中 華 大 學

中 華 大 學 碩 士 論 文

題目：

802.11 無線網路之動態頻道分配及 功率控制

Dynamic Channel Selection and Power Control in 802.11 wireless network

系 所 別：電機工程學系碩士班 學號姓名：M09201026 陳美琪 指導教授：李柏坤 博士

中華民國九十四年七月

Dynamic Channel Selection and Power Control in 802.11 Wireless Network

Student : Mei-Chen Chen Advisor : Dr. Bore-Kuen Lee

Institute of Electrical Engineering Chung Hua University

Abstract

In many power control paper, the utility function quantifies the level of satisfaction a user gets from using the system resources. Each player in the game maximizes function of net utility in a distributed system. Since users act selfishly, the equilibrium point is not necessarily the best operating point from a social point of view. Pricing of services in wireless networks emerges as an effective tool for radio resource management because of its ability to guide user behavior toward a more efficient operating point. However, in 802.11 wireless network system, it is a multi-channel system. Therefore, not only control power by net utility function, we also design a dynamic channel selection algorithm to reduce co-channel interference.

802.11 無線網路之動態頻道分配與功率控制

研究生：陳美琪 指導教授：李柏坤博士

中華大學

電機工程學系碩士班

中文摘要

在很多功率控制的論文中，使用者使用傳輸功率所得到的滿足 感，被表示成效用函數。而使用者需付出的成本，被表示成費用函數。

每個使用者都希望自己能得到最大的傳輸功率，但是使用較高的傳輸 功率，必須付出較大的成本，因此每個使用者都努力去達到自己最佳 的淨效用。然而在 802.11 的網路系統中，系統是多頻道系統，除了 調整功率外，另一個可控制的變數是頻道的選擇。我們不只使用淨效 用函數來控制功率，我們還設計了一個動態頻道分配的演算法，來降 低干擾。

Table of Contents

Table of Contents ...4

1. Introduction...6

1.1 Survey of related literature...6

1.2 Motivation and Objective ...7

2. System Model and Problem Formulation ...9

2.1 System Model and Assumption ...9

2.2 Dynamic Channel Selection Algorithm ...10

2.2.1 Fairness ...10

2.2.2 Max-min Fairness ...10

2.2.3 Algorithm Description ... 11

2.3 Power Control ...13

2.3.1 Transmitter Power Control (TPC) Method ...13

2.3.2 Utility-Based Transmitter Power Control (TPC) ...15

2.4 Dynamic Channel Selection and Power Control Procedure ...20

2.5 Synchronous Iterative Power Control...21

2.5.1 Iterative Power Control...21

2.5.2 General Interference Constraints ...21

2.5.3 Synchronous Iterative Power Control...23

3. Simulation ...25

3.1 Simulation Models ...25

3.2. Results and Analysis ...28

6. Conclusion and Future Works...34

Reference ...35

APPENDIX...37

1. 802.11 Network...37

A. 802.11 Network Structure ...37

B. 802.11 Media Access Control (MAC)...38

C. 802.11 Physical Layer ...39

D. 802.11 Management Architecture ...40

2 Dynamic Frequency Selection ...40

A. Fixed Channel Selection (FCS) and Dynamic Channel Selection (DCS) ...40

B. Dynamic Channel Selection (DCS) Mechanism ...41

1. Introduction

1.1 Survey of related literature

The channel models of wireless network can be categorized as single-channel and multi-channel. In single-channel, all mobile hosts operate on the single common channel for communication. In multi-channel model, the overall bandwidth is divided into several channels, and every host can operate on one or some of these channels for communication.

Bandwidth is rare and valuable for wireless networks, therefore, it is important to increase channels utilization. In wireless network, the most important factor that dictates channels utilization is contention/collision. In the single channel wireless network system, it is well known that using smaller radio transmission and design the MAC protocol of RTS/CTS-based and busy-tone-based can increase channel reuse.

One common problem with single-channel is that the network performance will degrade quickly as number of mobile hosts increase, due to high contention/collision.

One approach to reliving the contention/collision problem is to utilize multiple channels.

Using multiple channels has several advantages, for example, the total throughput may be increased and the collision probability may be reduced immediately. Therefore, it is important to design a dynamic channel algorithm to reduce contention/collision and increase channels utilization. However, it is a challenge, too. Many dynamic channel allocation algorithms have been proposed in wireless communication networks and cellular mobile networks such as “A New Adaptive Channel Assignment Algorithm in Cellular Mobile” [18], and “Channel Assignment for Time-Varying Demand Systems” [19]. In our paper, we design a dynamic channel selection algorithm to reduce co-channel interference and increase Signal to Noise Ratio (SNR) in multi-channel system. Another goal of our dynamic channel selection algorithm is to gives the most poorly treated station (i.e., the station who receives the lowest SNR) the largest possible SNR.

