Primary-Secondary Interaction Modeling inCellular Cognitive Radio Networks:

A Game-Theoretic Approach

Ehsan Pasandshanjani Babak H. KhalajAdvanced Communications Research Institute

Department of Electrical EngineeringSharif University of Technology

ehsan pasand@ee.sharif.edu khalaj@sharif.edu

December 15, 2010

Abstract

In this paper, we introduce a new distributed game-theoretic approach for dynamic spectrum

leasing in cellular cognitive radio networks. Underutilization and scarcity of available bandwidth

have led to the idea of spectrum sharing as primary users partially transfer their rights for spectrum

access to others in return for rewards. In such scenarios, it is necessary to define utility functions

for primary and secondary users according to their incentives in order to model such interaction

properly. We prove the existence of Nash equilibrium as well as its uniqueness for our model in the

framework of standard power control algorithms and demonstrate its high convergence rate through

numerical results. We also investigate the improvement in power consumption and SIR levels as

users are given the opportunity to switch between different base stations. Furthermore, we propose

a simple admission control scheme and show the resulting improvement on the performance of the

proposed algorithm through simulations.

Index Terms- Cognitive Radio Networks, Game Theory, Nash equilibrium, Spectrum Leasing,

Admission Control

1

1 Introduction

Scarcity of available bandwidth and its underutilization have led to the idea of implemen-

tation of secondary networks on top of original ones [1]. However, for such system to work

properly, primary users who have the right to use network resources must be protected from

extra interference. On the other hand, secondary users have to be served with a minimum

acceptable level of quality of service. Such QoS level may be defined based on the given

application scenario. The target signal-to-interference ratio achieved at an acceptable power

consumption level is usually a good criterion for service assessment [2], [3], [4], [5], [6].

Secondary networks may be implemented in three different ways [7]. The first option

is open sharing in which there is no difference between primary and secondary users for

resource access. Another option is hierarchical access by which secondary users tend to

use the spectrum without disturbing the performance of primary ones. The last option is

dynamic exclusive use in which primaries are willing to transfer their rights to others in

return for receiving rewards. In our problem, primary users are willing to offer services

to secondary ones in order to get rewards. However, the issue of imposing interference

introduces a conflict that should be addressed properly. On the other hand, secondary

users want to have acceptable services with the minimum power consumption level possible.

Therefore, it can be perceived that satisfaction of each users goals causes disturbance to

others in form of imposed interference.

In this paper, we focus on a game theoretic approach [8] to analyse primary-secondary

interaction in a cellular network based on the dynamic exclusive use approach. This problem,

also known as Dynamic Spectrum Leasing (DSL), was first addressed by [9]. In that ap-

proach, a game theoretic scheme is introduced in which primary users also participate in the

devised game and the effect of different receivers types on secondary base stations is consid-

ered. In [10], for the case of slow-varying fading channel, the trade-off between performance

and channel state information (CSI) update rate is addressed. [11] introduces new utility

2

functions for both secondary and primary users and analyses the effect of the new model on

system performance. A general structure for dynamic spectrum leasing considering different

variations of channel conditions is also introduced in [12]. In a recent paper [13], the algo-

rithm in [9] is generalized to a centralized method with linear and MMSE receivers. Finally,

[14] investigates interaction of two secondary operators competing for spectrum leasing. It

also compares performance of the non-cooperative method with the case in which operators

collaborate, and concludes that in some scenarios anarchy is more beneficial to end users.

In this paper, on the other hand, we focus on a cellular cognitive radio network as our

framework. In addition, we consider a new admission control as a way to improve the

performance of our algorithm. We investigate the effect of letting users switch over base

stations and show the resulting power saving efficiency. Also, we demonstrate how a simple

admission control algorithm can improve system performance in terms of power consumption,

SIR levels, and network capacity. The main point of this work is to introduce different trade-

offs which may be utilized to control network performance in various scenarios.

The outline of this paper is as follows. After introducing the system model in section II,

the convergence of the proposed algorithm and uniqueness of the resulting Nash equilibrium

point will be proved in section III. Numerical results are presented in section IV, and finally,

section V concludes the paper.

