Harmonic mean rate fairness for cognitive radio networks with heterogeneous traffic

TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIESTrans. Emerging Tel. Tech. (2012)

Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ett.2541

RESEARCH ARTICLE

Harmonic mean rate fairness for cognitive radionetworks with heterogeneous trafficMina Dashti1*, Paeiz Azmi2 and Keivan Navaie3

1 Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran2 Electrical and Computer Engineering Department, Tarbiat Modares University, Tehran, Iran3 School of Electronic and Electrical Engineering, University of Leeds, Leeds, UK

ABSTRACT

This paper proposes new utility-based fairness criteria for radio resource allocation in underlay cognitive radio networksbased on harmonic mean concept. Then, joint rate and power allocation problem is considered for secondary users subjectto the signal-to-interference-noise ratio and interference threshold constraints. It is shown that the proposed approachyields a better balance between throughput and fairness compared with conventional fairness criteria, namely max–minand proportional fairness. Furthermore, in order to address the requirements of future wireless applications a cross-layer resource-allocation is proposed for heterogeneous traffic. A combination of streaming (which requires a maximumguaranteed average delay) and elastic traffic (with flexible rate requirements) is considered. The optimization problemallocates the available resources to the streaming users such that the quality-of-service constraints of the streaming usersare satisfied. Through extensive simulations, the effect of streaming traffic on the total throughput of elastic users isinvestigated. Simulation results demonstrate the increase in throughput of elastic users achieved by decreasing number ofstreaming users. Copyright © 2012 John Wiley & Sons, Ltd.

*Correspondence

M. Dashti, Department of Electrical Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran.E-mail: [email protected]

Received 25 January 2012; Revised 14 March 2012; Accepted 27 March 2012

1. INTRODUCTION

The emerging high-speed wireless communication tech-nologies are expected to dramatically increase the demandsfor spectral resources. However, radio spectrum is a finiteresource. At the same time, it has been reported byFederal Communications Commission’s Spectrum PolicyTask Force that many parts of the spectrum are usedinefficiently [1]. This is really an artificial result of the wayspectrum is regulated. Cognitive radio, which enables spec-trum sharing between different wireless services, attemptsto overcome spectrum scarcity [2]. In cognitive radio net-works, unlicensed users, also known as secondary user(SU), are allowed to share spectrum with licensed users,referred to as primary user (PU).

As described in [3], there are three basic approachesthat allow SUs to use the licensed spectrum: interweaving,overlay and underlay. In the interweaving scheme, the sec-ondary detect-and-exploit portions of the spectrum that arenot being occupied by licensed services are referred to asspectrum holes or white space in order to avoid interferencewith the PUs. In the overlay principle, the SU detects thepresence of the PUs and changes its own transmitted signal

so that it does not disturb the PUs. In this approach, the SUutilizes the same spectrum simultaneously with the PUs. Inthe case of the underlay approach, the SU is allowed to uti-lize the spectrum of the PUs provided that the interferencereceived from the SU is less than the interference level thatis tolerable by the primary receiver.

This paper mainly focuses on the underlay spectrumsharing approach because it does not require complicatedspectrum-sensing technologies. A large number of studieshave considered the underlay spectrum sharing (e.g. [4–7]).The studies related to radio resource allocation (RRA) forSUs in the underlay spectrum sharing scheme are relevantto our work.

In [8], the problem of rate and power optimization wasformulated by considering the minimum required signal-to-interference-noise ratio (SINR) for different unlicensedusers and also interference threshold constraints for thePUs. The main objective here is to maximize the total trans-mission rate of the SUs by adjusting the transmit power ofthese users. A user removal algorithm based on the tree-pruning searching algorithm was presented. This algorithmhad, however, an exponential time complexity. Resourceallocation for spectrum underlay in code division multiple

Copyright © 2012 John Wiley & Sons, Ltd.

M. Dashti, P. Azmi and K. Navaie

access (CDMA) networks under SINR and interferenceconstraints was proposed in [9]. Joint power and rate allo-cation with max–min fairness and proportional fairness(PF) were presented and the impacts of different criteriaon the network performance were evaluated. Coexistenceof cognitive radio and CDMA networks were studied in[10], where a more flexible approach that exploits the aver-age channel gain was considered. It is also introduced theviolation probabilities against the short-term fading effectfor secondary and PU links.