In multiple channels system, a common and important method to increase users and decrease contention/collision is to control power of all users. It is well known that minimizing interference using power control increase capacity [29]-[31] and also extends battery life. Recently, an alternative approach to the power control problem in wireless systems based on an economic model has been offered in [27], [28],

[32]-[34]. In this model, service preferences for each user are represented by a utility function, such as “A Utility-Based Power-Control Scheme in Wireless Cellular Systems”[27]. As the name implies, the utility function quantifies the level of satisfaction a user gets from using the system resources. Game-theoretic methods are applied to study power control under this new model. And in our paper, we also apply game theory to model interactions between self-interested users and predicting their choice of strategies.

1.2 Motivation and Objective

With the ever-increasing need for capacity in wireless communication network system, allocation of resources has become a great challenge. In order to efficiently utilize the scarce spectrum resource, channel assignment is becoming one of the important issues for wireless communication researches and practicians.

In wireless network, a common and important method to increase users and decrease contention/collision is to control power of all users. Recently, an alternative approach to the power control problem in wireless systems based on an economic model has been offered in [27], [28], [32]-[34].

But, if two co-located Basic Service Sets (BSSs) operate at the same channel, which are referred to as overlapping BSSs, it is not easy to support Quality-of-Service (QoS) due to the possible contentions among overlapping BSSs. However, by having the dynamic channel selection been implemented, an Access Point (AP) can determine the best channel to work at, and initiate the switch of all the stations (STAs) associated with its BSS to the newly selected channel. We think it is an efficient method to prevent interference and increase the quality of service to STAs.

Besides prevent interference and providing SNR to all STAs as high as possible, another goal of our dynamic channel selection algorithm is to supply every station as fair as possible. Consequently, in our paper, we extend the power control utility-based power control (UBPC) and design a dynamic channel selection algorithm to reduce co-channel interference and increase Signal to Noise Ratio (SNR) to all STAs.

Another goal of our dynamic channel selection algorithm is to gives the most poorly treated station (i.e., the station who receives the lowest SNR) the largest possible SNR, that is to say, another goal of our dynamic channel selection algorithm is to supply every station as fair as possible. After designing a dynamic channel algorithm to reduce co-channel interference and increase channels utilization, we represented service preferences for each user by a utility function. The utility function quantifies the level of satisfaction a user gets from using the system resources. Each player in

the game maximizes some function of utility in a distributed system. Since users act selfishly, the equilibrium point is not necessarily the best operating point from a social point of view. Pricing the system resources appears to be a powerful tool for achieving a more socially desirable result. Pricing of services in wireless networks emerges as an effective tool for radio resource management because of its ability to guide user behavior toward a more efficient operating point.

The paper is organized as follows: In Section II, we explain a multi channel system model, and problem formulation of our system. Then we apply utility-based power control in our system model. In Section III, we present our dynamic channel selection algorithm, which is a max-min fairness concept algorithm. And apply power control game and dynamic channel selection to run our simulation and we present our simulation results in Section V. Then, we summarize our conclusions and future work in Section VI.

2. System Model and Problem Formulation

2.1 System Model and Assumption

If two co-located Basic Service Sets (BSSs) operate at the same channel, which are referred to as overlapping BSSs, it is not easy to support Quality-of-Service (QoS) due to the possible contentions among overlapping BSSs. By having the DCS function implemented, an Access Point (AP) can determine the best channel to work at, and initiate the switch of all the stations (STAs) associated with its BSS to the newly selected channel.

We apply IEEE 802.11h standard that there are nineteen 20 MHz channels between 5 GHz and 6 GHz allowed in Europe. Consequently, we consider channel number K={1,…,k}, and k=19. Consider in an environment such as office building, there is a 802.11 WLAN consisting of N infrastructure BSSs, N={1,…,n}. And we assume that the i-th BSS (i∈N) has a unique AP_i as a decision maker of the DCS and m(i) stations, M_i={STA_i,1, STA_i,2, …,STA_i,m(i)}, shown as figure 1. Denote Ω be the set containing all AP_i. Let all STAs in a set Γ, which equals M₁∪M₂∪…∪M_n. In the 802.11 WLAN system, we assume that there isn’t any centralized decision maker of the DCS among all BSSs, that is to say, APi is the unique decision maker of the DCS.

