A New Algorithm of Spectrum Allocation Based on ... - Semantic Scholar
JOURNAL OF NETWORKS, VOL. 7, NO. 7, JULY 2012
1017
A New Algorithm of Spectrum Allocation Based on the Balance between Supply and Demand in Cognitive Radio Networks Liang Ma
Jiangsu Key Laboratory of Wireless Communications, Nanjing University of Posts and Telecommunications Nanjing 210003, China. Key Lab on Wideband Wireless Communications and Sensor Network Technology of Ministry of Education Nanjing University of Posts and Telecommunications, Nanjing 210003, China. [email protected]
Qi Zhu
Jiangsu Key Laboratory of Wireless Communications, Nanjing University of Posts and Telecommunications Nanjing 210003, China. Key Lab on Wideband Wireless Communications and Sensor Network Technology of Ministry of Education Nanjing University of Posts and Telecommunications, Nanjing 210003, China. [email protected]
Abstract-- This paper investigates a new spectrum sharing algorithm based on price in cognitive radio networks. A supply and demand model between primary users and secondary users is established to decide the price of free spectrum leased by primary users. Considering the utility of primary users and the total bandwidth while deciding the price per bandwidth using supply and demand balance (the spectrum supplied by primary users is the same as which demanded by secondary users) and each secondary user competes with each other according to the price to share the available bandwidth allocated by primary users. At the same time, the battery capacity of secondary users and the reliance degree that how primary users trust secondary users in the spectrum sharing process are also considered ,a new secondary utility and spectrum sharing algorithm are designed based on above factors. The existence and uniqueness of Nash equilibrium are proved from both theoretic and simulated aspects. The simulation results show that primary users could receive maximum revenues by allocating spectrum in equilibrium price, and the spectrum shared by secondary users is directly affected by the reliance degree and remaining battery capacity of secondary users. Index Terms--cognitive radio; game theory; spectrum allocation; balance between supply and demand; battery capacity; reliance
I.
INTRODUCTION1
In recent years, an arisen technology called Cognitive Radio has becoming a hot research in wireless communication areas dues to its reusing of licensed spectrum, which reduces the conflicts between increasing 1This work is supported by National Natural Science Foundation of China
(61171094), National Science & Technology Key Project (2011ZX03001-006-02, 2011ZX03005-004-03) and Key Project of Jiangsu Provincial Natural Science Foundation（BK2011027）.
Corresponding author: Zhu Qi; E-mail：[email protected]
© 2012 ACADEMY PUBLISHER doi:10.4304/jnw.7.7.1017-1023
wireless services and the lack of spectrum caused by traditional static resource allocations. As a key technology to Cognitive Radio, the spectrum allocation methods can reuse the licensed spectrum reasonably to meet the demands of some cognitive secondary users while taking some revenues to primary users on the basis of Quality of Primary Services (Qos). Cognitive Radio changes systems parameters in real time to be adapted to wirelss communications by sensing the available spectrum [4]. Strategy selections are included in the spectrum allocations. Game theory, which is a helpful way to solve the strategy selections, can be used as an useful method to analyze how to allocation available spectrum effectively [5]. Nash equilibrium, which is a set of the best strategies of all the participants in the game, is an important notion in game theory. Game theory is proved to be the best effective method of spectrum allocation in distributed cognitive radio environments [6]. [7] analyzes a balance problem between supply and demand about commodity exchange in microeconomics, and the market model is given. A cognitive spectrum sharing problem including a primary user and multiple secondary users is discussed in [8]. However the utility and the available band of primary users are not considered and the price scheme is made randomly. [9] studies the impact taken by battery capacity to the wireless access in multiple-mode electric terminal. This paper improves the shortage of [8] by considering the primary user utilities and total spectrum bandwidths. The balance between supply and demand theory (the available shared spectrum is the same as the demands of secondary uses) in [7] is used to decide the price per bandwidth. New primary and secondary utilities and spectrum allocation methods are also established as follows. Firstly, we make the spectrum price according to the primary and secondary utility functions and balance
1018
JOURNAL OF NETWORKS, VOL. 7, NO. 7, JULY 2012
theory, which makes sure that the primary users can obtain the best profits. Then secondary users compete with each other to get access to available spectrum until all the secondary utilities reach the maximum. At last, we analyze the allocation results at the game equilibrium point. This paper is arranged as follows: establishing the spectrum sharing model in section 2; spectrum allocation method is given and the existence and uniqueness of Nash equilibrium are discussed in section 3; in section 4, the simulation results are given and analyzed; we make a summary of this paper in section 5. II. SPECTRUM SHARED MODEL A.
System model Similar to the spectrum sharing model [10], we consider a cognitive radio environment including a primary services system (the number of services is denoted by M ) and multiple secondary services systems (the number of services is denoted by N , and user i (∀i = 1,..., N ) has mi service requirements). Both the primary and secondary sides share the licensed spectrum of primary ones(primary users occupy some parts to meet their own services demands with the rest being allocated to needed secondary ones). The system model is show in Fig.1.
As produces, they wish a higher price while it is the opposite wish for consumers. Both sides should negotiate to make a reasonable price. Assuming the producer utility function is denoted by f 1( c, n ) , the consumer one is f 2( c, n ) , and the price and quantity are c、n separately . Defining the supply function is n1 = arg max( f1 ( c, n )) with the maximal primary utilities, and the demand function is n2 = arg max( f 2 ( c, n )) . The supply and demand reach a balance when n1 = n2 and the supply is more than demand ( n1 > n2 ) or less ( n1 < n2 ) will not
appear. The price under balance is the final exchange price for both sides. Comparing the market model in [7] and spectrum shared model given in Fig.1, we may establish the following corresponding relations: Commodity——Spectrum; Producers——Primary users; Consumers——Secondary users; Commodity price——Spectrum price; Commodity quantity——Spectrum bandwidth. A new supply and demand spectrum model is established according to the above relations. In this model, primary users supply available spectrum to demanded secondary users and both of them come to an agreement on the spectrum price and bandwidth to realize spectrum sharing. C.
Game model The game discussed in this paper is implemented among N secondary users which compete the available spectrum with each other to maximize their personal revenue. This game can be described as [11]: (1) G = {N ,{ A1 , A2 , AN },{u1 , u2 , uN }}
Fig.1 System model of spectrum sharing
As shown in Fig.1, primary users possess some licensed spectrum(total bandwidths is F ( Hz ) ). The grey parts are occupied by primary users to accomplish their own services, and the rest white ones are available which can be allocated temporarily to required secondary ones. B.
