B3G行動網際網路環境中位置導向多媒體資訊服務平台---子計畫三---行動網際網路中達成比例式差別服務之頻道導向排程策略及漏失管理(II)

行政院國家科學委員會專題研究計畫 成果報告

子計畫三:行動網際網路中達成比例式差別服務之頻道導向 排程策略及漏失管理(II)

計畫類別： 整合型計畫

計畫編號： NSC93-2219-E-011-008-

執行期間： 93 年 08 月 01 日至 94 年 07 月 31 日 執行單位： 國立臺灣科技大學資訊管理系

計畫主持人： 賴源正

報告類型： 完整報告

報告附件： 出席國際會議研究心得報告及發表論文 處理方式： 本計畫可公開查詢

中 華 民 國 94 年 9 月 6 日

I

中文摘要

最近幾年來在網路上提供服務品質的保證已成為一個重要的議題，比例式差別模式可 提供可控制及可預期的服務品質，然而，目前能提供比例式差別模式的研究大都應用於有 線網路，這些方法並不適用於無線網路，原因為無線網路具有高錯誤率、及頻寬會隨著行 動主機位置之不同及時間的變動而有所改變。在此報告中，我們提出一個在無線網路上提 供比例式延遲差別模式的方法。

在此報告中，我們提出一個框架式頻寬分配的方法，使得上行傳輸也能達到比例式延遲 差異性。當全部的頻寬需求大於上行通道的頻寬時，我們利用四種頻寬配置演算法: (1) 比 例式線性分配演算法; (2) 比例式多項式分配演算法; (3) 比例式最大最小分配演算法; (4) 比例式最小最大分配演算法，來達到比例式延遲。

而當全部頻寬需求小於通道的頻寬時，如果配置給使用者的頻寬剛好等於所預估需要 的，那麼可能仍有一些未被配置的頻寬浪費，為了避免此種浪費，在全部預估頻寬需求小 於通道頻寬情形下，我們也提出了四種頻寬配置演算法: (1) 公平性分配演算法; (2) 加權式 分配演算法; (3) 優先滿足需求參考先前式分配演算法; (4) 優先滿足需求預測未來式分配 演算法。

實驗結果顯示，比例式最小最大分配演算法的方法可以達到非常好的比例式延遲，效果 也比其他方法來得顯著。因此，在上行通道中能提供比例式延遲差異性的各種方法中，比 例式最小最大分配演算法是最好的選擇。

從避免頻寬浪費的觀點，實驗的結果顯示，公平性分配演算法、優先滿足需求參考先前 式分配演算法與優先滿足需求預測未來式分配演算法，這三個方法可以達到很好的節省頻 寬效果。除此之外，實驗的結果也顯示避免浪費頻寬的方法會影響比例式延遲的結果。

關鍵詞：比例式差別服務、服務品質保證、行動網際網路、等待時間優先權

II

Abstract

Providing the guarantee of the diverse Quality of Services (QoS) in the network has been an emerging issue in recent years. The Proportional Differentiated Model (PDM) was proposed to provide predictable and controllable QoS for different classes of connections. However, most of related works focused on providing this model in a wired network. These algorithms suffer difficulty when meet some distinct characteristics, such as high error rate, location dependent or time varying channel capacity, that exist in wireless networks. In this report, we proposed a method to provide the proportional delay differentiation in the wireless networks.

In this report, we proposed a frame-based allocation mechanism to provide the proportional delay differentiation in the uplink transmission. Four bandwidth allocation algorithms, proportional linear algorithm, proportional polynomial algorithm, proportional max-min algorithm, and proportional min-max algorithm, are proposed to achieve proportional delay when total predicted bandwidth requirements exceeds than the bandwidth supplied by uplink channel.

Besides, when total bandwidth requirements is less than the link bandwidth, if the allocated bandwidth for a connection is equal to its predicted requirement, there may be some un-used bandwidth. In order to avoid the waste of bandwidth, we also proposed four bandwidth allocation algorithms, fair requirement allocation (FRA), weighted requirement allocation (WRA), requirement+previous allocation (RPA), and requirement+next allocation (RNA), when total predicted bandwidth requirements is less than the link bandwidth.

The simulation results show that the proportional min-mix algorithm can achieve the proportional delay very well and outperform other algorithms. Therefore, to provide the proportional delay differentiation in uplink transmission, the proportional min-max algorithm is the best choice.