In order to avoid co-channel interference and support good Quality-of-Service (QoS), all APs must select the best channel to work at. Because inter-access point communications are not standardized, mobility of station between access points supplied by different vendors is not guaranteed, therefore, we assume any station can’t take hand-off between different APs.

Fig.1. APi and Mi={STAi,1, STAi,2, …,STAi,m(i)} in BSS

2.2 Dynamic Channel Selection Algorithm

2.2.1 Fairness

Besides providing SNR to all stations as high as possible, another goal of our dynamic channel selection algorithm is to supply every station as fair as possible.

Because the SNR of all stations in all channels are different, so it is impossible to provide a DCS algorithm that supplies every station the same SNR. That is to say, it is impossible to provide a DCS algorithm that satisfies SNR of every station fairly indeed.

2.2.2 Max-min Fairness

Even though we can’t provide a DCS algorithm that satisfy SNR of every station fairly indeed, but we can gives the most poorly treated station (i.e., the station who

receives the lowest SNR) the largest possible SNR.[8]. When it is impossible to increase the SNR of a station STAi,j without reducing the SNR of another user STAi,j’

with a SNR(STA_i,j,k) ≤ SNR(STA_i,j’,k) Roughly, this definition states that a max-min fair allocation gives the most poorly treated station the largest possible SNR, while not wasting any network resources.[8]

2.2.3 Algorithm Description

In this section, we focus on the case that APi would like to go the dynamic channel selection process. We describe an algorithm of how APi selects a channel. The steps are showed as following, which output the channel number that APi selects.

1. From the measurement stage, retrieve all SNR(γ, k) for all γ∈M_i and all k∈K.

2. For all k∈K, compute CSNR(k)=MIN{SNR(γ, k)| γ∈M_i}.

3. Choose c∈K such that CSNR(c)=MAX{CSNR(k)| k∈K } and let c be the output.

Give a simple example of our algorithm. If i=1, N=8, k=5, m1=4,

From Step 3, we can obtain 4 number of SNR(γ, k) tables for γ∈{STA_1,1,STA_1,2,STA_1,3,STA_1,4}, and all k∈{1,2,3,4,5} such as SNR(STA_i,j,k)

Table-2: SNR(STAi,j,k)

Channel 1 Channel 2 Channel 3 Channel 4 Channel 5

STA1,1 21 19 23 19 26

STA1,2 23 18 26 20 23

STA1,3 25 23 24 21 27

STA_1,4 23 25 25 24 28

STA_1,5 22 20 22 19 29

And from Step 4,

CSNR(1)=MIN{21,23,25,23,22}=21; CSNR(2)=MIN{19,18,23,25,20}=18 CSNR(3)=MIN{23,26,24,25,22}=22; CSNR(4)=MIN{19,20,21,24,19}=19 CSNR(5)=MIN{26,23,27,28,29}=23,

We obtain

Table-3: CSNR(k)

Channel 1 Channel 2 Channel 3 Channel 4 Channel 5

CSNR(k) 21 18 22 19 23

In Step 5, Choose c∈K such that CSNR(c)=MAX{CSNR(k) | k∈K },

MAX{21,18,22,19,23}=23, c=5. Consequently, the channel we choose in our algorithm is channel 5.

2.3 Power Control

2.3.1 Transmitter Power Control (TPC) Method

Extend reference paper [27], we apply a transmitter power control method in our system model. We consider a power-controlled wireless network system where the transmitted powers are continuously tunable. Within a BSS, every STA is associated with an AP. To maintain a reliable connection between the STA and the AP it belonged, the SIR at the receiver should be no less than some threshold that corresponds to a QoS requirement such as the bit error rate. We consider only downlink transmissions in this paper because the uplink case can be treated similarly [21], [22]. Assume that in the i-th BSS, there are m(i) stations, Mi={STAi,1, STA_i,2, …,STA_i,m(i)}, shown as figure 1. And let P_(b,γ,k), b∈Ω,γ ∈M_i,k∈Kbe the transmitted power level for the downlink of STA m(i). Let G_(b,γ,k) ,

K k M

b∈Ω,γ ∈ _i, ∈ denote the gain. Then, the received power at STA γ in the downlink channel k of other AP b is G_(b,γ,k)P_(b,γ,k). And the interference power received at STA γ in the downlink channel k of other AP d∈ ,Ω d ≠b is

∑

Ω ≠

∈ d b

d

k d k

d P

G

,

) , , ( ) , ,

( γ γ . Let υ_(γ,k) be the background noise received at STA γ in channel k.