2 System Model

We consider a cellular network in which there are B cells. In each cell there is a base station

at the center which serves a primary user. Generalization of model to scenarios in which

more than one primary user exist is straightforward. In addition, there are N cognitive users

distributed uniformly throughout the network. Each one of them is active with probability

p. Typical locations for active mobiles are shown in Fig. 1. It is assumed that secondary

3

access points are deployed in a co-located manner with primary base stations.

As stated earlier, primary users want to gain reward by allowing secondary ones to access

the network. They should also set a bound on imposed interference in order to be certain

about their quality of service level. The larger the bound, the more reward they achieve. So,

in the simplest form, we can model their utility with a linear function of maximum admissible

interference. On the other hand, we should take into account the following two issues when

addressing cost assignment. If interference is larger than the bound, they should be priced

because of violated quality of service level. On the other hand, if interference is lower than

the bound, they should be penalized too, since they obtain more services than they desire.

However, there is a difference between these two scenarios. The cost incurred in the case

of the first scenario should be much larger than the other case for an equal interference

deviation, since violation of QoS is not acceptable for primary users. Therefore, we consider

the following form for primary users utility function

ui(Q0, I0) = Q0 aiQ20 + bi

ciQ0 + di(1)

in which Q0 is the maximum admissible interference level adjusted at each stage and I0 is

the instantaneous value of imposed interference. It should be noted that at each level of the

game, primary users also adjust their uplink power with respect to Q0 in order to guarantee

their signal-to-interference ratio by

pi =tari (Q0 +

2)

hi(2)

Fig. 2 shows a typical graph of the utility function. As can be verified, the utility function

is concave and continuous. Consequently, a compact and convex strategy set guarantees

existence of Nash equilibrium point. Another issue to be mentioned is the effect of parameters

a, b, c, d on the shape of utility function.

4

For simplicity, we drop the parameters indices. a/c controls the rate of decrease in case of

extraneous service level. It is obvious that for a comparable decrease in utility with respect

to the left branch which declines asymptotically, a/c should be chosen much larger than

unity. Besides, if the instantaneous value of interference is equal to the threshold level, the

cost should be kept at a minimum level. As a result, by adapting b at each stage, we adjust

it such that the minimum value occurs at I0:

b = aI20 +2ad

cI0 (3)

d is used to set the vertical asymptote at a value that the difference between the threshold

and imposed interference values becomes significant. In addition, its value controls the rate

of decrease in the function when quality of service level is violated. Therefore, by choosing

a simple rational function we can adjust the cost value throughout different game levels.

For secondary users, we define a cost function which includes power consumption as well

as signal-to-interference ratio deviation from a target value. In general, the cost function

should be non-negative and convex in order to allow existence of a non-negative minimum.

In order to achieve such goal, it is sufficient to formulate the deviation from target SIR as a

non-negative and convex function. Therefore, we consider the following model

Ji(pi, i) = eipi + fi cosh(i tari ) (4)

in which ei and fi can be adjusted to put more emphasis on power level or SIR deviation.

If the value of fi is increased, achieving target SIR is more emphasized, otherwise power

consumption is prioritized.

5

3 Algorithm Convergence

In [15], Yates proposes the so-called standard power control framework, under which conver-

gence to a fixed point is demonstrated in synchronous and asynchronous cases, and for any

given initial power level for users. In a game-theoretic approach, a power control algorithm

is standard if its best response functions, p(k+1) = g(p(k)) meet three conditions:

1. positivity g(p) 0,

2. monotonicity p p g(p) g(p),

3. scalability 1;g(p) g(p).

Since the Nash equilibrium is a fixed point of best response functions, if the fixed point

becomes unique, there will be only one Nash equilibrium point. In this section, we show

that our method satisfies such condition.

First, we should derive best response functions of primary and secondary users in order

to show that all three conditions are met. For this purpose, we have to differentiate utility

and cost functions with respect to interference threshold and power, respectively, and equate

them with zero. For primary users, we have

ui(Q0, I0)

Q0= 0 = 1 2aQ0(cQ0 + d) c(aQ

20 + b)

(cQ0 + d)2(5)

Without loss of generality we set c = 1. After some manipulations,the best response function

of primary users will be in this simple form (see the Appendix)

g(p) = (I0 + d)

a

a 1 d (6)

Since a is larger than unity, the positivity condition is met. For monotonicity, if p p ,then I0 I 0, and therefore, g(p) g(p). It can also be verified easily that g(p) g(p).