From the saving link bandwidth point of view, the simulation results show that, FRA, RPA and RNA can achieve to avoid the waste of link bandwidth very well. Beside, the method we choice when total bandwidth requirement is less than the link bandwidth affects the performance of delay proportion.

Keywords: proportional differentiated services, quality of services, mobile Internet, waiting time priority.

III

Table of Contents

中文摘要... I Abstract ... II Table of Contents ...III List of Figures ... IV List of Tables...V

Chapter 1 Introduction ...1

Chapter 2 Preliminaries...4

2.1 Proportional delay differentiation model ...4

2.2 WTP scheduler ...4

2.3 The central-controlled bi-directional network ...4

2.4 The un-suitability of WTP in the uplink transmission ...6

Chapter 3 Proposed Algorithm...8

3.1 A frame-based bandwidth allocation...8

3.2 Allocation algorithms during congestion ...10

3.2.1 Proportional linear algorithm ...10

3.2.2 Proportional polynomial algorithm ...12

3.2.3 Proportional max-min algorithm...13

3.2.4 Proportional min-max algorithm...15

3.3 Allocation algorithm during non-congestion ...24

3.3.2 Weighted Requirement Allocation (WRA) ...26

3.3.3 Requirement + Previous Allocation (RPA) ...27

3.3.4 Requirement+Next allocation (RNA) ...28

Chapter 4 Simulation Results...30

4.1 Simulation Environment ...30

4.2 Simulation Results ...31

Experiment 0: ...31

Experiment 1: (Fairness) ...33

Experiment 2: (Proportion) ...34

Experiment 3: (Robustness) ...36

Experiment 4: (Fairness) ...37

Experiment 5: (Proportion) ...38

Experiment 6: (Frame size)...41

Experiment 7: (Propagation delay) ...42

Experiment 8: (Packet loss) ...43

Chapter 5 Conclusion...45

Reference...46

附件一：研究論文與系統實作成果………..……48

附件二：出席國際學術會議心得報告及發表之論文各一份……….…….49

IV

List of Figures

Figure 1 A central-controlled bi-directional network structure...5

Figure 2 A frame-based bandwidth allocation ...10

Figure 3 Diagram of proportional max-min algorithm ...14

Figure 4 Diagram of fair min-max algorithm ...16

Figure 5 Diagram of Proportional min-max...19

Figure 6 The topology of our simulation ...31

Figure 7 The probability of using congested algorithms ...32

Figure 8 The Proportion achieved by various congested algorithms...33

Figure 9 Proportion achieved by various congested algorithms ...35

Figure 10 Variance of proportion achieved by congested allocations in case of different number of class...37

Figure 11 Proportion achieved by various non-congested allocation algorithm....38

Figure 12 Proportion achieved by various avoiding waste allocation algorithm...39

Figure 13 The VP of the congested algorithms for various frame sizes ...41

Figure 14 The VP of the frame based bandwidth allocation for various propagation delays ...42

Figure 15 The packet loss of the frame based bandwidth allocation for various buffer sizes during non-congested under proportional allocation...43

V

List of Tables

Table 1 The probability of using congested algorithm for various non-congested algorithms ...39

Chapter 1 Introduction

With the tremendous growth of data transmission and the increasing popularity of new breed real-time multimedia applications, providing the function of quality of service (QoS) in the current Internet becomes more important.

In order to provide the QoS in a packet switch network, IETF first provided the Integrated Services (IntServ) model [1, 2], which uses the Resource Reservation Protocol (RSVP) [3] to reserve required bandwidths. RSVP requires each router to keep per-flow information and performs per-flow reservation for all flows passing through itself. However, it is a huge workload to process per-flow action with millions of simultaneously active flows, resulting in the impractical in implementing IntServ in a backbone network.

Because of the problem of scalability in IntServ model, IETF then offered an alternative approach, Differentiated Services (DiffServ) model [4, 5]. DiffServ guarantees the QoS for each service class and provides only a limited number of service classes. All flows with the same QoS requirement will be aggregated and mapped into a service class. Thus the router only manages per-class status and operates per-class scheduling, so DiffServ does not have any scalable problem.

DiffServ model has two categories: absolute service differentiation [6] and relative service differentiation [6, 7]. As absolute service differentiation is concerned, the performance metrics of service (such as delay, loss, throughput) demand precision. An admission control and a specific signal mechanism (semi-static reservations or “broker” agents) are required to provide the absolute differentiation services. On the other hand, relative service differentiation involves service models that offer assurance for the relative quality ordering between classes, rather than for actual service level in each class. It can be implemented without admission control, policing, or resource reservation. Compared with absolute differentiation, relative differentiation is an easy-deployed and easy-managed approach.