In [8–10], resource allocation in underlay spectrum shar-ing is studied in each time slot of packet transmissions. In[11], PF is achieved over multiple time slots. Consequently,those users who experience good channel conditions canincrease their transmission rate because a user who has apoor channel condition may delay its transmission. How-ever, this approach sometimes results in the lack of fair-ness to those users with poor channel conditions. In mostcases, fairness is in fact an important issue in designing andimplementing RRA schemes from both users and serviceproviders perspectives.

Fairness considerations have received enormous atten-tion in the related literature in recent years (e.g. [12–16]),the concept of fairness in specific domains of RRA in wire-less networks has been investigated in [17]. Two differenttypes of approaches are usually considered in defining theconcept of fairness in communications systems includingmax–min fairness and utility-based fairness.

In max–min fairness, the minimum data rate that adataflow achieves is maximized. Max–min fairness ignoresrevenue maximization for the network operators, but,rather, the focus is primarily on the satisfaction of ser-vice fairness from the users perspective. PF is a variantof utility-based fairness in which fairness is defined interms of maximizing total logarithmic user throughput. PFimproves the trade-off between the total network perfor-mance in terms of throughput and fairness experience ofindividual users.

In this paper, we first propose a new utility-based fair-ness which we refer to as harmonic rate fairness (HRF),to develop RRA schemes in the CDMA-based cognitiveunderlay network. In the treatment of the subject, homo-geneous traffic type is considered. The harmonic mean isdefined as a reciprocal of the arithmetic mean of the recip-rocals of the values. The harmonic mean of a set of num-bers is usually close to the least number in the set [18].Therefore, in RRA, adopting harmonic mean as the utilityfunction results in decreasing the possibility that users witha superior condition (i.e. high achievable rate) can overrideother users who experience inferior situations (i.e. deepfading). We believe that this characteristic can improvethe system fairness . We then consider the joint rate andpower allocation problem subject to SINR and maximumpower constraints for SUs. In our formulations, the inter-ference constraints is also considered at the PUs. We alsoprovide comparison analysis between our proposed utilityfunction and other two popular fairness concepts: max–minand PF.

In [8–11], resource allocation in underlay spectrum shar-ing is only studied for homogeneous traffic. In this paper,we further extend the related literature by incorporatinga combination of different traffic types. It is in line withthe fact that in the next generation of wireless networks,diversified services are provided for different traffic typeswith various delay requirements. The formulation basedon harmonic mean is then extended to the case of het-erogeneous traffic. The considered combination includesstreaming traffic, which requires a maximum guaranteedaverage delay, and elastic traffic with flexible rate require-ments. The main objective is to maximize the total through-put of elastic users where the average delay constraint forstreaming traffic and the maximum transmission power,rate and interference constraint are satisfied. A discussionof the effects of interference threshold, minimum processgain and SINR constraint on the throughput of SUs is alsoaccomplished.

In summary, the main contributions of our work are topropose novel utility-based fairness criteria based on theharmonic mean concept. In comparison with other relatedworks, our method provides a better compromise betweenefficiency and fairness. Moreover, we extend our modellingby considering the heterogeneous traffic model (a com-bination of streaming and elastic) and proposing an opti-mized cross-layer resource-allocation, which allocates theavailable resources to the elastic users so that the Qualityof Service (QoS) constraints of streaming users in termsof their delays are satisfied. Using simulations, we evalu-ate the impact of streaming traffic on the total throughputof elastic users. Simulation results are provided, whichshow the increase in throughput of elastic users achievedby decreasing number of streaming users. A discussion ofthe effects of interference threshold, minimum process gainand traffic load on the throughput and fairness of SUs arealso performed.

The remainder of the paper is organized as follows.Section 2 provides the system model and main assump-tions. The joint power and rate assignment problem for-mulation for different objective functions based on a fairand efficient framework is discussed in Section 3. Further-more, RRA based on heterogeneous delay requirementsis investigated in this section. Finally, simulation resultsare demonstrated in Section 4, followed by concludingremarks in Section 5.