And let SNR_th_(γ,k) be its desired SNR threshold of γ. Then the SNR for STA γ is given by

) , ( ,

) , , ( ) , , (

) , , ( ) , , ( )

, (

k d

b d

k d k d

k b k b

k G P

P SNR G

γ γ

γ γ γ

γ = +υ

∑

Ω ≠

∈

(1)

For the system considered the constraint SNR_(γ,k) ≥SNR_th_(γ,k) is enforced for each STA γ. The objective of a power-control scheme is to find the minimum power satisfying this constraint. To this end, there is a well-known power-control algorithm given by

) , , ( ) , (

) , ( )

, , 1(

_

k b l k

k k

l b P

SNR th

P SNR _γ

γ γ

γ =

+ (2)

Where P_l(b,γ,k) and SNR_(γ,k) correspond to the power level and the SNR for STA γ at the l-th iteration, respectively.

This algorithm and its stability have been studied extensively by Foschini and Miljanic [23], Mitra [24], and Bambos, Chen, and Pottie [25],[26]. The algorithm is distributed and autonomous because it relies only on locally available information. It allows each user to have different target SNR values. It has also been shown in [3] to be asynchronously convergent (with geometric rate) to the Pareto optimal power assignment, when the system is feasible (i.e., when there exists a power assignment such that SNR for all). However, if the system becomes infeasible, this algorithm diverges.

2.3.2 Utility-Based Transmitter Power Control

(TPC)

Extend the paper in reference [27]. Although achieving satisfactory QoS is important for users, they may not be willing to achieve it at arbitrarily high powerlevels, because power is itself a valuable commodity. This observation motivates a reformulation of the whole problem using concepts from microeconomics and game theory [27]. In this section, we will use such a reformulation to develop a mechanism for power control where the desire for increased SIR is weighed against the associated cost. Instead of enforcing the constraint SNR_(γ,k) ≥SNR_th_(γ,k) as in the hard constraint case, we use a utility function U_(γ) to represent the degree of satisfaction of STA to the service quality, and introduce a cost function C_(γ) to measure the cost incurred. The goal is to maximize the net utility UN_(γ) defined as

) )( ) ( )( ) ( , )(

( SNRP U SNR C P

UN_γ = _γ − _γ by adjusting the transmitted power P_(b,γ,k). Since each user in the system will try to maximize its own net utility, regardless of what happens to the other users, this problem is a typical non-cooperative game.

Generally, the QoS depends on SNR, so we let the utility U_(γ) be a function of SIR U(_γ)(SNR)satisfying:

1. 0

) 0 )(

(_γ =

U ; 1

) )(

(_{γ ∞} =

U ,

2. The utility function

) )(

( SNR

U _γ is increasing in SNR. Because SNR is a function of power, so utility function

) )(

( SNR

U _γ is a function of power.

This means that utility is more and more satisfied with power increasing.

We choose

) )(

( P

C_γ , the cost for user, as a function of power. As mentioned

before, power is itself a valuable commodity. The specific cost function should reflect the expenses of power consumption to the STA. There are at least two requirements for the cost function:

1. 0

) 0 )(

(_γ =

C ,

2. C(_γ)(P) increases in power. That is to say,

) )(

( P

C_γ is a function of power.

In this paper, we will use a linear cost function, i.e., P

C P _{( )}*

) )(

(γ =α γ (3) where α_(γ) is the “price” coefficient. Although α_(γ) is a constant independent of power P, it can be a function of environmental factors such as user location and received interference. In fact, the use of an adaptive price setting can be helpful in achieving fairness and robustness. As is the case of the utility function, other forms of cost functions would also work for our scheme.

The net utility function is

) )( ) ( )( ) ( , )(

( SNRP U SNR C P

UN_γ = _γ − _γ . The power-control

problem for user is formulated as

) , )( ( 0 )

max

( ^SNRP P

NU_γ

γ ≥

(4) This means that our power control goal is to maximum the net utility of each STA.

