6
Admission Control in Wireless Mesh Networks based on Game Theory Rong He, Xuming Fang Provincial Key Lab of Information Coding & Transmission Southwest Jiaotong University Chengdu 610031, China E-mail: [email protected], [email protected] Abstract—Admission control in Wireless mesh networks (WMNs) and Wireless local area network (WLAN) plays an important role in achieving fair resource schedule and load balance. Both Stations (STAs) and mesh AP (MAP) encounter the problem that how to select a proper access point in the access process. This paper addresses the problem of admission control game from the perspective of the stations and the service provider, and formulates the admission control process between STA and MAP as a nonzero-sum, non-cooperative and mixed strategy game. Three decisive factors and mixed strategy probability function for MAP and STA are defined. The calculated steps of the weight value for each decisive factor using Analytic Hierarchy Process (AHP) are discussed. Keywords—admission control; wireless mesh networks; game theroy; mixed strategy I. INTRODUCTION WMNs have experienced huge developments which expand the hot-spot of WLAN to hot-area using mesh architecture. Admission control scheme in WMNs and WLAN plays an important part in guaranteeing fair resource allocation and load balance of network. Both new STA and MAP have to cope with admission control problems, such as how to select a proper MAP, and whether to accept the request of a new STA. The goals of MAP and new STA are to seek the maximum of their utilities, which are different and often conflict with each other. Sharing much in common with the problem of selection in heterogeneous wireless networks, admission control problem can be modeled as a game process of optimal admission control strategy decision, which is determined by decisive factors. The main difference between admission control and heterogeneous network selection scheme lies in the game players and restrictive conditions. The selection strategy in heterogeneous network is implemented among new STA and several heterogeneous networks, while admission control scheme is carried out among new STAs and MAPs. Therefore, the characteristic of heterogeneous networks should not be taken into consideration. Also, network selection in the heterogeneous networks faces much more challenges, such as different interface, different pricing mechanism and vertical handoff problem. There is a significant amount of work in heterogeneous network selection mechanism. Current researches mainly focus on how to define the quality of service (QoS) criteria for different stations and jointly resource allocations [1][2][4][5] However, the current works on admission control in WMNs and WLAN are not rich. Most of these works aim at handling the utility maximum from the aspect of either stations or network providers. But these works seldom take the admission control issue into account both on station-centric benefit and network provider-centric benefit. Haris et al. consider different station requirements in [1] and map these requirements to the competing network conditions as a part of heterogeneous network selection model. The avoidance of frequent vertical handoffs to minimize the loss of revenue for network providers is also addressed. Jia Hu et al. formulate in [7] the admission control in WLAN as a non-cooperative non-zero-sum (n+1)-player game. The utility for stations in different classes is expressed as a function of the end-to-end delay and frame loss probability for real traffic and non-real traffic, respectively. It considers only single factor for decision making. On the other hand, most current research works use single factor as impacting the decision of admission control. However, both STA and MAP should take several factors into account for selecting a proper strategy. So it is a multi-objective decision- making problem for admission control. As far as the admission control scheme is concerned, the key issue is how to construct an efficient game model and decision-making function so as to maximize the utility of individual player. Apart from that, the tradeoff of the payoff obtained by accessing a new STA and the decrease of satisfaction of ongoing stations should be sorted out. Game theory and pricing mechanism have recently been applied to many areas of wireless communication, including power control, resource allocation, load balance, access control, congestion control and etc. On the other hand, analytic hierarchy process (AHP) has been widely applied to communication engineering design, especially in ability evaluation. It is an effective means of finding the optimal solution to a complex decision problem by synthesizing all problem-deciding factors [9]. Based on pricing mechanism and game theory, this paper attempts to investigate how to maximize the utility of individual player while meeting the QoS requirements of ongoing stations and new stations from the perspective of both station and network providers. We model the admission control process between new stations and MAP as a nonzero-sum, non-cooperative and mixed strategy game. Three decisive factors and mixed strategy probability This Work was supported by the Fundamental Research Funds for the Central Universities (No. SWJTU12CX097) and National Science Foundation (61071108). 978-1-4673-5939-9/13/$31.00 ©2013 IEEE 978-1-4673-5939-9/13/$31.00 ©2013 IEEE 2013 IEEE Wireless Communications and Networking Conference (WCNC): NETWORKS 2013 IEEE Wireless Communications and Networking Conference (WCNC): NETWORKS 1303