2. SYSTEM MODEL

A CDMA-based cellular network is considered as the pri-mary network, in which mobile stations (MSs) or PUscommunicate with the corresponding base station (BS)through uplink transmission. Different uplink and down-link frequencies are used. Spectrum utilization may bevery low for some periods and in some particular loca-tions. In other words, spectrum is available but inefficientlyutilized; therefore, some MSs may form a secondary adhoc network (cognitive radio network) and communicate

Trans. Emerging Tel. Tech. (2012) © 2012 John Wiley & Sons, Ltd.DOI: 10.1002/ett


directly with each other. Figure 1 illustrates the systemmodel considered in this paper.

Allowing ad hoc communications within a cellularcoverage has been investigated in the related literature(e.g. [11]). Point-to-point links are considered; that is,transmissions are made through a single hop. The one-hoptransmission range in the secondary network is usuallyshorter than the size of cellular network. Secondary trans-missions use the same frequency band as the uplink inthe primary network and interfere with the normal uplinktransmissions in the primary network. It is assumed thatthe spectrum band occupied by a PU may be utilized byan SU, as long as the latter adheres to interference limi-tations set by the PU. Ith is used to denote the maximuminterference limit tolerable at the primary receiving point(i.e. BS); in other words, the total experienced interferenceat the primary receiving point cannot exceed Ith.

For the aforementioned spectrum sharing scenario, rateand power allocations for SUs are performed such thatSINR requirement for each secondary link and the toler-able interference limit at the primary receiving point mustbe satisfied.

Transmitter i and receiver i are used to denote secondaryreceiver of link i .Ns is the total number of SUs. The trans-mission rate and the transmission power corresponding tothe i th secondary link areRi andPi , respectively. The totalinterference at the primary BS must not exceed the interfer-ence threshold .Ith/. Therefore, the interference constraint

Figure 1. The considered system model in this paper.

can be written as follows:

NsXiD1

Pig(s2p)i 6 Ith

where g(s2p)i is the link power gain from the transmitter

of i th secondary link to the primary receiving point. Weassumed that the BS has perfect knowledge about all chan-nel gains of the system. Some good approaches mentionedin [9] may be used to estimate these channel gains at theBS. The SINR requirement of each secondary link can beexpressed as follows:

SINRi DB

Ri

Pig(s2s)i i

NsPj¤i

Pj g(s2s)j i C �i

>Q�i ; i D 1; : : : ; Ns

where BRiD Gi is the processing gain, B indicates the

system bandwidth, g(s2s)j i denotes the link gain from trans-

mitter j to receiver i , and �i represent the total noise andinterference due to PU transmissions at the secondary linki . The SINR requirement of secondary link i can be satis-fied if the SINR of the i th secondary link is larger than acorresponding value Q�i .

3. FORMULATION OF RADIORESOURCE ALLOCATION IN THEUNDERLAY SPECTRUM SHARING

In this section, fair RRA subject to SINR constraint andinterference constraint described previously is studied.The RRA is implemented in centralized fashion at theprimary BS.

The SU rate allocation problem is formulated as the fol-lowing optimization problem, which maximizes a certainfunction of secondary link transmission rate

maxPi;Ri

f .Ri / (1)

s.t.NsXiD1

Pig(s2p)i 6 Ith (2)

GiPig

(s2s)i i

NsPj¤i

Pj g(s2s)j i C �i

>Q�i ; i D 1; : : : ; Ns

(3)

06 Pi 6 Pmaxi ; i D 1; : : : ; Ns (4)

Rmini 6Ri 6R

maxi ; i D 1; : : : ; Ns (5)

The objective function f .Ri / is designed so as to achieveefficiency in terms of throughput while keeping a cer-tain level of fairness. The following conventional fairness



criteria have been frequently used for different resourceallocation schemes (e.g. [9–11])

max–min fairness f .Ri /DmaxfminiRi g (6)

proportional fairness f .Ri /D

NsXiD1

ln.Ri / (7)

The max–min fairness criteria provides perfect fairness forall secondary links. However, the link with the worst chan-nel condition limits the transmission rate of all secondarylinks. Hence, the focus of max–min fairness is not on maxi-mizing the total throughput; instead, the emphasis is mostlygiven to the satisfaction of fairness from the users’ point ofview. On the other hand, PF, which is the well-known vari-ant of utility-based fairness, has been receiving increasingattention in recent years (e.g. [8–16]). The proportional fairallocation offers a better trade-off between max–min andmaximum throughput allocation.