Taking the derivative of

) , )( ( 0 )

max

( ^SNRP P

NU _γ

γ ≥

in (4), we have

) ( )

(

) ( )

(

)) ( ) ( ( ) , ( (

' ) ( '

) (

' ) ( '

) (

) ( )

( )

(

P P C

SNR SNR U

P dP C

SNR dSNR U

dP

P C SNR U

d dP

P SNR NU

d

γ γ

γ γ

γ γ

γ

−

=

−

=

= −

(5)

where '( )

U γ and C₍^'_γ) are the derivatives of U_(γ)and C_(γ), respectively. In the last step, we use the fact that SNR is linear with P, as defined by (1). If a positive power P is a local optimum for problem (4), we require ( _{( )}( , ) 0

dP = P SNR NU

d _γ

, i.e., P

P C SNR SNR

U₍^'_γ)( ) = ₍^'_γ)( )* (6) Considering (1) and (3), the above condition becomes

) , , (

) , , ( ) ( '

)

( ( )

k b

k b

G SNR R

U

γ γ γ

γ =α (7)

Where

∑

≠ Ω

∈

+

=

b d d

k k

d k d k

b G P

R

,

) , ( ) , , ( ) , , ( )

, ,

( γ γ γ υγ is the received interference of STA γ. Because the right-hand side of (11) is known or locally measurable, we find the solution to be

) (

) , , (

) , , ( 1 ) , ( ) , (

k b

k b k

k G

f R SNR

γ γ γ

γ = ⁻ α (8)

Where f_(γ,k)(SNR_(γ,k))=U₍^'_γ)(SNR_(γ,k)) in the concave part of U₍^'_γ,k), where a local maximum is possible. Then one candidate for the optimal power assignment is

) ˆ (

) , , (

) , , ( ) ( 1 ) , , ( ) , , (

) , , ( ) , , (

) , , ( ) , ( )

, , (

k b

k b k

b k b

k b

k b

k b k k

b G

f R G

R G

SNR R P

γ γ γ γ

γ γ γ

γ γ

γ = = ⁻ α (9) If we introduce the iteration index l, then (9) becomes

) , , ( ) , (

) , ( )

, , (

) , , ( ) , ( )

, , (

ˆ 1

k b l k k

k b

k b k k

b

l P

SNR SNR G

SNR R

P _γ

γ γ γ

γ γ

γ = =

+ (10)

Clearly, given the coefficients α_(γ),Pˆ_(b,γ,k), is a function of

) , , (

) , , (

k b

k b

G R

γ

γ . Now the

problem is whether ˆ_{( , , )}

k

Pb_γ defined above exists, and whether it is globally optimal (when it exists). Note that P_(b,γ,k) =0is on the constraint boundary of (4) and need not satisfy the above condition even when it is a maximum point. In fact, in some case

) 0

, ,

(_{b k} =

P _γ achieves the global optimum, though it corresponds to zero NU_(γ). Thus,

the optimal power P_(b,γ,k)^* is either ˆ_{( , , )}

k

Pb_γ or 0, whichever results in a larger net utility.

In order to satisfies our requirements for utility function, we choose the sigmoid utility function as mention in reference [27], which has been widely used in the study of neural networks.

The utility function for STA γ is

) ) (

, ( )

( _{( ) ( ) ( )}

1 ) 1

( γ αγ γ βγ

γ _k = + − _SNR −

e SNR

U (11) There are two tunable parameters in the Sigmoid utility (10): parameters α_(γ) and

) (γ

β , which can be used to tune the steepness and the center of the utility, respectively.

We illustrate the utility functions and the corresponding derivative functions for two different values of α_(γ) in Fig.2. When parameter α_(γ) increases, the utility becomes steep, and its derivative function becomes narrow and high.

Fig.2. Sigmoid utility versus SNR with different α

And in Fig. 3, we fix α_(γ) and vary β_(γ). The two utility functions and their

derivative functions have the same shape, but with different centers (atβ_(γ)). Based on the effect of the two parameters on our power control, we observe that the effect of the two parameters can be explained in integrated wireless systems with both voice users and data users, important in the third-generation wireless networks. For a voice user, the essential objective is low delay, and transmission errors are tolerable up to a relatively high point. Thus, the voice user does not want to be easily turned off, but its target SIR can be relatively low. This means that for a voice user, the cost should be

low, and the utility function should be steep and with low turnoff SIR.