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Page 1: [IEEE 2013 IEEE Wireless Communications and Networking Conference (WCNC) - Shanghai, Shanghai, China (2013.04.7-2013.04.10)] 2013 IEEE Wireless Communications and Networking Conference

Admission Control in Wireless Mesh Networks based on Game Theory

Rong He, Xuming Fang Provincial Key Lab of Information Coding & Transmission

Southwest Jiaotong University Chengdu 610031, China

E-mail: [email protected], [email protected]

Abstract—Admission control in Wireless mesh networks (WMNs) and Wireless local area network (WLAN) plays an important role in achieving fair resource schedule and load balance. Both Stations (STAs) and mesh AP (MAP) encounter the problem that how to select a proper access point in the access process. This paper addresses the problem of admission control game from the perspective of the stations and the service provider, and formulates the admission control process between STA and MAP as a nonzero-sum, non-cooperative and mixed strategy game. Three decisive factors and mixed strategy probability function for MAP and STA are defined. The calculated steps of the weight value for each decisive factor using Analytic Hierarchy Process (AHP) are discussed.

Keywords—admission control; wireless mesh networks; game theroy; mixed strategy

I. INTRODUCTION WMNs have experienced huge developments which expand

the hot-spot of WLAN to hot-area using mesh architecture. Admission control scheme in WMNs and WLAN plays an important part in guaranteeing fair resource allocation and load balance of network. Both new STA and MAP have to cope with admission control problems, such as how to select a proper MAP, and whether to accept the request of a new STA. The goals of MAP and new STA are to seek the maximum of their utilities, which are different and often conflict with each other. Sharing much in common with the problem of selection in heterogeneous wireless networks, admission control problem can be modeled as a game process of optimal admission control strategy decision, which is determined by decisive factors. The main difference between admission control and heterogeneous network selection scheme lies in the game players and restrictive conditions. The selection strategy in heterogeneous network is implemented among new STA and several heterogeneous networks, while admission control scheme is carried out among new STAs and MAPs. Therefore, the characteristic of heterogeneous networks should not be taken into consideration. Also, network selection in the heterogeneous networks faces much more challenges, such as different interface, different pricing mechanism and vertical handoff problem.

There is a significant amount of work in heterogeneous network selection mechanism. Current researches mainly focus on how to define the quality of service (QoS) criteria for different stations and jointly resource allocations [1][2][4][5]

However, the current works on admission control in WMNs and WLAN are not rich. Most of these works aim at handling the utility maximum from the aspect of either stations or network providers. But these works seldom take the admission control issue into account both on station-centric benefit and network provider-centric benefit.

Haris et al. consider different station requirements in [1] and map these requirements to the competing network conditions as a part of heterogeneous network selection model. The avoidance of frequent vertical handoffs to minimize the loss of revenue for network providers is also addressed. Jia Hu et al. formulate in [7] the admission control in WLAN as a non-cooperative non-zero-sum (n+1)-player game. The utility for stations in different classes is expressed as a function of the end-to-end delay and frame loss probability for real traffic and non-real traffic, respectively. It considers only single factor for decision making.

On the other hand, most current research works use single factor as impacting the decision of admission control. However, both STA and MAP should take several factors into account for selecting a proper strategy. So it is a multi-objective decision-making problem for admission control. As far as the admission control scheme is concerned, the key issue is how to construct an efficient game model and decision-making function so as to maximize the utility of individual player. Apart from that, the tradeoff of the payoff obtained by accessing a new STA and the decrease of satisfaction of ongoing stations should be sorted out.