The trade-off can be further improved by our proposedobjective function, which is based on the harmonic meanconcept. Here, this is referred to as HRF. On account ofthe fact that the harmonic mean of a set of numbers isvery biased toward the least numbers of the set, it tendsto decrease the influence of large numbers and exacerbatethe impact of small ones. Therefore, this characteristiccan be used to obtain better fairness. Moreover, compar-isons are made between the proposed HRF and two otherconventional fairness methods.

Equations (2) and (3) are the interference threshold con-straint and the SINR requirement of each secondary link,respectively, which are mentioned in Section 2. Further-more, it is assumed that the transmission power and rate ofMS is limited by Pmax

i and Rmaxi , respectively.

3.1. Harmonic rate fairness RRA

The harmonic mean is the reciprocal of the arithmetic meanof the reciprocals. If R1; R2; R3; : : : ; RNs are the ratevalues, then the harmonic mean of these values is definedas follows:

Ns1R1C 1R2C 1R3C � � � C 1

RNs

DNs

NsPiD1

1=Ri

Therefore, the rate and power allocation problem based onHRF can be stated as

Problem O1:

maxPi;Ri

f .Ri /DNs

NsPiD1

1=Ri

(8)

s.t.NsXiD1

Pig(s2p)i 6 Ith (9)

B

Ri

Pig(s2s)i i

NsPj¤i

Pj g(s2s)j i C �i

>Q�i ; i D 1; : : : ; Ns (10)

06 Pi 6 Pmaxi ; i D 1; : : : ; Ns (11)

Rmini 6Ri 6R

maxi ; i D 1; : : : ; Ns (12)

3.1.1. Obtaining solution.

It is observed that the objective function in Equation (8)

is equivalent to minNsPiD1

1=Ri . Hence, the above optimiza-

tion problem can be rewritten as follows:Problem O2:

minPi;Ri

f .Ri /D

NsXiD1

1=Ri

s.t.NsXiD1

Pig(s2p)i 6 Ith

B

Ri

Pig(s2s)i i

NsPj¤i

Pj g(s2s)j i C �i

>Q�i ; i D 1; : : : ; Ns

06 Pi 6 Pmaxi ; i D 1; : : : ; Ns

Rmini 6Ri 6R

maxi ; i D 1; : : : ; Ns

O2 is not a convex optimization problem. To solvethis problem, we obtain the equivalent standard geometricprogram (GP) [19], which is an optimization problem ofthe form

Problem O3:

minPi;Ri

NsXiD1

R�1i

s.t.NsXiD1

Pig(s2p)i

Ith6 1

RiP�1i

Q�i

Bg(s2s)i i

NsXj¤i

Pj g(s2s)j i CRiP

�1i

�iQ�i

Bg(s2s)i i

6 1

i D 1; : : : ; Ns�Pmaxi

��1Pi 6 1; i D 1; : : : ; Ns

Rmini R�1i 6 1; i D 1; : : : ; Ns�Rmaxi

��1Ri 6 1; i D 1; : : : ; Ns

As described in [19], in a standard form GP, the objectiveis posynomial (and it must be minimized), and the inequal-ity constraints can only have the form of a posynomial lessthan or equal to one. Therefore, O2 is a standard GP. Tosolve a standard GP efficiently, it is necessary to convert itto a convex optimization problem. Efficient solution meth-ods for convex optimization problems are well developedin [19].



This problem can, however, be transformed to a con-vex optimization problem by a logarithmic change of vari-ables and a logarithmic transformation of the objective andconstraint functions.