6

As a result, primary users best response functions are standard. For secondary users, we

have

Ji(pi, i)

pi= 0 = ei + fi

ipi

sinh(i tari ) (7)

It is clear that at Nash equilibrium point, the signal-to-interference ratio must be lower than

or equal to the target value for each user, otherwise the power update formula is not solvable.

Denoting the interference imposed on user i by Ii, and after some manipulations, we have

p(k+1)i =

I(k)i

hi(tari sinh1(

eiI(k)i

fihi)) If positive,

0 Otherwise.

(8)

Consequently, each user can update his/her power level only with the knowledge of his/her

own interference level and the method can be implemented in a distributed manner. It

should also be noted that if the calculated power level becomes negative, the user should be

silent during that stage.

From (8) we see that positivity requires

Ii hifiei

sinh(tari ) (9)

Since sinh(tari ) is usually large, if we choose a proper value forfiei

, the positivity condition

can be easily met. For monotonicity, it is sufficient to have an increasing best response

function with respect to I. If we differentiate (8) with respect to I, we will have

higi(p)

Ii= tari sinh1(

eiIifihi

) eiIifihi

sinh1(eiIifihi

)

eiIifihi

(10)

7

Using inequalities sinh1(x) x and sinh1(x)

x 1, for monotonicity, we should have:

tari 2eiIifihi

Ii fihi2ei

tari (11)

which is tighter than (9). Finally, we should prove the scalability of our method. Since is

larger than unity, we have sinh1(eiIi(p)fihi

) sinh1(eiIi(p)fihi

). As a result, the condition for

scalability can be written as

gi(p) gi(p) ( 1)2(tari sinh1(eiIifihi

)) (12)

So, for scalability, it is enough that positivity be met. Consequently, our method meets the

conditions of the standard power control framework and its convergence to a unique NE

point is guaranteed.

4 Numerical Results

In this section, we investigate performance of our proposed algorithm in a cellular network

through different simulation scenarios. For simplicity, we consider square cells of 2-km

length. Base stations are placed at the center of each cell and primary and secondary users

are located randomly based on uniform distribution throughout the network. In each cell,

there is only one primary user. The receiver background noise power level within users

bandwidth is 2 = 5 1015w. Channel gains are defined as [16]

hik =A

rnik(13)

where rik is user is distance to the base station k, n is path loss exponent which is set to 3.6

and A = 7.75103 denotes the attenuation factor. Maximum available power is Pmax = 20w

8

and target SIR of all users is considered to be tarPU = tarSU = 10dB. For primary users utility

function, we consider a = 106, b and d calculated in the same way discussed earlier. In

addition, cost parameters of secondary users are set using simulations such that minimum

power consumption occurs. Also, the cross correlation coefficient between secondary users

signalling waveforms and between secondary-primary waveforms is considered to have the

same value of 0.1. Furthermore, we update users strategies synchronously.

Fig. 3 shows convergence of the proposed algorithm to the Nash equilibrium point in

a typically loaded network for 100 secondary users with activity probability of 0.2. As can

be verified, the method succeeds in acquiring target SIR for primary and secondary users

in a few iterations. Another point to be mentioned is the possibility of users to switch

between base stations. If the nearest base for a secondary user becomes overloaded, it will

be better especially for boundary ones to switch to other base stations. Such situation can be

easily identified by tracking the assigned base station to each user through the convergence

process. In cases where switching to another base is not admitted, degradation in algorithm

performance is inevitable, especially when number of active secondary users increases. We

simulate both conditions for one hundred different users locations in the network and present

the average result. As the results show, 40.2% and 29.5% increase in secondary and primary

users power consumption level, respectively will be observed if no switching between bases

is allowed. Such difference is even more severe if we consider more heavily loaded scenarios.

Another issue to be considered, is dependency of the algorithms performance on sec-

ondary cost parameters. As discussed earlier, the cost value depends on the power level

and SIR deviation. In order to perceive efficacy of cost parameters, we simulated the model

for one hundred times with different network topologies and computed the average results.