Game theory and pricing mechanism have recently been applied to many areas of wireless communication, including power control, resource allocation, load balance, access control, congestion control and etc. On the other hand, analytic hierarchy process (AHP) has been widely applied to communication engineering design, especially in ability evaluation. It is an effective means of finding the optimal solution to a complex decision problem by synthesizing all problem-deciding factors [9]. Based on pricing mechanism and game theory, this paper attempts to investigate how to maximize the utility of individual player while meeting the QoS requirements of ongoing stations and new stations from the perspective of both station and network providers. We model the admission control process between new stations and MAP as a nonzero-sum, non-cooperative and mixed strategy game. Three decisive factors and mixed strategy probability

This Work was supported by the Fundamental Research Funds for the Central Universities (No. SWJTU12CX097) and National Science Foundation (61071108).

978-1-4673-5939-9/13/$31.00 ©2013 IEEE978-1-4673-5939-9/13/$31.00 ©2013 IEEE

2013 IEEE Wireless Communications and Networking Conference (WCNC): NETWORKS2013 IEEE Wireless Communications and Networking Conference (WCNC): NETWORKS

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function for MAP and STA are defined. The calculate steps of the weight value for each factor using AHP are discussed.

The rest of this paper is organized as follows. The non-cooperative game model is described in Section II. Section III presents three decisive factors and mixed strategies probability functions of MAP and STA, respectively. The calculation steps of the weight value for decisive factors based on AHP are discussed in Section IV. Section V analyses the Nash equilibrium Characteristic of this model. Finally, Conclusions are given in Section VI.

II. NON-COOPERATIVE GAME MODEL AND ASSUMPTION The admission control process is formulated as a non-

cooperative non-zero-sum game, where the MAP and n-STA (new stations) are players. Every access request of STA is associated with some parameters, such as the minimum required resource and expected price. On the other hand, MAP broadcasts the access price and the resources it can offer. The goals of STA and MAP are to maximize their utilities by selecting proper strategy.

We define an admission control game G={I,{SA, Si},{Hi}} in strategic form by following three elements. I is the set of players and there are n+1 players, that are one MAP and n new stations (STA1, STA2,…, STAn∈I). Note that, in this paper, the MAP can choose to either accept only one new station’s access request or reject all request. SA= {0, 1, 2,…, n} is the pure strategy set of MAP, where SA =0 represents to reject access requests of all new stations; SA =1 denotes to accept the access request of the first new station, while SA =n means to accept the access request of the n-th new station. The strategy set of the i-th new station is Si= {0, 1}, where Si =0 denotes to accept the service provided by MAP and Si =1 means to deny the service. To simplify the system model, we suppose that there is no information exchange among STAs.

We designate the joint set of the strategy space of all players as follows nA SSSS ×××= 1 . The set of chosen strategies within a game constitutes a strategy profile { }nA sssss ,,,, 21 = . Hi(s) is the payoff function of i-th player. In a nonzero-sum game, the aggregate payoff of all

players is nonzero, that is 0)(1

1≠∑

+

=

n

ii sH . The strategy of a STA

is related with that of MAP during an admission control game process. When MAP accepts the access request of i-th new station, only the strategy chosen by i-th new station is valuable while strategies selected by other new stations have no impact on the payoff. So the payoff matrices of the MAP and the i-th new station can be denoted as

22)1(][1 ×××+=

nSSSMAP nAmH

22)1(][1 ×××+=

ni

SSSSTA nAibH (1)

where nA SSSm

1and i

SSS nAb

1represent the payoff of the MAP

and the i-th STA respectively. If the strategy selected by MAP is SA =i (i=0, 1, … , n) and the i-th STA selects strategy bi, the

value of each element in payoff matrices HMAP and iSTAH are

defined respectively as

∑==≤≤

−+=0&&1

)(1

iA

nAbiSni

iiSSS LMMm

(2)

[ ])1()1(1

11 iiii

AiSSS USUS

niS

bnA

−+−•

+

−−=

(3)

where M is the total payoff by accessing the ongoing stations and Mi is the payoff acquired by accepting the i-th new station’s request. Li represents the decrease in payoff of the ongoing stations after accepting the i-th new station.