Spectrum shared model In the market economy, the participants are commodity producers and consumers. Both sides reach an agreements on the price and quantity, and the supply and demand model [7] describes this balanced relationships of price and quantity, which is also a balance between supply and demand. In this model, the supply function of producers shows commodity quantities they sell while the demand one of consumers reflects their purchasing commodity quantities, with the commodity price being an important factor to combine the participants together. © 2012 ACADEMY PUBLISHER
N is a finite set of all the participated secondary users; Ai , i ∈ N represents a strategy set of player i and corresponds to shared spectrum bi of secondary user i in this paper; secondary utility function is denoted by ui , i ∈ N which is affected by their own strategies（ Ai ） and the other strategies （ A− i ）of their competitors (all the secondary user expect user i ). III. A NEW SPECTRUM ALLOCATION ALGORITHM The spectrum management centre allocates a licensed band to primary users. The primary ones occupy some bandwidth to meet their own services demands while obtain several profits by leasing out the rest to required secondary users. This paper gives the primary and secondary utility functions based on the model in Fig.1 and relevant parameters in cognitive radio spectrum allocations. The Nash equilibrium of spectrum price is obtained by the supply and demand theory and this price is used in the spectrum game process.
JOURNAL OF NETWORKS, VOL. 7, NO. 7, JULY 2012
1019
A. Primary utility Primary utility is defined as [12](the required bandwidth is identical for each service): F−B 2 (2) ) u p = cB + λ1M − λ2 ( B req − M c is the spectrum price per bandwidth（1Hz）; B is spectrum bandwidth allocated to secondary users; λ1、λ2 are constants; the number of primary services is req
denoted by M ; B (Hz) represents required bandwidth of each primary user to accomplish their own service and the size of total bandwidths is denoted by F (Hz). We can find out from formulation (2) that the primary utility is made up of three parts:The first part cB is the revenue obtained by leasing out the available spectrum resource, the second one λ1M is the profit by
accomplishing their own services and the last part F − B 2 shows the positive effects to primary λ2 ( B req − ) M Qos caused by leasing out the spectrum.
services is denoted by mi ); in the secondary part,
1 − αi are the weight factor of band cost cost
ui = ri k i Fwi
bi
∑b
i
− (α i
bi p + pa bi + (1 − α i ) b ) − c(1 − α i )( ∑ bi )bi F pc
∀i = 1,..., N
，
(3)
ri is the profit per transmission rate for secondary user i ; bi is attained bandwidth; γ i is received signal to tar
noise; BER is the target bit error rate. The spectrum efficiency ki is defined as [13]:
ki = log 2 (1 + Kγ i )
while
K=
1.5 ln0.2/BER tar
the decreasing battery capacity which results in its spectrum demand reduction adaptively; the last part is the cost by purchasing spectrum from primary users. C. Nash equilibrium price C.1 Primary supply function According to balance theory, the supply function is defined when the primary utility u p reaches maximum. (2):
We can obtain the following based on formulation
c − 2λ2 ( B req − The supply function is
user i (
∑w i =1
i
(6)
cM )M 2λ2
(7)
C.2 Secondary demand function Assuming all the secondary users as a whole, and each user has the identical bandwidths per service. The N
total number of secondary services is denoted by
∑m i =1
i
according to system model in Fig.1. The whole secondary utility is N B (8) ) − cB us = ∑ mi ln( N i =1 ∑ mi i =1
(4)
Defining the shared spectrum under maximum us represents the demanded spectrum for secondary
(5)
users. N
∑m i =1
can transmit signals in the unused bands in a proper transmitted power. The secondary ones should exit the bands immediately without interfering primary users once the primary occupation are detected. αi is the percentage of remaining battery capacity for secondary terminal
i ; pb is background power loss( w )； pa is the density of power loss( w / Hz )； pc is the budget power loss ( w ).
The first part of formulation (2) represents the revenue of secondary user i by sharing the available spectrum to accomplish their services(the number of
i
B
= 1 ). In cognitive radio, secondary users
© 2012 ACADEMY PUBLISHER
F−B 1 ) =0 M M
n1 ( B ) = F − ( B req −
wi shows how the primary users trust the secondary N
bi and power F
pb + pa ∗ bi separately, and α i is increasing with pc
B. Secondary utility By considering the total bandwidth、 the reliance factor in spectrum accession for each secondary user and the remaining battery capacity of secondary terminal besides [8] , a new secondary utility is established :
αi and
−c =0
(9)
So the demand function is N
n2 ( B ) =
∑m i =1
c
i
(10)
C.3 Spectrum price Combining formulation (7) with (10) and considering the balance relation n 1 ( B ) = n2 ( B ) ,we can
obtain that
1020
JOURNAL OF NETWORKS, VOL. 7, NO. 7, JULY 2012
N
F − ( B req −
∑m
c∗ M ) M = i =1 ∗ c 2λ2
i
(11)
∗
and the Nash equilibrium price c is obtained as follows:
The dynamic game model is:
N
c* =
λ2 ( B req − F ) + (( F − B req )λ2 ) 2 + 2 M 2λ2 ∑ mi i =1
M2
(12)
D. The existence and uniqueness of Nash equilibrium Each player selects its own optimal strategy with given strategies of other players and all the strategies make up of a strategy set which is Nash equilibrium[14], and in this way, there exists no player which will change its strategy to maximize its own profit. Game aims to search for this optimal strategy set. Nash equilibrium is defined as: Definition 1: Assuming s = ( s1 , s2 ,..., sn ) is a set of n -persons game
G = {S1 , S2 ,..., Sn ;U1 ,U 2 ,...,U n } and player i , there exists: U i ( si , s− i ) ≥ U i ( si* , s−i )
for
E. Dynamic game In the practical cognitive radio environment, secondary user i cannot obtain the information of other secondary users j ( j ≠ i ) at present time t + 1 only expect getting the knowledge of the price and some information(e.g. ri 、 ki ) of other users at former time t .