The variables are defined as yi D logRi , so Ri D eyi

and zi D logpi , so pi D ezi . After the change of vari-ables, a posynomial becomes a sum of exponentials ofaffine functions. The GP .O3/ can be expressed in terms ofnew variables and by considering the logarithm as follows:

minpi;Ri

log

0@ NsXiD1

e�yi

1A

s.t. log

0@ NsXiD1

g(s2p)i

Ithezi

1A6 0

log

0@ Q�i

Bg(s2s)i i

NsXj¤i

e.yiCzj�zi /g(s2s)j i C

�iQ�i

Bg(s2s)i i

�e.yi�zi /

1A6 0; i D 1; : : : ; Ns

log��Pmaxi

��1ezi�6 0; i D 1; : : : ; Ns

log�Rmini e�yi

�6 0; i D 1; : : : ; Ns

log��Rmaxi

��1eyi

�6 0; i D 1; : : : ; Ns

Because log-exp and log-sum-exp functions are convex,the aforementioned problem is a convex optimizationproblem [19]. Hence, the global optimal solution can beachieved by interior point methods, which are methodsfor solving convex optimization problems that includeinequality constraints. They solve an optimization problemwith linear equality and inequality constraints by reduc-ing it to a sequence of linear equality constrained problems(as described in chapter 11 of [19]). Interior-point meth-ods solve this problem by applying Newton’s method to asequence of equality constrained problems [19].

3.2. Cross-layer radio resource allocationfor heterogeneous traffic

In this section, the optimal RRA based on heterogeneousdelay requirements is formulated. A mixture of two traf-fic types including streaming and elastic traffic is consid-ered. Elastic traffic is a model for delay-tolerant services;an example of such traffic is transferring digital docu-ments through FTP or TCP. In contrast to the elastic traffic,streaming traffic such as real-time applications has strictdelay requirements. Examples of such traffic include onlinegaming and Internet Protocol television (IPTV).

The total number of SUs, Ns, is divided into twogroups: (1) SUs with streaming traffic, where ks

i is the ith

secondary streaming user and us is the number ofsecondary streaming users; and (2) SUs with elastic traf-fic, where ke

i is the ith secondary elastic user and ue is thenumber of secondary elastic users so that ue C us D Ns.Hereafter, the term ‘user’ is utilized instead of ‘SU’ forbrevity, unless specified otherwise.

Corresponding to each user, a queue that followsM=G=1 model is considered. M/G/1 represents a singleserver that has unlimited queue capacity and infinite callingpopulation, while the arrival is Poisson process, meaningthat the statistical distribution of the inter-arrival times fol-lows the exponential distribution. The distribution of theservice time may follow any general statistical distribu-tion. This is primarily because of the fading channel, whichmakes the service process very hard for modelling. Thismodel is also appropriate for modelling various types oftraffic with different QoS requirements [20].

Here, the problem of resource allocation is consideredwith streaming traffic, which requires a maximum guaran-teed average delay, and elastic traffic with flexible rate ofrequirements. The main objective is to maximize the har-monic mean of transmission rate of elastic users while theaverage delay constraints for streaming traffic are satisfied.Moreover, SINR constraint, the maximum transmissionpower constraint and interference threshold constraint forboth elastic and streaming users are satisfied. The optimalcross-Layer RRA problem is as follows:

Problem O4:

maxPi;Ri

ue

uePiD1

1=Rkei

(13)

s.t. E�Wks

i

�6 �ks

i; i D 1; : : : ; us

(14)

ueXiD1

Pkeig

(s2p)kei

C

usXiD1

Pksig

(s2p)ksi

6 Ith (15)

BPksig

(s2s)ksii

Rksi

usPj¤i

Pksjg

(s2s)ksji

CuePj

Pkejg

(s2s)keji

C �i

! >Q�i ;

i D 1; : : : ; us

(16)

BPkeig

(s2s)keii

Rkei

uePj¤i

Pkejg

(s2s)keji

CusPj

Pksjg

(s2s)ksji

C �i

! >Q�i ;

i D 1; : : : ; ue

(17)

06 Pk 6 Pmax; k 2˚ks1; : : : ; k

sus ; k

e1; : : : ; k

eue

�(18)