+

−−

11

niS A is adopted to decide whether to accept the i-th

station’s request. If the MAP chooses to accept the the i-th

station’s request,

+

−−

11

niS A is valued 1, otherwise valued 0.

Ui is the normalized payoff of the i-th STA after it accepts the service and (1-Ui) is the payoff of the STA to deny the service.

There are several factors which should be taken into consideration for STA and MAP to select strategy. So it is a multi-objective decision-making problem for admission control. In this paper, STA makes decision that either accepts or denies the service offered by MAP according to three factors, which are QoS satisfaction degree u, access price p and signal to noise ratio (SNR) s detected by STA. We define function u(x) as a measure of station’s QoS satisfaction level, where x is the resource allocated to the station in normalized value. The resource may be bandwidth, transmission power, or time frequency block and etc. u(x) is usually defined as a sigmoid function, thus should meet the law of diminishing marginal utility [14], that is

0)(≥

dxxdu , 0)(lim =

∞→ dxxdu

x (4)

We employ the bandwidth occupied time proportion as the resource allocated to station. The value of x is normalized, that is 10 ≤≤ x . We define the station's QoS satisfaction degree function as [10], which can facilitate the representation of the station’s satisfaction degree from the lowest to highest level.

)( minmax11)( rxre

xu −−+= (5)

where the value of rmax implies the station’s sensitivity to the change of QoS performance and rmin indicates the threshold of QoS performance value that station can tolerate. Both rmax and rmin are positive number and related with the inherent QoS requirement of different traffic classes. Different traffic classes have diverse QoS preference. Furthermore, the value of rmax and rmin can impact the slope degree and center of the curve respectively. Since the QoS perceived by station has a threshold, the value of u(x) increases with the incremental of x until reaching the threshold. Therefore, u(x) should match the following condition:

gxux

=∞→

)(lim (6)

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where g is a positive constant. Thus, the value of u(x) is between zero and g.

We adopt the price function in [4] to characterize the access price as follows:

ε)1(

)( 0

xpxp−

= (7)

where p0 is the original price constant. Its value can be adjusted as different QoS requirements. Parameter ε determines the increased rate in price with the incremental of resource x allocated to station. The larger the value of ε is, the steeper the price curve will rise.

On the other hand, MAP should make a decision that either accepts one request among n new stations or rejects all requests. It depends on three factors, which are revenue r, load l and quitting probability q of ongoing stations. We measure the load via the number of allocated resource, which can reflect the available resource of the MAP. The more the resource has been allocated, the heavier the load was. Because the resource provided by MAP is limited, accepting a new station may degrade the QoS satisfaction degree of the ongoing stations, then quit the network. Therefore we introduce the quitting probability as one of the decisive factors.

III. MIXED STRATEGIES FUNCTION According to related theorems of game theory, there may

exists no pure-strategy Nash equilibrium for non-cooperative n-player game; while there always exists at least a mixed strategy Nash equilibrium[13]. Therefore, we employ a mixed strategy model and define *

AS and *iS as the mixed strategy

sets of SA and Si. A player can select each pure strategy with some probabilities. *

ASB∈ and *iSA∈ are the mixed strategy of

MAP and the i-th STA respectively. MAP selects pure strategy i (means accepting one station’s requests, ni ≤<0 ) with probability B, and selects pure strategy 0 (means rejecting all access requests) with probability 1-B. Similarly, the i-th STA selects pure strategy 0 with probability A and selects pure strategy 1 with probability 1-A. The value of A and B are determined by mixed strategy probability function A(u, p, s) and B(r, l, q) respectively.