Dovrolis et al. proposed the proportional differentiation services model, which is a subset of relative DiffServ model [8]. Under this model, a differentiation factor

is assigned to each class, and the performance of each class is proportional to its corresponding differentiation factor. The performance metrics may be delay, loss, or jitter. In this report, we focus on the proportional delay differentiation model.

A famous scheduler, waiting time priority (WTP) [7], was proposed to achieve the proportional delay differentiation very well. The WTP scheduler determines which packet is the next to be sent according to the waiting times of heading of line (HOL) packets. Thus, the scheduler has to perceive the waiting times of HOL packets.

A central-controlled bi-directional network, such as HFC, WATM, and wireless infrastructures, consists of a central controller and several remote hosts. The traffic communication between any two hosts is controlled by the central controller. The data transmissions in this bi-directional network have two directions, downlink transmission (from the central controller to the remote host) and uplink transmission (from the remote host to the central controller).

In a central-controlled bi-directional network, WTP well performs the proportional delay differentiation in downlink transmission because the scheduler easily obtains the waiting times of all HOL packets. However, WTP is not suitable in uplink transmission, because of two main limitations. First, WTP operating in the central controller does not easily perceive the information of packet’s waiting times, because this information exists in the remote hosts. If we want that WTP normally operate, the waiting time of HOL packet in each individual remote host must be a transmitted to the central controller, but this is a huge overhead. Second, there is a propagation delay between the central controller and the remote host. Thus, even the waiting times of HOL packets can be received by the controller host, these information will be out of date, resulting in the incorrect operation of WTP.

Therefore, we proposed a new approach, a frame-based bandwidth allocation to achieve the proportional delay differentiation for uplink packets and to eliminate the huge overhead for the central controller. A frame is a time period of fixed or variable number of time slots that packet can be transmitted between the different remote hosts and the central controller. At the beginning of each frame, the remote host has to transmit the predicted packet arrival rate and the current number of backlogged packets to the central controller, and the central controller determines how much uplink bandwidth each remote host can obtain by using some allocation algorithms, and finally inform this allocation to these remote hosts.

When total predicted bandwidth requirements in a frame exceeds the link bandwidth of uplink channel in a frame (congested state), in order to achieve the proportional delay, four bandwidth allocation algorithms, proportional linear algorithm, proportional polynomial algorithm, proportional max-min algorithm, and proportional min-max algorithm, are proposed for the central controller to allocate bandwidth.

Besides, when total predicted bandwidth requirements is less than the link bandwidth (non-congested state), if the allocated bandwidth for a remote host is equal to its predicted requirement, there may be some un-used bandwidth in a frame.

In order to avoid the waste of bandwidth, we also proposed four bandwidth allocation algorithms, fair requirement allocation (FRA), weighted requirement allocation (WRA), requirement+previous allocation (RPA), and requirement+next allocation (RNA) when total predicted bandwidth requirements in a frame is less than the link bandwidth. These proposed allocations in congested state and non-congested state will be compared to get their advantages and drawbacks in this report.

The rest of this report is organized as follows. The proportional delay differentiation model, the WTP scheduler, the architecture of the central-controlled bi-directional network, and the un-suitability of uplink transmission are presented in Section II. Section III formally presents our proposed algorithms in congested and non-congested states. In Section IV, some simulations are conducted to compare our proposed algorithms. Finally, we summarize this study in Section V.

Chapter 2 Preliminaries

2.1 Proportional delay differentiation model

Under the proportional delay differentiation model, the ratios of the average queuing delays between any two classes are proportional to their corresponding differentiation factors, which may be chosen by the network operator. Formally, this model wants to achieve

j i j i

c c d

d = , where diis the average delay of departed packet for class i and c is the delay differentiation factor of class i . _i

2.2 WTP scheduler

Some schedulers to achieve the proportional delay differentiation have been proposed for the past years [2]. The wait time priority (WTP) scheduler, which increases the priority of a packet in proportional to its waiting time, has a splendid performance.

Let )d_i(t be the waiting time of the HOL packet in classiat t . The WTP scheduler determines the priority of HOL packet of class i at the time t as

i i

i c

t t d

P ( )

)

( = , and selects the packet with highest priority to be served, that is, selects the packet of class j , where j argmaxP_i(t)

i

= . The WTP scheduler must

record the arriving time of each packet, and then calculates its corresponding waiting time to determines which packet will be sent next.