Fig.3. Sigmoid utility versus SNR with different β

Conbining (7) and (11), we can derive ˆ_{( , , )}

k

Pb_γ

⎥⎥

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢⎢

⎢

⎣

⎡

−

⎟⎟

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜⎜

⎜

⎝

⎛

−

−

= 1 1

2 ln ˆ *

2

) , , (

) , , (

) ( )

, , (

) , , ( )

, , (

) ( ) , , ( ) , , (

k b

k b i k

b k b

k b

i k b k b

G G R

R G

P R

γ γ γ γ

γ γ

γ γ γ

α α

β (12)

2.4 Dynamic Channel Selection and Power Control Procedure

1. From the measurement stage, retrieve all SNR(γ, k) for all γ∈M_i and all k∈K.

2. For all k∈K, compute CSNR(k)=MIN{SNR(γ, k)| γ∈M_i}.

3. Choose c∈K such that CSNR(c)=MAX{CSNR(k)| k∈K } and let c be the output.

4. Update power _{( , , )}

) , (

) , ( )

, , (

) , , ( ) , ( )

, , (

ˆ 1

k b l k k

k b

k b k k

b

l P

SNR SNR G

SNR R

P _γ

γ γ γ

γ γ

γ = =

+ in the channel k=c.

2.5 Synchronous Iterative Power Control

2.5.1 Iterative Power Control

For the most part, analytical methods have derived convergence results for iterative power control algorithms that meet an SNR requirement for each user. In this work, we will see that for a broad class of power controlled systems in uplink, the users' SIR requirements can be described by a vector inequality of interference constraints of the form:

) ( p I

p≥ (13) In this case, p=(p₁,...,p_N), where p denotes the transmitter power of STA j and _j

)) ( ),..., ( ( )

(p I₁ p I p

I = _N where I_j =( p) denotes the effective interference of other users that STA j must overcome. We will say that a power vector p≥0 is a feasible solution if p satisfies the constrains (13) and that an interference function I(p) If (13) has a feasible solution. In addition, if under power vector p , )p_j ≥I_j(p , then we say STA j has an acceptable connection.

For a system with interference constraints (13), we will examine the iterative power control algorithm

)) ( ( ) 1

(t I p t

p + = (14)

2.5.2 General Interference Constraints

We assume N users, M base stations and a common radio channel. The transmitted power of STA j is p_{( j)}. Let G_(b,j) denote the gain of a STA j to AP b. At

AP b, the received signal power of STA j is G_(b,j)p_(j) while the interference seen by STA j at AP b is

∑

≠ Ω

∈

+

b d d

j d j

d P b

G

,

) , ( ) ,

( σ( ), where σ(b) denotes the receiver noise power at AP b. Hence, under power vector p , the SNR of STA j is p μ

Where

∑

≠

= +

j b

j b j

b G b

p G

λ λ σ

μ ( ) ( )

) , (

) , ( )

,

( (15) We now express the interference constraints of a number of systems in the form of (13).

․Fixed Assignment: We will denote by b_j the assigned AP of STA j, which we assume to be fixed or specified by outside means such as the received signal strength of AP pilot tone signals. The SNR requirement of STAj at its assigned AP bj can be

written _{( )}

) , ) ( ,

( ( ) _j

j j b

b p

p

j μ j ≥γ . That is, we can write

) ) (

( ^{( )}

) ) (

,

( I p p

p

j b

j j

FA j b

j

j μ

= γ

≥ (16)

․M^Minimum Power Assignment (MPA): At each step of this iterative procedure, STA j is assigned to the base station at which its SNR is maximized. The convergence of the MPA iteration has been analyzed by Yates and Huang [32,33] and Hanly [34] for continuous power adjustments, and by Stolyar [35] for discrete power adjustments.

MPA can also be considered a generalization of soft handoff; see [36]. The SNR constraint of STA j is _bj p_{b j b j} p _j

j

j μ ( )≥γ

max _{( , ) ( , )} , which can be written

) ) (

( ^{( )}

) ) (

,

( ^{I p}

min

_p

p

j b

j

b j

MPA j

b

j j

j μ

= γ

≥ (17)

In the MPA iteration p_(t+1) =I^MPA(p(t)), STA j is assigned to the AP k where minimum power is needed to attain the target SNR γj, under the assumption that the other users hold their powers fixed.

For an arbitrary interference function I p

( )

=

(

I1

( )

p ,L ,I_N

( )

p

)

, we make the following definition.

Definition: Interference function ^{I p}

( )

is standard if for all p≥0 the following properties are satisfied.

•Positivity ^{I p}

( )

^>0

•Monotonicity If

p ≥ p′

, then ^{I p}

( ) ( )

^{≥I p′}

•Scalability For all

α

>1,

^α

^{I p}

( ) ( )

^>I

^α

^p

We adopt the convention that the vector inequality

p ≥ p′

is a strict inequality in all components. The positivity property is implied by a nonzero background receiver noise. The scalability property implies that if

p

j

≥ I

j

( ) p

then

( ) ( )

j j j

p I p I p

α ≥ α > α

for

α

>1. That is, if user j has an acceptable connection under power vector p, then user j will have a more than acceptable connection when all power are scaled up uniformly. Note that positivity and convexity of

I

j

( ) p

for all j implies scalability; however, the converse does not hold.

