continue with the use of their OFs, the dimension of some burst left

cannot be placed into some of these rectangles, which is the reason why OBBP results

Chapter 6:

Conclusion

This work presents a novel downlink burst allocation algorithm for IEEE 802.16e

Mobile WiMAX networks. Similar to eOCSA, the proposed algorithm meets the

rectangle shape allocation constraint, achieves high throughput by considering

mapping the larger resources first, and optimizes energy consumption at MS by

minimizing its receive time. The basic idea of our proposed algorithm is to find the

maximum rectangle in the un-allocated space for allocation in each iteration. An efficient method for finding the maximum rectangle is provided. Furthermore, the

algorithm is enhanced to improve the efficiency. The performance of the proposed

algorithm is compared with that of eOCSA and OBBP. Simulation results show that

the proposed algorithm, enhanced maximum rectangle-based burst allocation

algorithm, outperforms eOCSA and OBBP.

Mapping bursts column wise or row wise can reduce the complexity of packing

algorithms but may not result in the best efficient way of packing. A packing

algorithm that strictly tries to maximize the utilization of the frame can result in a

violation of the agreed QoS guarantees or unfair distribution of the resources.

Therefore, a trade-off between radio resource usage maximization, QoS and fairness

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Biography

Jimmy Yau Zhong was born in July1, 1985 in Panama City, Panama. Jimmy entered

Technological University of Panama as an undergraduate in 2004 after completing his

elementary and high school studies at Saint George International School of Panama. In

2008, he completed his Bachelor of Computer Networks. Later that same year, he