Inspired by Cobb-Douglas requirement curve which is widely used in economics, we define the mixed strategy probability function A(u, p, s) as (8)

θβα spcuespuA−−−= 1),,( (8)

where c is a positive constant, α, β and θ are variable parameters within 0 and 1, which can be used as influential factors for function A(u,p,s). These parameters indicate station’s sensitive degree for QoS satisfaction degree, access price and link quality. The value of function A(u,p,s) goes up along with the incremental of u(x) and SNR; while it decreases along with the incremental of access price. Thus, function A(u,p,s) should meet with the conditions as follows:

0,0,0 ≥∂∂

≤∂∂

≥∂∂

sA

pA

uA

1),,(,0),,(,0,0 limlim0

==>>∀∞→→

spuAspuAspuu

0),,(,1),,(,0,0 limlim0

==>>∀∞→→

spuAspuAsupp

1),,(,0),,(,0,0 limlim0

==>>∀∞→→

spuAspuApuss

(9)

Similarly, we define the mixed strategy probability function B(r, l, q) as

ηκφ −−−−= qlreqlrB 1),,( (10) where Φ, κ and η can be regarded as influential factors for B(r, l, q) and represents MAP’s sensitive degree to revenue r, load status l and the quitting probability q. The restricting conditions of B(r, l, q) share much in common with those of A(u, p, s). The quitting probability q is related to the average QoS satisfaction degree of the ongoing stations. The higher the average satisfaction degree is, the lower the quitting probability is. We define the quitting probability q as (11)

=

=

−= k

ii

k

ii

xk

xuq

1

11 (11)

where k is the number of ongoing stations and xi is the resource allocated to the i-th STA. )(

1∑=

k

iixu reflects the QoS

satisfaction degree of k ongoing stations with the sum of resource allocated to them.

We define the load function as xxlk

ii +=∑

=1

, and

revenue function as follows

qkfxpxprk

ii −+= ∑

=

)()(1

(12)

where ∑=

k

iixp

1)( is the revenue obtained by accessing the

ongoing stations and p(x) is the new added revenue acquired by accepting a new station’s request with allocated resource x. The unit lost revenue due to one ongoing station quitting the network is denoted by a constant number f and its value will be discussed later. The total lost revenue is qkf.

IV. CALCULATION STEPS OF THE WEIGHT VALUE BASED ON AHP

The parameters in A(u,p,s) and B(r,l,q), such as α, β and θ, impact the probability of selecting different strategy for STA and MAP. We apply the hierarchy single sorting method of AHP to determine the weight value of each decisive factor. There are three levels from top to bottom according to different criteria characteristic. Fig.1 and Fig.2 represent the hierarchical structures on admission control process for STA and MAP respectively. The overall objective is placed at the topmost layer of the hierarchy and the player’s preference criteria are set at the middle layer. The strategy alternatives are located at the bottom layer.

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The relative importance of each criteria objective, namely, the weight value of related parameters in mixed strategy function A(u ,p ,s) and B(r, l, q), can be computed via square root means. The detail steps are as follows:

Fig. 1. Hierarchical structures for STA

Fig. 2. Hierarchical structures for MAP

The relative importance of each criteria objective, namely, the weight value of related parameters in mixed strategy function A(u ,p ,s) and B(r, l, q), can be computed via square root means. The detail steps are as follows:

∙ Every criterion is compared against all criteria within the same parent to decide the relative importance of each criterion. The comparisons with the same parent result in a square matrix ( )

nnijaM×

= , where n is the

number on criterion and ija represents the relative degree of importance for the i-th criterion against the j-the criterion,

ijji a

a 1= Authors and Affiliations

∙ Compute the geometric means of all criteria in each row of the matrix, that is ( )Tnwwww ,, 21= . The element

of iw can be expressed as

niaw n

n

jiji ,,2,1

1

== ∏=

(13)

∙ Normalize iw as niw

ww n

ii

ii ,,2,1

1

==

∑−

, thus

obtain the correlative weight vector ( )Tnwwww ,, 21=

∙ Compute the maximum eigenvalue λmax of matrix according to (14)

∑=

=n

i i

i

nwMw

1max

)(λ (14)

where ( )iMw is the i-th component of vector Mw.