each
(13)
s = ( s1 , s2 ,..., sn ) is a Nash equilibrium set and s− i
represents a set including other strategies of secondary users except users i . For the set s = ( s1 , s2 ,..., sn )
bi (t + 1) = bi (t ) + β i bi (t )
this set is the only Nash equilibrium of this game. In this paper, multiple secondary users compete with each other to share the primary available spectrum and the game is not stable until no secondary revenue will be increased any longer with the equilibrium solution {b1 , b2 ,..., bN } .Next the existence and
uniqueness will be proved. Proof: According to the secondary utility given in formulation (3), we will obtain bj ∑ ∂ui α p j ≠i = rk − ( i + (1 − α i ) a ) − c(1 − α i )(bi + ∑ bi ) Fw i i i （∑ bi）2 ∂bi F pc i i
∂ 2ui ∂bi 2
bi =bi*
= rk i i Fwi
−∑ b j j ≠i
( ∑ b j + bi* )4
− 2c(1 − αi )
(15)
(16)
j ≠i
Considering (15) and (16): −∑ b j ∂ 2ui j ≠i − 2c(1 − αi ) < 0 * = rk i i Fwi ∂bi 2 bi =bi ( ∑ b j + bi* )4
(17)
j ≠i
Therefore, there exists the unique Nash equilibrium in this paper. © 2012 ACADEMY PUBLISHER
(18)
∂ui (t ) can be obtained by formulation (3). ∂bi (t ) bi (t ) represents the size of bandwidth allocated to secondary user i at time t ; β i is a convergence speed adjusted parameter for secondary user i (i.e. learn factor). We can find that the spectrum is allocated dynamically and each secondary user can only get partial and past information of others by communicating with primary users. The shared spectrum of each secondary user will not be changed at Nash equilibrium and the dynamic game is over. bi (t + 1) = bi (t ) (19)
Combining formulation (19) and (18), we will have
∂ui (t ) =0 ∂bi (t )
which satisfies
∂ui ( s ) ∂ 2ui ( s ) = 0, < 0, i = 1, 2,..., n (14) ∂si ∂si 2
∂ui (t ) ∂bi (t )
(20)
i.e.
b j (t ) ∑ 1 p j ≠i rk − (αi + (1 − αi ) a ) − c(1 − αi )( ∑ bi (t ) + bi (t )) = 0 i i Fwi F pc （∑ bi (t )）2
(21) Next the stability of the dynamic game is discussed in this paper on the basis of Jacobian matrix shown in the following[15]:
∂b1 (t + 1) ⎡ ∂b1 (t + 1) ∂b1 (t + 1) ⎢ ∂b (t ) , ∂b (t ) ,..., ∂b (t ) 1 2 N ⎢ ⎢ ∂b2 (t + 1) ∂b2 (t + 1) ∂b2 (t + 1) ⎢ ∂b t , ∂b (t ) ,..., ∂b (t ) . 1( ) 2 N ⎢ J =⎢ ....... ⎢ . ⎢ ⎢ ∂bN (t + 1) ∂bN (t + 1) ∂b (t + 1) ,..., N , ⎢ b t b t ∂ ∂ ∂bN (t ) ( ) ( ) ⎢ 1 2 ⎢ ⎣
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(22) The eigenvalues of the above matrix are less than 1 to ensure the stability. We may obtain the relations among β i ( ∀i = 1,..., N ) and then achieve the stable region of the dynamic game. Taking two secondary users
JOURNAL OF NETWORKS, VOL. 7, NO. 7, JULY 2012
1021
( N = 2 ) for example: ⎡ ∂b1 (t + 1) ∂b1 (t + 1) ⎤ , ∂b2 (t ) ⎥ ⎡ j1,1 , j1, 2 ⎤ ⎢ ∂b1 (t ) ⎢ ⎥ = J =⎢ ⎥ ⎣ j2,1 , j2, 2 ⎦ ⎢ ∂b2 (t + 1) , ∂b2 (t + 1) ⎥ ⎢ ∂b (t ) ∂b2 (t ) ⎥⎦ 1 ⎣
(23)
⎡ ⎤ ∂u1 (t ) ∂ 2 u1 (t ) ∂ 2 u1 (t ) , β1b1 (t ) ⎢1 + β1 ∂b (t ) + β1b1 (t ) ⎥ 2 ( ) ( ) ( ) b t b t b t ∂ ∂ ∂ 1 1 1 2 ⎥ =⎢ 2 2 ⎢ ∂ u2 (t ) ∂u2 (t ) ∂ u2 (t ) ⎥ ,1 + β 2 + β 2 b2 (t ) ⎢ β 2 b2 (t ) ⎥ 2 ∂b2 (t ) ∂b2 (t )∂b1 (t ) ∂b2 (t ) ⎥⎦ ⎢⎣
According to the standard function[16] 2
λ − λ ( j1,1 + j 2,2 ) + ( j1,1 j 2,2 − j1,2 j 2,1 ) = 0 (24)
spectrum dem and of secondary user 2(Hz)
12
x 10
6
secondary user secondary user secondary user secondary user secondary user secondary user
10
8
6
4
2
0
0
1
2 3 4 5 6 7 spectrum demand of secondary user 1(Hz)
we can have
λ1 , λ2 =
( j1,1 + j 2,2 ) ± 4 j1,2 j 2,1 +( j1,1 − j 2,2 ) 2
Letting
2
λi < 1(i = 1, 2)
stable region can be obtained.
β1、β 2
(25)
and the
5
,B
req
= 0.6( MHz ) .
F = 20( MHz ) ，the revenue per transmission rate r1 = r2 = 10 / bps ，background power loss pb = 262.74 μ w , the density of power loss pa = 1.22 μ w / kHz, power loss budget pc = 10μ w, learn factor β1 = β 2 = 0.14 .Constants w1、w2、α1、α 2 ∈ [0,1] 。 The total bandwidth
Fig.2 shows We can learn change with capacity αi , ∀i
6
a static game between secondary users. that the secondary spectrum demands the percentage of terminal battery = 1,..., N . To some secondary user, for
example user 1, comparing the three curves under α1 = 0.7， 0.6,0.4 separately and we will find out that spectrum demands are reducing with decreasing α1 . The
intersect point of two curves (for example α1 = 0.6，α 2 = 0.3 ) corresponds to the Nash
7
spectrum supply function sepctrum demand function
4 spectrum bandwidth(Hz)
users and the number of primary services is M = 30 .
x 10
4.5
3.5 3 2.5 2 equilibrium
1.5 1 0.5 0 0.2
0.4
0.6
0.8
1 1.2 1.4 spectrum price/Hz
1.6
1.8
2 x 10
-6
Fig.3 Balance between supply and demand and equilibrium price
5.8
x 10
6
secondary user 1 secondary user 2
5.6 spectrum of secondary users(Hz)
In this paper, we consider a cognitive radio environment including a primary service system and two secondary users ( N = 2 ). The service requirements are m1 = m2 = 5 separately for primary −11
9 x 10
Fig.3 gives the relations between spectrum supply and demand. There shows increasing supply and decreasing demand with increasing spectrum price. The intersect point corresponds to the equilibrium price which ensures the leased bandwidths are in the same size as the one demanded by secondary users.
IV. SIMULATION RESULTS AND ANALYSIS
Constants λ 1 = 1, λ2 = 5.5 ⋅ 10
8
Fig.2 Static game and Nash equilibrium
and combining formulation
(23) and (25), the relations between
1(α1=0.7,α2=0.7) 2(α1=0.7,α2=0.7) 1(α1=0.4,α2=0.4) 2(α1=0.4,α2=0.4) 1(α1=0.6,α2=0.3) 2(α1=0.6,α2=0.3)
5.4 5.2
SNR1=8dB
5 4.8 4.6 4.4 4.2
5
5.5
6
6.5
7 SNR2(dB)
7.5
8
8.5
9
Fig.4 The shared spectrum under different signal to noise
equilibrium of spectrum bandwidths.
