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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 [email protected] [email protected]
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
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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
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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
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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.
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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.
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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).
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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)
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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