∙ Compute the coincidence indicator CI and coincidence ratio CR to verify the coincidence of the matrix.

1max

−−

=n

nCI λ (15)

If λmax=n and CI=0, the matrix has the complete coincidence. Since it is difficult to meet the complete coincidence due to some complicated reasons, there is λmax >n. With the increase of dimension n, the coincidence becomes worse. The bigger the value of CI is, the worse the coincidence of matrix is.

We introduce CR=CI/RI as the coincidence indicator of matrix, where RI is random indicator. According to [13], RI is set to 0.58 and n is set to 3 in the paper; and if CR<0.1, the estimated value of aij is considered as acceptable.

V. PERFORMANCE ANALYSIS AND EVALUATION To simplify the performance analysis, we consider 2-player

game including one STA and one MAP. The matrices for STA and MAP are defined in Table 1 and 2. Using square root means, we can compute the relative weight value of three parameters α, β and θ in A(u, p, s), which are QoS satisfaction degree, access price and SNR for STA. The values are 0.636986, 0.258285 and 0.104729 respectively. Similarly, the relative weight value of three parameters Φ, κ and η in B(r, l, q) are 0.614411,0.117221and 0.268369 respectively.

TABLE I. MATRIX OF STA

QoS satisfaction degree

Access price SNR

QoS satisfaction degree 1 3 5

Access price 1/3 1 3

SNR 1/5 1/3 1

TABLE II. MATRIX OF MAP

Revenue Load The quitting probability

Revenue 1 3 5

Load 1/3 1 3 The quitting probability 1/5 1/3 1

Goal Decide whether to accept the service

Criteria

QoS Satisfaction

degree Access price SNR

Accept the service

Reject the service

Strategy

Goal Decide whether to accept request of STA

Criteria Revenue Load The quitting probability

Accept one STA Reject all

STA

Strategy

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Our next step is to verify the coincidence and compute the maximum latent root of the matrix of STA

=

=

318222.0784802.0935488.1

104729.0258285.0636986.0

13/15/1313/1531

Mw (16)

Then calculate

038511.33

)(3

)(3

)()(

3

3

2

2

1

1

1max =++==∑

= wMw

wMw

wMw

nwMwn

i i

019256.02

3038511.31

max =−

=−−

=n

nCI

λ

when n=3, RI=0.58,there are

1.0033199.058.0

019256.0<===

RICICR

Similarly, we can calculate the value of maxλ and CR for MAP, that are 073514.3max =λ and 1.0063374.0 <=CR

We set ε=1,rmin=0.2 and p0=1. In order to simplify the analysis, SNR is assumed to be a constant value 20. During the process of admission control game, we can obtain the change tendency of probability function A(u, p, s) according to different resource number x allocated to new station in terms of (8). Fig.3 illustrates the variation of probability A for STA selecting acceptance strategy under different rmax, which is 10, 20 and 30. It can be observed that the probability curve is a convex parabola. At first, it increases with the increment of x. Then it decreases with the increment of x after reaching a maximum value. In addition, station’s QoS satisfaction degree and access price will increase with the increment of resource number allocated to STA. The former part of probability curve is in an ascending tendency due to the increment of QoS satisfaction degree which plays dominative role. However, access price becomes a dominating factor after reaching a maximum point. So the probability curve began to decrease. The value of rmax indicates the slope degree of probability curve. The greater the value of rmax is, the more precipitous the curve is. It conforms to the characteristic of station’s QoS satisfaction degree function.

Fig. 3. Probability A with different rmax.

Given rmax=20, rmin=0.2, the number of ongoing stations is fixed to 2 and the allocated resource number is 0.2. Fig. 4 shows the changing tendency of probability B for MAP selecting acceptance strategy with resource number x allocated to new station under different value of f in (12), which is 1, 1.5, and 2. We can see that the curve of probability B increases

according to the enhancement of x. Under the same value of x, the bigger the value of f is, the smaller the probability B is.