Fig.4 shows the spectrum game under different received signal to noise with α1 = α 2 = 0.5 and w1 = w2 = 0.5 . © 2012 ACADEMY PUBLISHER
1022
JOURNAL OF NETWORKS, VOL. 7, NO. 7, JULY 2012
x 10
10
6
4.5 secondary user1 secondary user2
8
secondary user 1 secondary user 2
3.5
7 6 5 4 3
3 remaining battery capacity percentage secondary user 1:50% secondary user 2:30%
2.5 2
2
1.5
1 0
0
0.2 0.4 0.6 0.8 Remaining battery capacity percentage of secondary user 2
1 20
1
Fig.5 The shared spectrum under different secondary battery capacity
80% battery capacity. The spectrum bandwidth shared by user 2 is increasing with its capacity percentage increasing from 0 to 100%. 6
x 10
6
secondary user 1 secondary user 2
5
4
3
2
1
0
0
0.2
0.4 0.6 0.8 reliance factor of secondary user 1
21 21.5 22 22.5 23 Total bandwidth of primary users (MHz)
23.5
24
1
V. SUMMARY A cognitive radio spectrum allocation problem including a primary service system and multiple secondary users is discussed using game theory in this paper. The primary utility is considered. Firstly, the Nash equilibrium of spectrum price is established according to supply and demand theory and primary users have the maximal utility under this equilibrium price; then a reliance factor which shows how the primary users trust secondary ones is introduced, and the spectrum allocation under different secondary received signal to noise and terminal battery capacities is analyzed; next the existence and uniqueness of Nash equilibrium is proved; finally we find out that the allocated bandwidth size is increasing with the increased reliance factor、signal to noise and remaining battery capacity by analyzing the simulation results.
Fig.6 The spectrum under different primary reliance
REFERENCE
[1]
In cognitive spectrum sharing problems, secondary users will have a prior opportunity to get access to primary available spectrum if they establish good relationships with primary users. This good relation requires that the secondary users can exit the spectrum immediately without interfering primary ones when they need to occupy the spectrum to meet their service demands. We use a reliance factor to describe this relation. Fig.6 shows that secondary users which have higher reliance factors will be allocated more available spectrum. In Fig.7, the remaining battery capacities are 50% and 30% for each secondary user separately. The shared bandwidth size is increasing with the increasing total primary bandwidth, and the secondary utilities are increasing accordingly.
[2] [3] [4] [5]
[6]
[7] [8]
© 2012 ACADEMY PUBLISHER
20.5
Fig.7 The spectrum under different bandwidth of primary users
In Fig.5, assuming identical received signal to noise SNR1 = SNR2 = 10dB .Secondary user 1 remains
spectrum of secondary users(Hz)
7
4
secondary utility
spectrum of secondary users(Hz)
9
x 10
Haykin S, “Cognitive radio:brain-empowered wireless Communications,” IEEE Journal on Selected Areas in Communications, 23(2),pp.201-220, 2005. Jovicic,A.,Viswanath,P, “Cognitive Radio:An Information-Theoretic Perspective,”IEEE Transactions on Information Theory, 55(9),pp.3945-3958, 2009. BRUCE F, “Cognitive Radio Technology,”New York: Academic Press,2007. Wang,S.,Zheng,H., “A resource management design for cognitive radio ad hoc networks,”. MILCOM,pp.1-7, 2009. Attar,A.,Nakhai,M.R,Aghvami,A.H..,“Cognitive Radio game for secondary spectrum access problem,”IEEE Transactions on Wireless communications, 8(4),pp. 2121-2131,2009. Enrico D.R., Gherardo G., Luca S.R., Rosalba.S., “Resource Allocation in Cognitive Radio Networks: A Comparison Between Game Theory Based and Heuristic Approaches,” Wireless Personal Communications,49(3),pp.375-390,2009. Mceachern W A.,“Microeconomics:a contemporay Introduction,”Cincinnati,OH:South-Western Colledge Publishing,2005. Dusit Niyato,Ekram Hossain, “Competitive Spectrum Sharing in Cognitive Radio Networks: A Dynamic Game
JOURNAL OF NETWORKS, VOL. 7, NO. 7, JULY 2012
[9] [10]
[11] [12]
[13] [14] [15]
[16]
Approach,” IEEE Transactions On Wireless communications, 7(7):2651-2660, 2008. Huiling Jia, “Access Selection and Call Admission Control in Heterogeneous Wireless Networks,”Zhe Jiang University, 2007. Lin Yu-qing,Zhu Qi and Zhu Hong-bo, “A Three-Level Market-Based Scheme of Spectrum Allocation for Cognitive Ad hoc Networks,” Journal of Electronics & Information Technology,vol .32,pp.643-648,2010. Peters H., “Game Theory: A Multi-Leveled Approach,” USA: Springer,pp.133-137, 2008. Niyato,D.,Hossain,E., “ Competitive Pricing for Spectrum in Cognitive Radio Networks: Dynamic Game,Inefficiency of Nash Equilibrium,and Collusion,”IEEE Journal Selected Areas in Communications,26(1),pp.192-202,2008. Goldsmith A.J.and Chua.S.G., “Variable rate variable power MQAM for fading channel,”IEEE Trans.Common,45(10),pp.1218-1230,1997. Drew Fudenberg, Jean Tirole., “Game Theory,”The MIT Press, August 29,1991. H.N.Agiza,G.-I.Bischi,and M.Kopel, “Multistability in a dynamic cournot game with three oligopolists,” Mathematics and Computers in Simulation, vol. 51, pp. 63-90, 1999. Han Y C,M Y, “Eigenvalue Problem of Doubly Stochastic Hamilitonian Systems with Boundary Conditions,” Communications in Mathematical Research, 25(1),pp. 0024-0030,2009
© 2012 ACADEMY PUBLISHER
1023
Liang Ma was born in Anhui, China, on September 5 1987. She received the Bachelor Degree from Nanjing University of Posts and Telecommunications in 2009 and now is a post graduate student in communication and information systems from Nanjing university of Posts and Telecommunications. She has published 4 papers in journals or conference proceedings. Her main research interests include mobile communication system and wireless technology. Qi Zhu was born in Jangsu, China, on June 18 1965. She received the Bachelor Degree from Nanjing Institute of Posts and Telecommunications in 1986 and received Master Degree in communication and electrical systems from Nanjing Institute of Posts and Telecommunications in 1989. She is a full professor Ph.D. supervisor in school of Telecommunications & Information Engineering at Nanjing University of Posts and Telecommunications, Nanjing, China. She has published more than 80 papers in journals or conference proceedings. Her main research interests include mobile communication system and wireless technology.