We note that _(b,j)(p)

μ j satisfies

) ( )

( _{( , )}

) ,

(_{b j} p _{b j} p

j

j μ

μ ≥ (p≥ p^') (17)

α α μ

μ_{( , )}( ) ^{( , )(}p⁾

p ^{b j}

j b

j

j ≥ (α >1) (18)

2.5.3 Synchronous Iterative Power Control

When I(p) is a standard interference function, the iteration (14) will be called the standard power control algorithm. In this section, we examine the convergence properties of standard power control under the assumption that I(p) is feasible. When I(p) is infeasible, we have a call admission problem [18, 23, 24] in finding a subset of users that can obtain acceptable connections. In addition, the feasibility of I(p) is highly dependent on the underlying wireless system implementation while this work emphasizes the common properties of interference based systems.

Starting from an initial power vector p, n iterations of the standard power control algorithm produces the power vector Iⁿ( p). We now present convergence results for the sequence Iⁿ( p)

Theorem 1 If the standard power control algorithm has a fixed point, then that fixed point is unique.

Proof: Theorem 1 Suppose p and

p′

are distinct fixed points. Since ^{I p}

( )

^>0

for all p, we must have p_j >0 and p ′_j >0 for all j. Without loss of

generality, we can assume there j such that p_j < p ′_j . Hence, there exist

α

>1 such that

α p ≥ p ′

and that for some j,

α

p_j = p_j^′. The monotonicity and scalability properties imply

( ) ( ) ( )

j j j j j

p ^′ =I p′ ≤ I

α

p <

α

I p =

α

p (19) Since p_j′ =

α

p_j, we have found a contradiction, implying the fixed point must be unique.

Lemma 1 if ( )I p is feasible, then staring from z ,the all zero vector, the standard power control algorithm produces a monotone increasing sequence of power vectorsIⁿ( )z that converges to the fixed point p^*.

Proof：Lemma 2 Let ( )z n =Iⁿ( )z . We observe that z(0)< p^* and that (1) ( )

z =I z ≥ . Suppose z z≤z(1)≤ ≤... z n( )≤ p^*,monotonicity implies

* *

( ) ( ( )) ( ( 1)) p =I p ≥I z n ≥I z n−

That is, p^*≥z n( + ≥1) z n( ) .Hence the sequence of ( )z n is nondecreasing and bounded above by p^*.Theorem 1 implies ( )z n must converge to p^*.

Theorem 2 if ( )I p is feasible, then for any initial power vector p , the standard power control algorithm converges to a unique fixed point p^*.

Proof： Feasibility of ( )I p implies the existence of the unique fixed pointp^*. Since

*_j 0

p > for all j , for any initial p , we can find α ≥1 such that αp^* ≥ . By the p scalability property, αp^* must be feasible . Since z≤ ≤p αp^*, the monotonicity property implies

( ) ( ) ( *)

n n n

I z ≤I p ≤I αp

Lemmas 1 and 2 imply lim ⁿ( ) ^*

n I z p

→∞ = and the claim follows.

We have shown that for any initial power vector p , the standard power control algorithm converges to a unique fixed point whenever a feasible solution exists.

3. Simulation

3.1 Simulation Models

The simulation environment we considered is a wireless network such as Fig.4.

There are 37 BSSs in our simulation system, and there are 10 STAs in each BSS. In each connection, AP transmits data to one STA only. The available channel numbers are 6, 12 (in 802.11a) or 19 (802.11h). In the system, all APs use omnidirectional antennas and are located at the centers of the BSSs. The distance of each AP is about 17 meters. We assume that the locations of all STAs in a BSS are uniform distribution.

We consider that implement dynamic channel selection period is after 10 times power update iteration. We consider the case that implement by power control and dynamic channel selection is after 10 times power update iteration, dynamic channel selection will be performed. That is to say, in our simulation, we record 10 times dynamic channel selection and 100 times power control update results in one experiment. In each dynamic channel selection, if the SNR value of connection STAs in BSS is blew down

2

) (γ

β , then the AP will perform the dynamic channel selection. The path gain

is modeled as

) , , (

) , , ( ) , , (

k b

k b k

b d

G A

γ γ

γ = , where d_(b,γ,k) is the distance between AP b and STA

γ, and the attenuation factor A_(b,γ,k) models the power variation due to shadowing.