Rk 6Rmax; k 2˚ks1; : : : ; k

sus ; k

e1; : : : ; k

eue

�(19)



where Equation (13) is the harmonic mean of transmis-sion rate of elastic users; delay constraint for streaming

users is shown in Equation (14), in which E�Wks

i

�is the

average waiting time of user ksi in the queue plus the ser-

vice time (which indicates how long the service will take).�ksi

is the required average maximum delay of streaminguser ks

i ; Equation (15) represents the interference thresholdconstraint for both elastic and streaming users as definedin previous sections; Equations (16) and (17) state SINRconstraint for streaming and elastic traffic, respectively.Equations (18) and (19) are the total transmission powerand rate constraints. In the aforementioned optimizationproblem, it is assumed that users have enough trafficwaited in the queue and ready to be transmitted (i.e. fullbuffer assumption).

In order to solve the cross-layer resource-allocationproblem, O4, the delay constraint (14) is transformed intoa constraint in terms of physical-layer parameters. To trans-form the constraint in Equation (14), according to [20],an equation is obtained which models the relationshipbetween the scheduled data rate of the user ks

i and its traffic

characteristic��ksi; �ks

i

�. It is assumed that the arrival traf-

fic for user ksi is modelled via a Poisson distribution with

the average �ksi. As mentioned in [21] for M=G=1 model

E�Xks

i

�C

�ksiE�X2ksi

�2�1� �ks

iE�Xks

i

�� 6 �ksi

(20)

where Xksi

is the service time of user ksi . By mathematical

derivations based on Equation (20), which is provided inAppendix I, the following constraint, which is a relaxationof Equation (20) is obtained:

Rksi>

�E.z/; �ks

i; �ks

i

�(21)

where �E.z/; �ks

i; �ks

i

�is

2�ksiE.z/�

2C 2�ksi�ksi

��

r�2C 2�ks

i�ksi

�2� 8�ks

i�ksi

(22)

and E.z/ is the average packet size. Therefore,

�E.z/; �ks

i; �ks

i

�D Rm is the minimum rate require-

ment of streaming users. As a result, the optimizationproblem O4 is transformed into the following:

Problem O5:

maxPi;Ri

ue

uePiD1

1=Rkei

(23)

s.t. Rksi>Rm; i D 1; : : : ; us

(24)

ueXiD1

Pkeig

(s2p)kei

C

usXiD1

Pksig

(s2p)ksi

6 Ith (25)

BPksig

(s2s)ksii

Rksi

usPj¤i

Pksjg

(s2s)ksji

CuePj

Pkejg

(s2s)keji

C �i

! >Q�i ;

i D 1; : : : ; us (26)

BPkeig

(s2s)keii

Rkei

uePj¤i

Pkejg

(s2s)keji

CusPj

Pksjg

(s2s)ksji

C �i

! >Q�i ;

(27)

i D 1; : : : ; ue

06 Pk 6 Pmax; k 2˚ks1; : : : ; k

sus ; k

e1; : : : ; k

eue

�(28)

Rk 6Rmax; k 2˚ks1; : : : ; k

sus ; k

e1; : : : ; k

eue

�(29)

where constraints (14) in O4 are substituted byEquation (24) in this optimization problem. The originalproblem formulation presented in O4 is a cross-layer opti-mization problem, which considers different delay require-ments. In other words, in this formulation, we translatedMAC-layer parameters into physical-layer parameters.

Here, similar to Section 3.1, we adopt geometricprogramming to obtain the solution of the optimizationproblem in O5.

4. SIMULATION RESULTS

A single-cell CDMA system is considered. The total band-width of the system is 5 MHz. The maximum transmissionpower on secondary links is 0.1 W. As previously men-tioned in Section 2, R is inversely proportional to PG.Therefore, because of the fact that B is a constantvalue, the maximum transmission rate can be written asRmax D B=PGmin , where PGmin is the minimumprocessing gain.

The channel gains are modelled as K010�=10.d�4/ [9]where d is the distance between the transmitter and thereceiver, � is random Gaussian variable with zero meanand the standard deviation equals 6 dB, K0 D 103, whichcaptures system and transmission effects such as antennagain and carrier frequency. The total noise and interferenceat the receiving nodes of all secondary links is chosen to be10�10 W.