1017
A New Algorithm of Spectrum Allocation Based on the Balance between Supply and Demand in Cognitive Radio Networks Liang Ma
Jiangsu Key Laboratory of Wireless Communications, Nanjing University of Posts and Telecommunications Nanjing 210003, China. Key Lab on Wideband Wireless Communications and Sensor Network Technology of Ministry of Education Nanjing University of Posts and Telecommunications, Nanjing 210003, China. [email protected]
Qi Zhu
Jiangsu Key Laboratory of Wireless Communications, Nanjing University of Posts and Telecommunications Nanjing 210003, China. Key Lab on Wideband Wireless Communications and Sensor Network Technology of Ministry of Education Nanjing University of Posts and Telecommunications, Nanjing 210003, China. [email protected]
Abstract-- This paper investigates a new spectrum sharing algorithm based on price in cognitive radio networks. A supply and demand model between primary users and secondary users is established to decide the price of free spectrum leased by primary users. Considering the utility of primary users and the total bandwidth while deciding the price per bandwidth using supply and demand balance (the spectrum supplied by primary users is the same as which demanded by secondary users) and each secondary user competes with each other according to the price to share the available bandwidth allocated by primary users. At the same time, the battery capacity of secondary users and the reliance degree that how primary users trust secondary users in the spectrum sharing process are also considered ,a new secondary utility and spectrum sharing algorithm are designed based on above factors. The existence and uniqueness of Nash equilibrium are proved from both theoretic and simulated aspects. The simulation results show that primary users could receive maximum revenues by allocating spectrum in equilibrium price, and the spectrum shared by secondary users is directly affected by the reliance degree and remaining battery capacity of secondary users. Index Terms--cognitive radio; game theory; spectrum allocation; balance between supply and demand; battery capacity; reliance
I.
INTRODUCTION1
In recent years, an arisen technology called Cognitive Radio has becoming a hot research in wireless communication areas dues to its reusing of licensed spectrum, which reduces the conflicts between increasing 1This work is supported by National Natural Science Foundation of China
(61171094), National Science & Technology Key Project (2011ZX03001-006-02, 2011ZX03005-004-03) and Key Project of Jiangsu Provincial Natural Science Foundation（BK2011027）.
Corresponding author: Zhu Qi; E-mail：[email protected]
© 2012 ACADEMY PUBLISHER doi:10.4304/jnw.7.7.1017-1023
wireless services and the lack of spectrum caused by traditional static resource allocations. As a key technology to Cognitive Radio, the spectrum allocation methods can reuse the licensed spectrum reasonably to meet the demands of some cognitive secondary users while taking some revenues to primary users on the basis of Quality of Primary Services (Qos). Cognitive Radio changes systems parameters in real time to be adapted to wirelss communications by sensing the available spectrum [4]. Strategy selections are included in the spectrum allocations. Game theory, which is a helpful way to solve the strategy selections, can be used as an useful method to analyze how to allocation available spectrum effectively [5]. Nash equilibrium, which is a set of the best strategies of all the participants in the game, is an important notion in game theory. Game theory is proved to be the best effective method of spectrum allocation in distributed cognitive radio environments [6]. [7] analyzes a balance problem between supply and demand about commodity exchange in microeconomics, and the market model is given. A cognitive spectrum sharing problem including a primary user and multiple secondary users is discussed in [8]. However the utility and the available band of primary users are not considered and the price scheme is made randomly. [9] studies the impact taken by battery capacity to the wireless access in multiple-mode electric terminal. This paper improves the shortage of [8] by considering the primary user utilities and total spectrum bandwidths. The balance between supply and demand theory (the available shared spectrum is the same as the demands of secondary uses) in [7] is used to decide the price per bandwidth. New primary and secondary utilities and spectrum allocation methods are also established as follows. Firstly, we make the spectrum price according to the primary and secondary utility functions and balance
1018
JOURNAL OF NETWORKS, VOL. 7, NO. 7, JULY 2012
theory, which makes sure that the primary users can obtain the best profits. Then secondary users compete with each other to get access to available spectrum until all the secondary utilities reach the maximum. At last, we analyze the allocation results at the game equilibrium point. This paper is arranged as follows: establishing the spectrum sharing model in section 2; spectrum allocation method is given and the existence and uniqueness of Nash equilibrium are discussed in section 3; in section 4, the simulation results are given and analyzed; we make a summary of this paper in section 5. II. SPECTRUM SHARED MODEL A.
System model Similar to the spectrum sharing model [10], we consider a cognitive radio environment including a primary services system (the number of services is denoted by M ) and multiple secondary services systems (the number of services is denoted by N , and user i (∀i = 1,..., N ) has mi service requirements). Both the primary and secondary sides share the licensed spectrum of primary ones(primary users occupy some parts to meet their own services demands with the rest being allocated to needed secondary ones). The system model is show in Fig.1.
As produces, they wish a higher price while it is the opposite wish for consumers. Both sides should negotiate to make a reasonable price. Assuming the producer utility function is denoted by f 1( c, n ) , the consumer one is f 2( c, n ) , and the price and quantity are c、n separately . Defining the supply function is n1 = arg max( f1 ( c, n )) with the maximal primary utilities, and the demand function is n2 = arg max( f 2 ( c, n )) . The supply and demand reach a balance when n1 = n2 and the supply is more than demand ( n1 > n2 ) or less ( n1 < n2 ) will not
appear. The price under balance is the final exchange price for both sides. Comparing the market model in [7] and spectrum shared model given in Fig.1, we may establish the following corresponding relations: Commodity——Spectrum; Producers——Primary users; Consumers——Secondary users; Commodity price——Spectrum price; Commodity quantity——Spectrum bandwidth. A new supply and demand spectrum model is established according to the above relations. In this model, primary users supply available spectrum to demanded secondary users and both of them come to an agreement on the spectrum price and bandwidth to realize spectrum sharing. C.
Game model The game discussed in this paper is implemented among N secondary users which compete the available spectrum with each other to maximize their personal revenue. This game can be described as [11]: (1) G = {N ,{ A1 , A2 , AN },{u1 , u2 , uN }}
Fig.1 System model of spectrum sharing
As shown in Fig.1, primary users possess some licensed spectrum(total bandwidths is F ( Hz ) ). The grey parts are occupied by primary users to accomplish their own services, and the rest white ones are available which can be allocated temporarily to required secondary ones. B.