2.3 The central-controlled bi-directional network

In a central-controlled bi-directional network, as shown in Figure 1, a central

controller manages all delivery of packets in the network both in upstream and downstream. There is no direct communication between two remote hosts, that is, the transmission between two remote hosts must go through the central controller.

The central-controlled bi-directional network usually uses a time-slot system to increase link utilization. A time slot is equal to a packet (cell) transmission time, and is usually a fixed size to simplify switch design and increase processing speed. The user data is packeted and transmitted in the time slots on a slot-by-slot basis.

Since during each one timeslot only one packet can be transmitted between the central controller and one of remote hosts, the central controller has to decide which host can use a slot to send its packet to the central controller via a uplink channel, and which packet can be transmitted to its destination host via a downlink channel.

Each remote host has a buffer (uplink buffer) for uplink transmission and has its corresponding buffer (downlink buffer) in the central controller for downlink transmission. The packets transmitted from the remote host to the central controller are first stored in the uplink buffer. Similarly, a packet in downlink transmission will be stored in the downlink buffer.

central controller

uplink buffer

host 1 uplink buffer

host 2

uplink buffer

host N

uplink

downlink

..

.

downlink buffer

Figure 1: A central-controlled bi-directional network structure

In downlink transmission, the central controller directly perceives the information of downlink buffers, and easily performs its bandwidth allocation and packet scheduling. However, it is difficult that the central controller manage the packet transmission in uplink transmission. The common procedure is that the

central controller first perceives how much bandwidth the remote hosts require (bandwidth requirement), then allocates time slots for each remote host, and finally informs the bandwidth allocation information to each remote host.

There are different methods that the central controller perceives the requirement bandwidth of remote hosts in various networks. The First Transmit Rule Protocol (FTP)[12, 13, 14] is considered for the newcomer host with first requirement in HFC network. In Wireless ATM network, the methods, which are provided for the remote host to inform the central controller (base station) about its bandwidth requirement, are classified into three categories: fixed assignment (TDMA [15] and FDMA), random access [16, 17] (ALOHA [18], CSMA/CD) and demand assignment (which combines fixed assignment and random access) [19].

2.4 The un-suitability of WTP in the uplink transmission

In downlink transmission, the packet scheduler in central controller is able to directly and immediately perceive the information of downlink buffers (packet waiting times, the amount of backlog). Thus the WTP scheduler has the ability to determinate which packet is to be sent in the next downlink time slot. Therefore, it is easy to achieve the proportional delay differentiation in downlink transmission.

However, in uplink transmission, the uplink buffers exist in each remote host, and the WTP scheduler does not directly gain the information of uplink buffers.

Suppose that the propagation delayτ is 100 units, and the network has two classes of class A and class B , where c_A =2 and c_B =1. At time t , the class A and class B have only one backlogged packet. The waiting times of HOL packet in class A and B are 5 and 2, respectively. Because of the propagation delay, the WTP scheduler decides to transmit the HOL packet of class B first according to the result of

B B A

A c

d c

d = < =

1021

1052 . But the practical waiting times of HOL packet in class A and class B perceived by the central controller are 5 and 2, respectively. The WTP decides to transmit the HOL packet of class A first by according to the

equation

B B A

A c

d c

d = > =

21

52 . It leads to the incorrect operation for WTP.

Moreover, if we want that WTP normally operate in uplink transmission, the waiting time of HOL packet in each individual remote host must be transmitted to the central controller and the selection of WTP are also need to be transmitted to the

corresponding remote host. Thus, it is a huge overhead to transmit these information.

Therefore, WTP is not suitable for providing proportional delay differentiation in uplink transmission.

Chapter 3

Proposed Algorithm

3.1 A frame-based bandwidth allocation

In order to achieve the proportional delay differentiation in the uplink transmission and to eliminate the huge overhead for the central controller, a frame-based bandwidth allocation is proposed. A frame is a time period of fixed or variable number of time slots that packet can be transmitted between the remote hosts and the central controller (In this report, we assume that the number of time slots in a frame is fixed).

Let us denote the number of available time slots in a frame by Λ , that is, available link bandwidth provided by the uplink channel. Also, let us denote the predicted packet arrival rate for the i th host in kth frame by λ^k_i . This is the component of a host's bandwidth requirement. Let x be the buffer occupancy of _i^k the i th host at frame k.