Primary users communicate with its BS in theuplink direction. Transmitting nodes corresponding to thesecondary links are randomly located in a coverage areawith the BS of primary network located at the centre. Thesize of coverage area is 4 km2. Furthermore, the receiv-ing node of each secondary link is generated randomly ina 1 km � 1 km rectangle with its transmitting node beingat the centre. Because the channel power gains can be dif-ferent for different channel realizations, all the numerical



results presented in this part are obtained by averaging over2000 independent simulation runs.

4.1. Comparison of HRF, PF andmax–min fairness

In Figure 2, the total throughput of SUs for HRF, PFand max–min fairness versus the minimum processinggain under different SINR and interference constraints areillustrated. It is seen that PF and HRF criteria achievehigher throughput than of the one with max–min fairnesscriterion. Moreover, it is observed that the PF criterion onlyobtains a slightly higher throughput than the HRF criterion.Note that, for the minimum processing gain grater than 20,both PF and HRF criteria accomplish similar performancein terms of throughput. In addition, when the PGminincreases, the performance gap in terms of total through-put under different fairness criteria become smaller. Thisresults from the fact that, when the PGmin increases,the maximum rate decreases, which essentially reduces thefeasible region. Hence, the gaps among different through-put curves become smaller. Moreover, the total through-put of SUs for different sets of constraints are evaluated.Two sets are plotted in Figure 2. In the first set, we con-sider Q� D 10 dB and Ith D 20N0. However for themore stringent constraints, we assume that both SINR andinterference constraints are more stringent (Q� D 15 dBand Ith D 5N0), which means that SUs should satisfyhigher SINR and also cause less interference at the PUs.As expected, it can be observed that the more stringentthe SINR and interference constraints are, the lower thethroughput that can be achieved for all above fairnesscriteria is.

In order to examine the effectiveness of our proposedfairness criteria, we use Jain metric for comparing PF and

10 15 20 25 30 35 400.5

1

1.5

2

2.5x 10

6

Minimum processing gain

Tot

al th

roug

hput

(bp

s)

PF, Q*=10dB, Ith=20NoHRF, Q*=10dB, Ith=20NoPF , Q*=15dB, Ith=5NoHR, Q*=15dB, Ith=5NoMaxMin, Q*=10dB, Ith=20NoMaxMin, Q*=15dB, Ith=5No

Figure 2. Comparison of the SUs throughput under differentfairness criteria and different sets of constraints versus the

minimum processing gain, Ns D 5.

HRF criteria, which is define as follows:

FIJain D

NsPiD1

Ri

!2

Ns

NsPiD1

R2i

(30)

This index is applicable to any resource-allocationproblem and has been used variously in the literature(e.g. [9, 11]). It is independent of the amount of availableresources. In other words, it is dimensionless and inde-pendent of scale and is also a continuous parameter. Itis bounded between 0 (for the worst case) and 1 (for thebest case). If all users get the same amount, then the fair-ness index is equal to one (when all users receive thesame resource allocation), and the system is 100% fair.Max–min fairness is an instance of the systems with thefairness index of 100%.

It is apparent from Figure 3 that resource allocation solu-tions are fairer for the HRF case in comparison with the PFcase. Furthermore, Fairness improves when the minimumprocessing gain increases (i.e. due to smaller feasible rateregion). Moreover, as is observed from Figure 3, the morestringent the SINR constraint is, the lower the fairness thatcan be achieved is. However, fairness curves get closerto each other as the minimum processing gain increasesbecause of smaller feasible rate region.