Spectrum shared model In the market economy, the participants are commodity producers and consumers. Both sides reach an agreements on the price and quantity, and the supply and demand model [7] describes this balanced relationships of price and quantity, which is also a balance between supply and demand. In this model, the supply function of producers shows commodity quantities they sell while the demand one of consumers reflects their purchasing commodity quantities, with the commodity price being an important factor to combine the participants together. © 2012 ACADEMY PUBLISHER
N is a finite set of all the participated secondary users; Ai , i ∈ N represents a strategy set of player i and corresponds to shared spectrum bi of secondary user i in this paper; secondary utility function is denoted by ui , i ∈ N which is affected by their own strategies（ Ai ） and the other strategies （ A− i ）of their competitors (all the secondary user expect user i ). III. A NEW SPECTRUM ALLOCATION ALGORITHM The spectrum management centre allocates a licensed band to primary users. The primary ones occupy some bandwidth to meet their own services demands while obtain several profits by leasing out the rest to required secondary users. This paper gives the primary and secondary utility functions based on the model in Fig.1 and relevant parameters in cognitive radio spectrum allocations. The Nash equilibrium of spectrum price is obtained by the supply and demand theory and this price is used in the spectrum game process.
JOURNAL OF NETWORKS, VOL. 7, NO. 7, JULY 2012
1019
A. Primary utility Primary utility is defined as [12](the required bandwidth is identical for each service): F−B 2 (2) ) u p = cB + λ1M − λ2 ( B req − M c is the spectrum price per bandwidth（1Hz）; B is spectrum bandwidth allocated to secondary users; λ1、λ2 are constants; the number of primary services is req
denoted by M ; B (Hz) represents required bandwidth of each primary user to accomplish their own service and the size of total bandwidths is denoted by F (Hz). We can find out from formulation (2) that the primary utility is made up of three parts:The first part cB is the revenue obtained by leasing out the available spectrum resource, the second one λ1M is the profit by
accomplishing their own services and the last part F − B 2 shows the positive effects to primary λ2 ( B req − ) M Qos caused by leasing out the spectrum.
services is denoted by mi ); in the secondary part,
1 − αi are the weight factor of band cost cost
ui = ri k i Fwi
bi
∑b
i
− (α i
bi p + pa bi + (1 − α i ) b ) − c(1 − α i )( ∑ bi )bi F pc
∀i = 1,..., N
，
(3)
ri is the profit per transmission rate for secondary user i ; bi is attained bandwidth; γ i is received signal to tar
noise; BER is the target bit error rate. The spectrum efficiency ki is defined as [13]:
ki = log 2 (1 + Kγ i )
while
K=
1.5 ln0.2/BER tar
the decreasing battery capacity which results in its spectrum demand reduction adaptively; the last part is the cost by purchasing spectrum from primary users. C. Nash equilibrium price C.1 Primary supply function According to balance theory, the supply function is defined when the primary utility u p reaches maximum. (2):
We can obtain the following based on formulation
c − 2λ2 ( B req − The supply function is
user i (
∑w i =1
i
(6)
cM )M 2λ2
(7)
C.2 Secondary demand function Assuming all the secondary users as a whole, and each user has the identical bandwidths per service. The N
total number of secondary services is denoted by
∑m i =1
i
according to system model in Fig.1. The whole secondary utility is N B (8) ) − cB us = ∑ mi ln( N i =1 ∑ mi i =1
(4)
Defining the shared spectrum under maximum us represents the demanded spectrum for secondary
(5)
users. N
∑m i =1
can transmit signals in the unused bands in a proper transmitted power. The secondary ones should exit the bands immediately without interfering primary users once the primary occupation are detected. αi is the percentage of remaining battery capacity for secondary terminal
i ; pb is background power loss( w )； pa is the density of power loss( w / Hz )； pc is the budget power loss ( w ).
The first part of formulation (2) represents the revenue of secondary user i by sharing the available spectrum to accomplish their services(the number of
i
B
= 1 ). In cognitive radio, secondary users
© 2012 ACADEMY PUBLISHER
F−B 1 ) =0 M M
n1 ( B ) = F − ( B req −
wi shows how the primary users trust the secondary N
bi and power F
pb + pa ∗ bi separately, and α i is increasing with pc
B. Secondary utility By considering the total bandwidth、 the reliance factor in spectrum accession for each secondary user and the remaining battery capacity of secondary terminal besides [8] , a new secondary utility is established :
αi and
−c =0
(9)
So the demand function is N
n2 ( B ) =
∑m i =1
c
i
(10)
C.3 Spectrum price Combining formulation (7) with (10) and considering the balance relation n 1 ( B ) = n2 ( B ) ,we can
obtain that
1020
JOURNAL OF NETWORKS, VOL. 7, NO. 7, JULY 2012
N
F − ( B req −
∑m
c∗ M ) M = i =1 ∗ c 2λ2
i
(11)
∗
and the Nash equilibrium price c is obtained as follows:
The dynamic game model is:
N
c* =
λ2 ( B req − F ) + (( F − B req )λ2 ) 2 + 2 M 2λ2 ∑ mi i =1
M2
(12)
D. The existence and uniqueness of Nash equilibrium Each player selects its own optimal strategy with given strategies of other players and all the strategies make up of a strategy set which is Nash equilibrium[14], and in this way, there exists no player which will change its strategy to maximize its own profit. Game aims to search for this optimal strategy set. Nash equilibrium is defined as: Definition 1: Assuming s = ( s1 , s2 ,..., sn ) is a set of n -persons game
G = {S1 , S2 ,..., Sn ;U1 ,U 2 ,...,U n } and player i , there exists: U i ( si , s− i ) ≥ U i ( si* , s−i )
for
E. Dynamic game In the practical cognitive radio environment, secondary user i cannot obtain the information of other secondary users j ( j ≠ i ) at present time t + 1 only expect getting the knowledge of the price and some information(e.g. ri 、 ki ) of other users at former time t .
each
(13)
s = ( s1 , s2 ,..., sn ) is a Nash equilibrium set and s− i
represents a set including other strategies of secondary users except users i . For the set s = ( s1 , s2 ,..., sn )
bi (t + 1) = bi (t ) + β i bi (t )
this set is the only Nash equilibrium of this game. In this paper, multiple secondary users compete with each other to share the primary available spectrum and the game is not stable until no secondary revenue will be increased any longer with the equilibrium solution {b1 , b2 ,..., bN } .Next the existence and
uniqueness will be proved. Proof: According to the secondary utility given in formulation (3), we will obtain bj ∑ ∂ui α p j ≠i = rk − ( i + (1 − α i ) a ) − c(1 − α i )(bi + ∑ bi ) Fw i i i （∑ bi）2 ∂bi F pc i i
∂ 2ui ∂bi 2
bi =bi*
= rk i i Fwi
−∑ b j j ≠i
( ∑ b j + bi* )4
− 2c(1 − αi )
(15)
(16)
j ≠i
Considering (15) and (16): −∑ b j ∂ 2ui j ≠i − 2c(1 − αi ) < 0 * = rk i i Fwi ∂bi 2 bi =bi ( ∑ b j + bi* )4
(17)
j ≠i
Therefore, there exists the unique Nash equilibrium in this paper. © 2012 ACADEMY PUBLISHER
(18)
∂ui (t ) can be obtained by formulation (3). ∂bi (t ) bi (t ) represents the size of bandwidth allocated to secondary user i at time t ; β i is a convergence speed adjusted parameter for secondary user i (i.e. learn factor). We can find that the spectrum is allocated dynamically and each secondary user can only get partial and past information of others by communicating with primary users. The shared spectrum of each secondary user will not be changed at Nash equilibrium and the dynamic game is over. bi (t + 1) = bi (t ) (19)
Combining formulation (19) and (18), we will have
∂ui (t ) =0 ∂bi (t )
which satisfies
∂ui ( s ) ∂ 2ui ( s ) = 0, < 0, i = 1, 2,..., n (14) ∂si ∂si 2
∂ui (t ) ∂bi (t )
(20)
i.e.