Assuming that the propagation delay is equal to τ frames and the bandwidth requirement of the remote host at the next frame will be the same as bandwidth requirement at the current frame k, the i th host requires

(

)

k

xi +λ for frame 1

k+ to clear the backlog of packets x in the uplink buffer for _i^k i th host and the predictive amount of new arrival packet in a frame. The assumption here is that the packet rates λ^k_i change with a larger time constant compared to that of the service time slots.

Since the propagation delay is τ , the bandwidth allocation for the frame k+1 is actually applied at the frame k+ +τ 1. Let F_i^k+τ+1 be the amount of allocated

bandwidth to the host i in frame k+τ +1. Assuming all the buffers are infinite, the dynamics of the system can be described by

⎪⎪

⎩

⎪⎪⎨

⎧

=

=

Λ

≤

=

− +

=

∑

+ + + +

+

N i

X x

F

N i

F x

x

i i

N i

k i

k i k i k i k

i

,..., 1 ,

(1)

,..., 1 , ] [

0 1

1

1 1

τ

τ

τ λ

where the notation X to represent the quantity⁺ max(0,X . )

At the beginning of each frame, the remote host would have to transmit the predicted packet arrival rate and the current number of backlogged packets to the central controller, and the central controller determines how much uplink bandwidth each remote host can obtain in the next τ frame, and finally inform this allocation to these remote hosts.

When the amount of total bandwidth requirements in a frame exceeds the amount of link bandwidth that provide by uplink channel in a frame (congested state), four bandwidth allocation algorithms, proportional linear algorithm, proportional polynomial algorithm, proportional max-min algorithm, and proportional min-max algorithm, are proposed for the central controller to achieve the proportional delay differentiation.

Since the bandwidth of uplink channel is usually small, when the amount of bandwidth requirements is less then the link bandwidth (non-congested state), then there may be some un-allocated bandwidth in the frame, in order to avoid waste of bandwidth, we also proposed four bandwidth allocation algorithms in a non-congested state: fair requirement allocation (FRA), weighted requirement allocation (WRA), requirement+previous allocation (RPA), and requirement+next allocation (RNA).

As shown in the Figure 2, there are N remote hosts and one central controller in the network. The buffer occupancy x and predicted packet arrive rate _i^k

k

λi in a frame are used as the bandwidth requirement of host i . In this report, for clearer explanation, we use the weighted value w for host _i^j i , j means that the host i belongs to class j , and where

j j

i c

w = 1 . The up index j of w_i^j was eliminated

for assuming that if the weighted value w_i^j for the host i is equal to the weighted value w_k^j of the host k, then we say that the host i and the host k are belong to the same class j. so the up index of w_i^j. w_i means the quality of service for host i . The larger w_i is, the lesser delay the host i has, and the better quality of service its owns.

k

λ2

k

λN k

λ1

x₂k

x₁k

k

xN

1 1

+ +τ

Fk

Λ

1 2

+ +τ

Fk

+1 +τ k

FN

w1

w2

wN

Figure 2: A frame-based bandwidth allocation.

3.2 Allocation algorithms during congestion

Firstly, four allocation algorithms used in the congested state are proposed.

3.2.1 Proportional linear algorithm

From the methods for the packet scheduler providing the fair scheduling in a wireless network, the first method come to our mind is linear allocation [20, 21].

The linear allocation algorithm calculates the ratio of the each bandwidth requirement over the total bandwidth requirement, and then allocates the link bandwidth for each host according to its corresponding ratio. The larger ratio the host has, the more bandwidth it will be assigned.

To achieve the proportional delay differentiation, we expect that the host i with larger weighted value w_i would gain more bandwidth. Thus we modify the linear allocation with adding the concept of proportion as

1

1

( )

(2)

( )

k k

k i i i

i N

k k

j j j

j

F w x

w x

τ λ

λ

+ +

=

= + Λ

∑

The advantage of this method is that it is very simple to implement, and is very efficient [22]. Due to the low complexity, this algorithm is suitable to the system without high computing ability.