Fig. 5 and Fig. 6 show that the probability function B (r, l, q)and the quitting probability of MAP vary with loads respectively. The minimum requested resource x of STA is set to 0.2. As seen in Fig. 5 and Fig. 6, the quitting probability of the ongoing stations for MAP increases along with the increment of loads; while the probability for MAP selecting acceptance strategy is in drop tendency according to the rise of load. This is because that if load increases, the revenue of MAP will also increase but the QoS satisfactory degree of station may reduce. This will lead to the increment of the quitting probability of ongoing stations for STA. Since the impact degree of revenue and the quitting probability on probability function B(r, l, q) is different, the curve of B(r, l, q) presents a decreasing tendency with little fluctuation.

Fig. 4. Probability B with different f

Fig. 5. Probability B vary with load of MAP

Fig. 6. The quiting probability of onging stations vary with load of MAP

Note that Fig.3~Fig.6 are obtained under the assumed parameters. We can calculate and draw corresponding curve of probability according with different parameters.

0.2 0.4 0.6 0.8x

0.55

0.6

0.65

0.7概概 AProbability A

x 0.8 0.6 0.4 0.2

0.55

0.6

0.65 0.7

rmax=20 rmax=30

rmax=10

0.2 0.4 0.6 0.8x

0.6

0.7

0.8

0.9

概概 BProbability B x

0.9

0.8

0.7

0.6

0.8 0.6 0.4 0.2

f=1 f=1.5 f=2

0.710.720.730.740.750.760.770.78

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8The Load of MAP

Prob

abili

ty B

0.65

0.7

0.75

0.8

0.85

0.9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8The Load of MAP

The

Qui

tting

Pro

babi

lity

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VI. NASH EQUILIBRIUM CHARACTERISTIC According to Nash theorems, there always exists at least

one mixed strategy Nash equilibrium for non-cooperative n-player games [13]. In finite nonzero-sum two-player game, there exist *

1* SA ∈ , *

2* SB ∈ , which meet equation (17) and

(18)

*1

*** SABHABHA STAT

STAT ∈≥ (17)

*2

*** SBBHABHA STAT

MAPT ∈≥ (18)

where HSTA and HMAP are payoff matrix defined as equation (1). In order to obtain the equilibrium point, we should know the value of each element of each payoff matrix. According to [13], we can conclude the sufficient and necessary conditions for profile (A, B) being the equilibrium point are as follows

≥−≤−−−

≥−≤−−−

00)1()1(

00)1()1(

rBRABBrBRA

qxQABAqBAQ

(19)

where Q=m11+m22-m21-m1,, q=m22-m12, R=b11+b22-b21-b12 and γ=b22-b11. mij (1≤i, j≤2) and bij (1≤i, j≤2) are elements of payoff matrix for MAP and STA respectively. If the value of each payoff matrix is given, we can obtain the equilibrium points by solving equation (19). Furthermore, the expectation payoff of STA and MAP at the equilibrium points can be calculated as ESTA=ATHSTAB and EMAP=ATHMAPB, respectively. However, the equilibrium points might not be unique. Certainly, there are different equilibrium points for different payoff matrix. Since it is non-cooperative game, Nash equilibrium is not necessarily Pareto-optimality.

VII. CONCLUSIONS This paper investigates the admission control problem

between STAs and MAP from both stations’ and network provider’s perspective. We model the admission control process between new stations and MAP as a nonzero-sum, non-cooperative and mixed strategy game. Three decisive factors and mixed strategy probability function for MAP and STAs are defined. In addition, the calculation steps of the weight value for each factor using AHP are described. Furthermore, we discuss the change tendency of mixed strategy probability function under different parameters based on the numeral analysis of one example. Finally, this paper

analyzes the Nash equilibrium in the admission control game and concludes that there always exist one or more equilibrium points.

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