b j (t ) ∑ 1 p j ≠i rk − (αi + (1 − αi ) a ) − c(1 − αi )( ∑ bi (t ) + bi (t )) = 0 i i Fwi F pc （∑ bi (t )）2
(21) Next the stability of the dynamic game is discussed in this paper on the basis of Jacobian matrix shown in the following[15]:
∂b1 (t + 1) ⎡ ∂b1 (t + 1) ∂b1 (t + 1) ⎢ ∂b (t ) , ∂b (t ) ,..., ∂b (t ) 1 2 N ⎢ ⎢ ∂b2 (t + 1) ∂b2 (t + 1) ∂b2 (t + 1) ⎢ ∂b t , ∂b (t ) ,..., ∂b (t ) . 1( ) 2 N ⎢ J =⎢ ....... ⎢ . ⎢ ⎢ ∂bN (t + 1) ∂bN (t + 1) ∂b (t + 1) ,..., N , ⎢ b t b t ∂ ∂ ∂bN (t ) ( ) ( ) ⎢ 1 2 ⎢ ⎣
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(22) The eigenvalues of the above matrix are less than 1 to ensure the stability. We may obtain the relations among β i ( ∀i = 1,..., N ) and then achieve the stable region of the dynamic game. Taking two secondary users
JOURNAL OF NETWORKS, VOL. 7, NO. 7, JULY 2012
1021
( N = 2 ) for example: ⎡ ∂b1 (t + 1) ∂b1 (t + 1) ⎤ , ∂b2 (t ) ⎥ ⎡ j1,1 , j1, 2 ⎤ ⎢ ∂b1 (t ) ⎢ ⎥ = J =⎢ ⎥ ⎣ j2,1 , j2, 2 ⎦ ⎢ ∂b2 (t + 1) , ∂b2 (t + 1) ⎥ ⎢ ∂b (t ) ∂b2 (t ) ⎥⎦ 1 ⎣
(23)
⎡ ⎤ ∂u1 (t ) ∂ 2 u1 (t ) ∂ 2 u1 (t ) , β1b1 (t ) ⎢1 + β1 ∂b (t ) + β1b1 (t ) ⎥ 2 ( ) ( ) ( ) b t b t b t ∂ ∂ ∂ 1 1 1 2 ⎥ =⎢ 2 2 ⎢ ∂ u2 (t ) ∂u2 (t ) ∂ u2 (t ) ⎥ ,1 + β 2 + β 2 b2 (t ) ⎢ β 2 b2 (t ) ⎥ 2 ∂b2 (t ) ∂b2 (t )∂b1 (t ) ∂b2 (t ) ⎥⎦ ⎢⎣
According to the standard function[16] 2
λ − λ ( j1,1 + j 2,2 ) + ( j1,1 j 2,2 − j1,2 j 2,1 ) = 0 (24)
spectrum dem and of secondary user 2(Hz)
12
x 10
6
secondary user secondary user secondary user secondary user secondary user secondary user
10
8
6
4
2
0
0
1
2 3 4 5 6 7 spectrum demand of secondary user 1(Hz)
we can have
λ1 , λ2 =
( j1,1 + j 2,2 ) ± 4 j1,2 j 2,1 +( j1,1 − j 2,2 ) 2
Letting
2
λi < 1(i = 1, 2)
stable region can be obtained.
β1、β 2
(25)
and the
5
,B
req
= 0.6( MHz ) .
F = 20( MHz ) ，the revenue per transmission rate r1 = r2 = 10 / bps ，background power loss pb = 262.74 μ w , the density of power loss pa = 1.22 μ w / kHz, power loss budget pc = 10μ w, learn factor β1 = β 2 = 0.14 .Constants w1、w2、α1、α 2 ∈ [0,1] 。 The total bandwidth
Fig.2 shows We can learn change with capacity αi , ∀i
6
a static game between secondary users. that the secondary spectrum demands the percentage of terminal battery = 1,..., N . To some secondary user, for
example user 1, comparing the three curves under α1 = 0.7， 0.6,0.4 separately and we will find out that spectrum demands are reducing with decreasing α1 . The
intersect point of two curves (for example α1 = 0.6，α 2 = 0.3 ) corresponds to the Nash
7
spectrum supply function sepctrum demand function
4 spectrum bandwidth(Hz)
users and the number of primary services is M = 30 .
x 10
4.5
3.5 3 2.5 2 equilibrium
1.5 1 0.5 0 0.2
0.4
0.6
0.8
1 1.2 1.4 spectrum price/Hz
1.6
1.8
2 x 10
-6
Fig.3 Balance between supply and demand and equilibrium price
5.8
x 10
6
secondary user 1 secondary user 2
5.6 spectrum of secondary users(Hz)
In this paper, we consider a cognitive radio environment including a primary service system and two secondary users ( N = 2 ). The service requirements are m1 = m2 = 5 separately for primary −11
9 x 10
Fig.3 gives the relations between spectrum supply and demand. There shows increasing supply and decreasing demand with increasing spectrum price. The intersect point corresponds to the equilibrium price which ensures the leased bandwidths are in the same size as the one demanded by secondary users.
IV. SIMULATION RESULTS AND ANALYSIS
Constants λ 1 = 1, λ2 = 5.5 ⋅ 10
8
Fig.2 Static game and Nash equilibrium
and combining formulation
(23) and (25), the relations between
1(α1=0.7,α2=0.7) 2(α1=0.7,α2=0.7) 1(α1=0.4,α2=0.4) 2(α1=0.4,α2=0.4) 1(α1=0.6,α2=0.3) 2(α1=0.6,α2=0.3)
5.4 5.2
SNR1=8dB
5 4.8 4.6 4.4 4.2
5
5.5
6
6.5
7 SNR2(dB)
7.5
8
8.5
9
Fig.4 The shared spectrum under different signal to noise
equilibrium of spectrum bandwidths.