Example 1:

We assume that there are two remote hosts (host A, host B) in the network, the link bandwidth capacity is 2 (Λ =2), and there is no propagation delay (τ =0). We also assume that the weighted values for host A and host B are 1 and 1, respectively (w_A = ,1 w_B =1), and the additional bandwidth requirements of host A and host B in a frame are 1 and 3, respectively (λ_A =1, λ_B =3). At the beginning the numbers of buffer occupancy of host A and of host B are 0 (x⁰_A = x_B⁰ = ). For the first frame, the 0 bandwidth allocation for host A is F_A¹= 1 1

1 1 1 3 2

× ×

× + × = 1

2 and the bandwidth allocation for host B is F_B¹ = 1 3

1 1 1 3 2

× ×

× + × = 3

2 . After the first frame was completely transmitted, the number of buffer occupancy for host A is

1 0 1 1

A A A A 2

x =x +λ −F = , and for the host B is ^{1 0 1} 3

B B B B 2

x =x +λ −F = , the ratio of the number of buffer occupancy for host A to the number of buffer occupancy for host B after the 1st frame is

1 1^A 3

B

x x = . For the second frame allocation:

2

2 1 2 3 2 1 2 2 1 2

2, , 2 1, 3, and 3.2^A

A B A A A A B B B B

B

F F x x F x x F x

λ λ x

= = = + − = = + − = =

For the third frame allocation:

3

3 1 3 3 3 2 3 3 3 2 3 9

2, , 2 2, 2, and 33^A

A B A A A A B B B B

B

F F x x F x x F x

λ λ x

= = = + − = = + − = = .

For easy exhibition of the un-achievement of targeted delay proportion for proportional linear algorithm, we use an example of proportional linear algorithm in a fair case (w_A = , 1 w_B =1). Example 1 exhibits the backlog number of hostA is almost three times as large as the backlog number of host B. However, their weights are same and we expect they have the same delay. It is very obvious that the

proportional linear algorithm cannot achieve the proportional delay differentiation in a fair case, of course also in a proportional case.

3.2.2 Proportional polynomial algorithm

A nonlinear allocation scheme is also considered to study whether it can achieve the proportional delay differentiation. Herein, the polynomial allocation scheme is selected to be the representative of a nonlinear allocation. The proportional polynomial algorithm allocate the bandwidth for the host i is:

1

1

( )

(3)

( )

k n

k i i i

i N

k n

j j j

j

F w x

w x

τ λ

λ

+ +

=

= + Λ

∑

The more n is the more bandwidth, the host with larger requirement gains.

Because of this property, compared to a proportion linear algorithm, the proportion polynomial (n>=2) is an algorithm more sensitive to the bandwidth requirement.

Example 2:

Assume that a network with link capacity 10 has 2 hosts, host A and host B, and there is no propagation delay (τ =0). The weighted values of host A and B are 1 and 2, respectively (w_A =1, 2w_B = ), and the additional bandwidth requirements in a frame for host A and B are 8 and 4, respectively (λ_A =8, λ_B =4). At the beginning the buffer occupancy of host A and B are empty (x⁰_A =0, 0x⁰_B = ).

When adopting proportional linear scheme (n=1), the bandwidth allocation and the number of backlogs are as:

1 1

1 8 2 4

10 5, 10 5, 3, max(0 4 5, 0) 0.

1 8 2 4 1 8 2 4

A B A B

F = × × = F = × × = x = x = + − =

× + × × + ×

When adopting proportional polynomial scheme (n=2), the bandwidth allocation and the number of backlogs are as:

2 2

1 1

2 2 2 2

1 8 2 4

10 6.7, 10 3.3, 1.3, 0.7.

1 8 2 4 1 8 2 4

A B A B

F × F × x x

= × = = × = = =

× + × × + ×

When adopting proportional polynomial scheme (n=3), the bandwidth allocation and the number of backlogs are as:

3 3

1 1

3 3 3 3

1 8 2 4

10 8, 10 2, 0, 2.

1 8 2 4 1 8 2 4

A B A B

F × F × x x

= × = = × = = =

× + × × + ×

When adopting proportional polynomial scheme (n=4), the bandwidth allocation and the number of backlogs are as:

4 4

1 1

4 4 4 4

1 8 2 4

10 8.8, 10 1.2, max(0 4 8.8, 0) 0, 2.8.

1 8 2 4 1 8 2 4

A B A B

F × F × x x

= × = = × = = + − = =

× + × × + ×

An obvious fact can be observed in this example that the allocated bandwidth to the host with large requirement increases as n increases.

3.2.3 Proportional max-min algorithm

It is unavoidable that the allocated bandwidth to a certain host will exceed its requirement when either using proportional linear scheme or proportional polynomial scheme. From Example 2 (n=1), the host B only requires 4 units of bandwidth but it is assigned to 5 units. The extra one unit would waste. We expect to avoid the waste of link bandwidth and to achieve the delay proportion simultaneously. A new algorithm is proposed, named proportional max-min algorithm, which idea comes form the well-known fair resources allocation, max-min algorithm. An allocation is max-min fairness if there is no way to give more bandwidth to a host without decreasing the allocation of lesser or equal bandwidth.