We assume that A_(b,γ,k) are independent and identically log-normally distributed random variables with 0-dB expectation and 8-dB log-variance.

.

Fig.4. Cell wireless network system

In our simulation, we compare the case only implement by power control with the case that implement by power control and dynamic channel selection. We investigate the factors β_(γ), and channel number. The factor β_(γ) is target SNR,

which implies that the higher the β_(γ) is, the more difficult the STAs connection to the network is. The channel number implies that the fewer channel number is, the more interference the simulation system is.

And we observe the major results in average SNR of STAs, sverage net utility of STAs, channel assignment result, and total connection rate of STAs:

1. Average SNR of STAs: In our simulation, we observe the average SNR of STAs in each power update iteration. From average SNR of STAs, we can comprehend the signal strength in each power update iteration. It is also an important index of quality of service.

2. Average Net Utility of STAs: In our simulation, average net utility is also a meaningful observation. From average net utility in each power update iteration, we can understand the average net utility, maybe can be called as satisfaction of users during connection period.

3. Channel Assignment Result: From channel assignment result, we can observe the channel change trend chart.

4. Total Connection Rate of STAs: In order to compare the case only implement by power control with the case that implement by power control and dynamic channel selection, the total connection rate of STAs in each power update iteration is an important index, because of it presents the use rate of the network. The higher the total connection rate is, the more users can be serviced by the system provided.

Therefore, the total connection rates of STAs are more and more better.

3.2. Results and Analysis

We show some cases of our experiment results in the section. Extend the Utility-based distributed power control in the paper[27], and compared with our proposed dynamic channel selection and power control.

Case 1. The BSSs number in our simulation system are 37, and there are 10 STAs in each BSS. The β_(γ)=15 and the channel number is 6.

Fig.5. β_(γ)=15 and the channel number is 6

Fig.6. β_(γ)=15 and the channel number is 6

Case 2. The β_(γ)=20 and the channel number is 6.

Fig.7. β_(γ)=20 and the channel number is 6

Fig.8. β_(γ)=20 and the channel number is 6

Compared Case 1 in Fig.5 and Fig.6 with Case 2 in Fig.7 and Fig.8, we can observe that average net utility and connection rate go down obviously with β_(γ)increasing.

This is because that as β_(γ) increasing, there are more STAs that will be dropped, then connection rate will go down, so the average net utility and connection rate go down obviously. In Case 1 and in Case 2, we can observe that the average net utility and connection rate are improved significantly by performing dynamic channel selection scheme.

Case 3. The β_(γ)=20 and the channel number is 12.

Fig.9. β_(γ)=20 and the channel number is 12

Fig.10. β_(γ)=20 and the channel number is 12

Case 4. The β_(γ)=20 and the channel number is 19.

Fig.11. β_(γ)=20 and the channel number is 19

Fig.12. β _γ =20 and the channel number is 19

Compare with Case 3 in Fig.9 and Fig.10 with Case 4 in Fig.11 and Fig.12, we can observe that average SNR, average net utility and connection rate rise with channel number increasing obviously. This is because that with channel number increasing, there are more channel that APs can dynamic select, so interfere by other APs can be improved, then average SNR increase. Therefore, the average net utility and connection rate increase obviously. In Case 3 in Fig.9 and Fig.10 with Case 4 in Fig.11 and Fig.12, we can observe that the average SNR, average net utility and connection rate are improved significantly by performing dynamic channel selection scheme.

6. Conclusion and Future Works

In this paper, we provided a dynamic channel selection algorithm and compared utility-base power control schemes with and without this dynamic channel selection algorithm into a multi-channel wireless network. We analyzed the two schemes by the three points of view, average SNR, average net-utility, and connection rate. In our discussion and experiment results, we discovered several remarkable results. Firstly, we discovered no matter how many channels we have, the scheme with DCS is never worse than the scheme without DCS in a long run in all three points of view. Secondly, we discovered that in different SNR thresholds, the scheme with DCS is never worse than the scheme without DCS in a long run in all the points of view, either. Finally, the same result was discovered when we try different numbers of STA of a BSS. In another words, the utility-base power control scheme with DCS never worse than that without DCS in all of our three different environments.

The feature study may head to the following possible directions. The centralized algorithm could be considered to develop the dynamic channel selection. It could shorten the number of iterations before stabilization. Considering the building effects on a wireless network could have different results. Applying DCS to time-division wireless networks or power-division wireless networks. It could have interesting results.

Reference

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