Fig.4 shows the spectrum game under different received signal to noise with α1 = α 2 = 0.5 and w1 = w2 = 0.5 . © 2012 ACADEMY PUBLISHER
1022
JOURNAL OF NETWORKS, VOL. 7, NO. 7, JULY 2012
x 10
10
6
4.5 secondary user1 secondary user2
8
secondary user 1 secondary user 2
3.5
7 6 5 4 3
3 remaining battery capacity percentage secondary user 1:50% secondary user 2:30%
2.5 2
2
1.5
1 0
0
0.2 0.4 0.6 0.8 Remaining battery capacity percentage of secondary user 2
1 20
1
Fig.5 The shared spectrum under different secondary battery capacity
80% battery capacity. The spectrum bandwidth shared by user 2 is increasing with its capacity percentage increasing from 0 to 100%. 6
x 10
6
secondary user 1 secondary user 2
5
4
3
2
1
0
0
0.2
0.4 0.6 0.8 reliance factor of secondary user 1
21 21.5 22 22.5 23 Total bandwidth of primary users (MHz)
23.5
24
1
V. SUMMARY A cognitive radio spectrum allocation problem including a primary service system and multiple secondary users is discussed using game theory in this paper. The primary utility is considered. Firstly, the Nash equilibrium of spectrum price is established according to supply and demand theory and primary users have the maximal utility under this equilibrium price; then a reliance factor which shows how the primary users trust secondary ones is introduced, and the spectrum allocation under different secondary received signal to noise and terminal battery capacities is analyzed; next the existence and uniqueness of Nash equilibrium is proved; finally we find out that the allocated bandwidth size is increasing with the increased reliance factor、signal to noise and remaining battery capacity by analyzing the simulation results.
Fig.6 The spectrum under different primary reliance
REFERENCE
[1]
In cognitive spectrum sharing problems, secondary users will have a prior opportunity to get access to primary available spectrum if they establish good relationships with primary users. This good relation requires that the secondary users can exit the spectrum immediately without interfering primary ones when they need to occupy the spectrum to meet their service demands. We use a reliance factor to describe this relation. Fig.6 shows that secondary users which have higher reliance factors will be allocated more available spectrum. In Fig.7, the remaining battery capacities are 50% and 30% for each secondary user separately. The shared bandwidth size is increasing with the increasing total primary bandwidth, and the secondary utilities are increasing accordingly.
[2] [3] [4] [5]
[6]
[7] [8]
© 2012 ACADEMY PUBLISHER
20.5
Fig.7 The spectrum under different bandwidth of primary users
In Fig.5, assuming identical received signal to noise SNR1 = SNR2 = 10dB .Secondary user 1 remains
spectrum of secondary users(Hz)
7
4
secondary utility
spectrum of secondary users(Hz)
9
x 10
Haykin S, “Cognitive radio:brain-empowered wireless Communications,” IEEE Journal on Selected Areas in Communications, 23(2),pp.201-220, 2005. Jovicic,A.,Viswanath,P, “Cognitive Radio:An Information-Theoretic Perspective,”IEEE Transactions on Information Theory, 55(9),pp.3945-3958, 2009. BRUCE F, “Cognitive Radio Technology,”New York: Academic Press,2007. Wang,S.,Zheng,H., “A resource management design for cognitive radio ad hoc networks,”. MILCOM,pp.1-7, 2009. Attar,A.,Nakhai,M.R,Aghvami,A.H..,“Cognitive Radio game for secondary spectrum access problem,”IEEE Transactions on Wireless communications, 8(4),pp. 2121-2131,2009. Enrico D.R., Gherardo G., Luca S.R., Rosalba.S., “Resource Allocation in Cognitive Radio Networks: A Comparison Between Game Theory Based and Heuristic Approaches,” Wireless Personal Communications,49(3),pp.375-390,2009. Mceachern W A.,“Microeconomics:a contemporay Introduction,”Cincinnati,OH:South-Western Colledge Publishing,2005. Dusit Niyato,Ekram Hossain, “Competitive Spectrum Sharing in Cognitive Radio Networks: A Dynamic Game
JOURNAL OF NETWORKS, VOL. 7, NO. 7, JULY 2012
[9] [10]
[11] [12]
[13] [14] [15]
[16]
Approach,” IEEE Transactions On Wireless communications, 7(7):2651-2660, 2008. Huiling Jia, “Access Selection and Call Admission Control in Heterogeneous Wireless Networks,”Zhe Jiang University, 2007. Lin Yu-qing,Zhu Qi and Zhu Hong-bo, “A Three-Level Market-Based Scheme of Spectrum Allocation for Cognitive Ad hoc Networks,” Journal of Electronics & Information Technology,vol .32,pp.643-648,2010. Peters H., “Game Theory: A Multi-Leveled Approach,” USA: Springer,pp.133-137, 2008. Niyato,D.,Hossain,E., “ Competitive Pricing for Spectrum in Cognitive Radio Networks: Dynamic Game,Inefficiency of Nash Equilibrium,and Collusion,”IEEE Journal Selected Areas in Communications,26(1),pp.192-202,2008. Goldsmith A.J.and Chua.S.G., “Variable rate variable power MQAM for fading channel,”IEEE Trans.Common,45(10),pp.1218-1230,1997. Drew Fudenberg, Jean Tirole., “Game Theory,”The MIT Press, August 29,1991. H.N.Agiza,G.-I.Bischi,and M.Kopel, “Multistability in a dynamic cournot game with three oligopolists,” Mathematics and Computers in Simulation, vol. 51, pp. 63-90, 1999. Han Y C,M Y, “Eigenvalue Problem of Doubly Stochastic Hamilitonian Systems with Boundary Conditions,” Communications in Mathematical Research, 25(1),pp. 0024-0030,2009
© 2012 ACADEMY PUBLISHER
1023
Liang Ma was born in Anhui, China, on September 5 1987. She received the Bachelor Degree from Nanjing University of Posts and Telecommunications in 2009 and now is a post graduate student in communication and information systems from Nanjing university of Posts and Telecommunications. She has published 4 papers in journals or conference proceedings. Her main research interests include mobile communication system and wireless technology. Qi Zhu was born in Jangsu, China, on June 18 1965. She received the Bachelor Degree from Nanjing Institute of Posts and Telecommunications in 1986 and received Master Degree in communication and electrical systems from Nanjing Institute of Posts and Telecommunications in 1989. She is a full professor Ph.D. supervisor in school of Telecommunications & Information Engineering at Nanjing University of Posts and Telecommunications, Nanjing, China. She has published more than 80 papers in journals or conference proceedings. Her main research interests include mobile communication system and wireless technology.