The idea of class max-min algorithm is to maximize the service of the host to receive the poorest services [23]. According to this concept, if there are N hosts in the network and the link bandwidth isΛ , then an equal share for each host is

Λ . N If an equal share exceeds the actual requirement of one host, this host is classified as satisfied. Let M be the number of satisfied hosts and the amount of total bandwidth requirement of these satisfied hosts be B_s . Thus the residual bandwidth is

s

R B

M N

B = ×(Λ )− , and should be fairly allocated to other unsatisfied hosts, this is, each one can get extra^BR (^{N −M}) unit of bandwidth.

The proportional classical max-min algorithm is derived form the classic max-min algorithm by additionally considering the weighted valuew_i. First we replace the equal share with the proportional share, which is calculated as

1

( _N )

i

j j

w

w

=

Λ

∑

bandwidth requirement with its proportional share. Now the residual bandwidth is

∑ ∑

∈

=

+ Λ −

×

=

SC i

i N i

j j i

R x

w w

B ( ( ) ( ))

1

λ , where SC means the set the satisfied hosts,

and is allocated to other unsatisfied hosts. Thus, the unsatisfied host i additionally

gets _i

UC j

j

R w

w

B

∑

∈

)

( , where UC means the set of the unsatisfied hosts. The

following example is used to explain the proportional max-min algorithm.

Example 3:

The network has four hosts, host A, host B, host C, and hosts D. The link bandwidth is 20 units, and there is no propagation delay(τ =0). Suppose that the weighted values are w_A =2 ,w_B =1,w_C =1, and w_D =1, and the bandwidth requirements are x_A +λ_A =12, x_B +λ_B =10, 3x_C +λ_C = , and x_D+λ_D =2.

Λ i

B C D

A (c)

(10,5) (12,10)

(2,2) (3,3)

Λ i

B C D

A

( a )

12 10

3 2

Bandwidth Available

Bandwidth Request 20

k i k

xi +λ

20 20

(b)

un-satisfied

Remaining Bandwidth

Λ i

B C D

A

(10,4) (12,8)

(2,2) (3,3)

satisfied

20

( b )

k i k

xi +λ

k i k

xi +λ

Figure 3 : Diagram of proportional max-min algorithm

The proportional share for host i is

∑

Λ

×

N

j j i

w w

1

unit of bandwidth, so, in

Figure 3(b), the host A can be assigned to 8 units of bandwidth and other hosts can be assigned to 4 units. However, the host C and the host D actually require amount of bandwidth are 3 units and 2 units, respectively, and those are below than the proportional share. Therefore, the host C and the host D are classified to the satisfied hosts. The residual bandwidth, which is 3 units, will be allocated to the host

A and the host B .

As shown in Figure 3(c), the residual bandwidth is allocated to the unsatisfied host i according to ratio of the weighted valuew_iover the sum of the weighted values for all unsatisfied hosts. Therefore, the host A gets additionally 3 2

1 2

2 × =

+ units of

bandwidth, and the host B gets additionally 3 1 1 2

1 × =

+ unit of bandwidth. Thus, the host A , B, C, and D are allocated 10, 5, 3, and 2 units of bandwidth, respectively.

However, as stated in the following theorem, if the bandwidth requirement of each host always exceeds its proportional share, the proportional max-min algorithm does not have any residual bandwidth to reduce the number of over-proportional backlogs in the unsatisfied host, and thus does not achieve the proportional delay differentiation.

Theorem 1: Suppose there are N hosts in the system and system capacity is Λ , if ∀i

1

i N

i i

j j

x w

w λ

=

+ > × Λ

∑

congested state, then each host i will get

1

( _N )

i

j j

w

w

=

× Λ

∑

Proof: trivial.

3.2.4 Proportional min-max algorithm

Basing on the queuing theory, the average delay is affected by the average backlog number in a queuing network. The greater number of backlog is, the longer the average delay is. The min-max fairness is motivated by this concept [24]. After

amount of unallocated backlog of each host. Therefore, theoretically, the delay of each host will be the same.