Figs. 4 and 5 show the power consumption level and the average achieved SIR levels with

respect to the ratio of f/e. As expected, the more priority is given to the power consumption

level, the more cautious user will be about its transmission power. Consequently, there is

9

higher probability that it will switch between base stations in order to find less interfering

situations. On the other hand, if SIR deviation is more emphasized, the user will not put

much emphasis on its power consumption and consumes more power to get closer to the

target SIR level. However, such situation leads to more interference imposed on primary

users and their performance degrades. Therefore, a trade-off between power consumption

level and SIR deviation exists which is adjusted by setting the value of secondary users cost

parameters. It should be noted that in order to observe such situation better, the network

was loaded twice as much as the first case.

We also applied a simple admission control to the network in order to improve our

algorithm performance. Naturally, if network is feasible, convergence to the target point

is achieved [15]. As a result, a decline in a users instantaneous SIR value during the

convergence process can be interpreted as a sign that the offered load is too heavy to be

feasible. Thus, adopting a proper admission control algorithm to decrease network load is a

way to improve the algorithm performance. Since users with decline in their SIR values can

not attain their target SIR, it is rational to reduce their target value in order to mitigate

network load. The scheme works as follows.

At each step that strategies are updated, we decrease target SIR of at most n users who

observe decline in their instantaneous SIR values, by a factor of . This reduction can be

repeated until the target value reaches a level equal to % of its initial value. Although this

method may appear unfair at the first glance, but we mitigate such unfair behavior using

linear proportional pricing to penalize users according to their positions in the network.

In this approach, all users have the same SIR deviation factor, but the power factor is

proportional to their distances to base stations. Based on this method, users become more

cautious with their power consumption, especially the closer they are to the base. In addition,

if there are more than n users whose SIR levels have decreased, we randomly select n out

of them in order to avoid repeated target SIR level reduction for the same users. Figures

10

6 and 7 show users average power consumption and average SIR level, respectively at NE

point with respect to n for 100 users with activity probability of 0.4, = 70%, = 0.05 and

SIR deviation cost factor of 103. As can be observed, the larger the n, the more decrease

on power consumption occurs. Also, the SIR levels for primary users increases due to the

reduction in interference level caused by secondaries. However, there is a slight decrease

in secondary average SIR levels which is too low to be considered. As an illustration, for

n = 5 there are 30.4% and 20.5% decrease in secondary and primary power consumption

levels, respectively, and also 11% improvement in primary average SIR levels, but only 6%

degradation in average SIR levels of secondary ones. It should be noted that as we choose

smaller values for or more number of users under admission control, more power saving at

the cost of more reduction in secondary SIR levels occurs, which introduces a design trade-off

criteria. In order to obtain a better judgement about performance of the admission control

method, comparisons on SIR levels for the case of n = 5 users are presented. By averaging

over a number of scenarios, it can be verified that admission control is beneficial to more

than 69% of secondary users in terms of improved SIR level. In addition, the maximum level

of improvement is at 83% while maximum degradation is at 23% level. Furthermore, the

minimum achieved SIR level at NE point is increased by 57% and 39%, for secondary and

primary users, respectively. As it can be verified, use of admission control and proportional

pricing is totally beneficial in saving power and improving performance of the proposed

algorithm.

The final issue to be considered is the capacity of the proposed algorithm in serving

secondary users. In order to obtain more reliable results, at each activity probability value,

the scenario is simulated for several times and the average result is computed. Fig. 8 shows

the percentage of secondary users that achieve their target SIR with respect to their activity

probability with and without applying the admission control algorithm. As it was also

observed in Fig. 3, the algorithm is completely successful in typically loaded networks. But

11

as the network load increases, less number of users succeed to achieve their incentives tightly.

The important point in Fig. 8 is that application of admission control is not recommended

in typically loaded scenarios. This is due to the fact that network still has the capacity

to tolerate interaction of users and reduction of their target SIR values subjugates their

incentives. However, as the load is gradually increased, the capacity declines and applying

the admission control on users becomes of higher importance. The admission control effect

on capacity improvement in heavily loaded networks is another evidence which strengthens

our conclusion about the performance of the proposed method.

5 Conclusion

We proposed new utility and cost functions for primary and secondary users, respectively,

in order to model their interaction as a non-cooperative game in a cellular cognitive radio

network. Through simulation results, we demonstrated that when network load is not too

heavy, our method succeeds in serving users at their desired objective levels after a few

iterations. An important point to be noted is that allowing secondary users to switch between

base stations leads to approximately the same SIR levels but with additional power saving.