As mentioned earlier in example 1, the proportional linear algorithm can’t prevent the problem of unfair growing buffers (Since the problem of unfair growing buffers will cause the un-achievement of the proportional delay differentiation.). For the fair scheduling, the Fair Min-mix algorithm can solve this similar endless growing problem, the min-max algorithm achieves that the remaining backlog of all hosts will maintain the same length after each bandwidth arrangement. Thus it also achieves that the largest number of backlog to be as smaller as possible. Figure 4 is an example explaining how the min-max algorithm works.

i Λ B C D

A

( a )

i Λ A

D B C ( b )

( d ) A i

D B C Λ

( e ) ^Λ

A i D B C

A i

D B C Λ ( c )

d1

d2

d3

k i k

xi +λ

k i k

xi +λ

k i k

xi +λ

k i k

xi +λ

k i k

xi +λ

Figure 4: Diagram of fair min-max algorithm

From Figure 4(a) to 4(b), the Fair min-max algorithm sorts bandwidth requirement of all hosts in a decreasing order. As shown in Figure 4(c), the fair min-max algorithm firstly considers the one who has the maximal bandwidth requirement in the system. Allocate bandwidth d₁ to the host D (D is the host with the maximal bandwidth requirement), where d₁ is denoted as the gap between the

maximal bandwidth requirement of host. From Figure 4(d), the host D and the host B became the hosts, which have maximal bandwidth requirement, is considered to for central control to allocated bandwidth. Therefore, it assigns d₂ bandwidth to both of hosts D and B . In Figure 4(e), the residual bandwidth that the system has can’t be assigned again, since the remaining bandwidth is not enough. In this case, this algorithm assigns the remaining bandwidth fairly to host D , B , and A .

Since the min-max fairness algorithm can maintain the amount of buffers to be as equal as possible. We combine the fair min-max algorithm and concept of proportion to provide a new allocation algorithm, named proportional min-max allocation algorithm. The proposed algorithm expects the amount of backlog of host

i is proportional to

wi

1 (in other word, the average delay between any tow hosts is proportional to its corresponding w_i).

The Proportional min-max bandwidth allocation algorithm:

Suppose there are N hosts in the network and capacity of uplink channel is Λ . Initial:

The amount of remanding bandwidth is denoted to be Λ′. Λ′=Λ. First Step:

It is expected that “the larger weighted value the host i has, the smaller the amount of backlog it owns”. By above reason, at the beginning the weighted bandwidth request for host i is calculated as:

. ..., , 2 , 1 ), 1 (

) ) (

( w x i N

w

x x _{i i}^{k k}_i

i k i k i i

i + = × + =

′=

+λ λ λ

Second Step:

The weighted bandwidth requests (x_i +λ_i)′are sorted in a descending order, where (x₁+λ₁)′≥(x₂ +λ₂)′≥....≥(x_N +λ_N)′.

Third Step:

Suppose that the set of hosts with the maximal unallocated weighted request is

(*), } ) (

,..., ) (

) (

| ) (

,..., ) (

, )

{( _{1 1 1 1}

1 = x_i + _i ′ x_i+ + _i+ ′ x_i+k + _i+k ′ x_i + _i ′= x_i+ + _i+ ′= = x_i+k + _i+k ′

T λ λ λ λ λ λ

and the set of hosts with the second maximal unallocated weighted request is

}. ) (

,..., ) (

) (

| ) (

,..., ) (

, )

{( _{1 1 1 1}

2 = x_m+ _m ′ x_m+ + _m+ ′ x_m+n + _m+n ′ x_m+ _m ′= x_m+ + _m+ ′= = x_m+n + _m+n ′

T λ λ λ λ λ λ

The difference between T₁ and T₂ is denoted as d, where (d=(x_i+λ_i)′−(x_m+λ_m)′).

The ratio of the value

j

wi₊

1 for the i+jth host to the value

∑

= +

1

0 k 1

j wi j is

denote to be Ω , j=1,…,k , where the subindex i of _i+j wi

1 is the same as the

subindex i in (*).

Then to check whether 1 ) (

1

0

w d

k

j i

∑

=

exceeds than Λ′: If the value of 1 )

(

1

0

w d

k

j i j

∑

= +

does not exceed Λ′, then it allocated the i+jth hosts with d

w_i+j× units of bandwidth. (j=1,…,k) , then there are still available

bandwidth (suppose the remaining bandwidth is Λ ′′ ), then it replace Λ′ with Λ ′′ and repeats the third step again.

If the value of 1 ) (

1

0

w d

k

j i

∑

=

exceeds Λ′, then the i+jth host is allocated

j i j

w1i₊ ×Ω⁺

, j=1,…,k and finish the bandwidth allocation.