Also, our scheme introduces a trade-off between power consumption and SIR levels at NE

point by changing secondary cost parameters, that may be properly utilized in the design

process.

Furthermore, we applied a simple admission control to our algorithm and have shown

adopting such scheme leads to considerable power saving at the cost of slight reduction in SIR

levels. In addition, admission control leads to higher capacity in heavily loaded networks.

An interesting topic for future research is developing a game theoretic method for the

downlink side of cellular cognitive radio networks. Adopting a game-theoretic approach in

the downlink leads to a quite different framework in comparison with the one addressed in

12

this paper. Such difference is due to the fact that the target SIR levels must be satisfied at

all primary and secondary user nodes and not just at base station locations which has been

the case addressed in this paper.

Appendix

proof of (6): Solving (5) for c = 1 we will have:

Q0 =

d2(a 1)2 + (a 1)(d2 + b) d(a 1)

a 1

= d

1 +

d2 + b

d2(a 1) d

= d

1 +

1

a 1(1 + aI20 + 2dI0

d2) d

= d

a

a 1 + (a

a 1)(I20 + 2dI0

d2) d

= (I0 + d)

a

a 1 d

(14)

Acknowledgement

This work was supported in part by Iran National Science Foundation under Grant 87041174

and in part by Iran Telecommunication Research Center.

13

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14

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15

2000 1500 1000 500 0 500 1000 1500 20002000

1500

1000

500

0

500

1000

1500

2000

X(meters)

Y(

meter

s) Network Topology

BS SU PU

Figure 1: Typical locations of active mobiles for 100 users at p = 0.2

0.9 0.95 1 1.05 1.1x 1015

2.2

2.18

2.16

2.14

2.12

2.1

2.08

2.06

2.04

2.02

2

x 1010

Interference Threshold

Prim

ary

User

Utili

ty

Figure 2: Primary users utility function, I0 = 1015w

16

0 5 10 15 200

1

2

3 Secondary Users Power Update

Po

wer

(Watt

s)

0 5 10 15 200

2

4

6

8

10 Secondary Users SIR Update

SI

R

0 5 10 15 200

0.5

1

1.5

2 Primary Users Power Update

iteration

Po

wer

(Watt

s)

0 5 10 15 200

2

4

6

8

10 Primary Users SIR Update

iteration

SI

R

Figure 3: Performance of the proposed algorithm for 100 users at p = 0.2 and tarPU = tarSU = 10

104 103 102 101 100 101 1020

2

4

6

8

10

12

f/e

Aver

age

Powe

r Con

sum

ptio

n at

NE

Poin

t(Watt

s) SUPU

Figure 4: Dependency of average power consumption on secondary cost parameters for 100users at p = 0.4

17

104 103 102 101 100 101 1023

4

5

6

7

8

9

10

11

f/e

Aver

age

SIR

Leve

l at N

E Po

int(W

atts)

SUPU

Figure 5: Dependency of average SIR achievement on secondary cost parameters for 100users at p = 0.4 and tarPU =

tarSU = 10

0 1 2 3 4 57.5

8

8.5

9

9.5

10

10.5

11

11.5

12

Number of Users Susceptible to Admission Control in Each Cell

NE

Poin

t Ave

rgae

Pow

er C

onsu

mpt

ion(w

atts)

SUPU

Figure 6: The effect of adding the admission control algorithm on average power consumptionfor 100 users at p = 0.4, = 70%, SIR cost factor of 103, and = 0.05

18

0 1 2 3 4 55

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

Number of Users Susceptible to Admission Control in Each Cell

NE

Poin

t Ave

rage

SIR

Lev

el

SUPU

Figure 7: The effect of adding the admission control algorithm on average SIR levels for 100users at p = 0.4, = 70%, SIR cost factor of 103, = 0.05, and tarSU =

tarPU = 10

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.4

0.5

0.6

0.7

0.8

0.9

1

Activity Probability

Co

vera

ge P

erce

ntag

e

No ACWith AC

Figure 8: Percentage of secondary users meeting their target levels for the case of 100 userswith tarSU = 10

19