The sum power of SUs and the total interference ofSUs at the PU versus the minimum processing gain underNs D 5, Q� D 10 dB and Ith D 20N0 for HRF, PFand max–min fairness are demonstrated in Figures 4 and 5,respectively. As it is seen, the sum power of SUs in HRFis less than that of PF as a result of lower rate and powerallocated to the HRF. Therefore, the HRF causes less inter-ference to the primary receivers than that of the PF, whichis observed from Figure 5. In addition, the sum power ofHRF is higher than the sum power of the max–min fairness

10 15 20 25 300.8

0.85

0.9

0.95

1


Fai

rnes

s In

dex

PF, Q*=15dB,Ith=5No HRF,Q*=15dB,Ith=5No PF, Q*=10dB,Ith=20NoHRF,Q*=10dB,Ith=20No

Figure 3. Fairness index under proportional fairness (PF) andharmonic rate fairness (HRF) and under different sets of con-

straints versus the minimum processing gain, Ns D 5.



10 15 20 25 30 35

0.08

0.09

0.1

0.11


Sum

pow

er (

W)

PFHRFMax−Min

Figure 4. Comparison of the SUs’ sum power for HRF, PF andmax–min fairness versus the minimum processing gain under

Ns D 5, Q� D 10 dB and Ith D 20N0.

10 15 20 25 30 353.4

3.6

3.8

4

4.2

4.4x 10

−10


Tot

al in

terf

eren

ce a

t the

prim

ay u

ser

PF HRF MaxMin

Figure 5. Total interference of SUs at the primary user versusthe minimum processing gain under Ns D 5, Q� D 10 dB and

Ith D 20N0 for HRF, PF and max–min fairness.

as a result of higher rate and power allocated to the usersin HRF. Hence, HRF is more efficient than PF in terms ofsum power of SUs.

4.2. Effect of heterogeneous traffic

Here, we consider a mixture of elastic and streaming trafficfor the HRF case. In the following figures, the number ofSUs has been set to 10, and the minimum rate requirementof streaming users is 128 kbps.

To investigate the effect of streaming traffic on the totalthroughput of elastic users, we consider different num-bers of streaming traffic (Us D 2; 4; 6; 8). The results aredemonstrated in Figure 6, which shows the total through-put of elastic users versus interference threshold. In thisfigure, the minimum processing gain has been set to 10and Q� D 5 dB. As can be seen, increasing the number of

10−11

10−10

10−9

10−8

10−7

0.5

1

1.5

2

2.5

3x 10

6

interference threshold

Thr

ough

put o

f ela

stic

use

rs (

bps)

Us=2Us=4Us=6Us=8

Figure 6. Total throughput of elastic users versus interfer-ence threshold for two, four, six and eight streaming users(Us=2,4,6,8), under minimum process gain = 10 and Q� D 5 dB.

streaming traffic results in lower throughput of elastic usersbecause of the fact that fewer elastic users are compet-ing with more streaming users for the network resources.The figure also reveals that as the interference thresholdincreases, the allocated power for elastic users; hence, theirachievable rate and, consequently, throughput of elasticusers also increase. However, the rate of throughput growthis not the same for all amount of interference thresholdowing to the fact that as the interference threshold furtherincreases, the elastic users make more interference to PUs.Then, the SU increases its transmission power, whereasthis is finally limited by its maximum transmission power.Consequently, further increase of the interference thresh-old does not help the elastic users to transmit higher power.Hence, the higher level of the interference threshold(more than 10�8) have negligible effects on total through-put of elastic users.

We also performed simulations for streaming users toexamine whether Equation (24) is satisfied; Figure 7 illus-trates that the rate of streaming users is almost constant fordifferent interference threshold levels, which is very impor-tant for keeping the QoS of streaming users consistently.For instance, when the number of streaming is 2, the totalthroughput should be at least 256 kbps, which is apparentfrom this figure. It can be also observed from these plotsthat as the number of streaming users increases, the totalthroughput also increases.

We further examine the effect of different SINR con-straint on the total throughput of elastic users. Plots forthe total throughput of elastic users versus the interfer-ence threshold under various SINR constraints (Q� D5; 10; 15) dB with Us D 4 for HRF are provided inFigure 8. We observe that by increasing the SINR con-straint, the rate of elastic users decreases; thereby, thetotal throughput of elastic users decreases; however, gapin the throughput graphs decreases for lower values ofinterference threshold due to smaller feasible rate region.



10−11

10−10

10−9

10−8

10−7

2

4

6

8

10

12x 10

5


Thr

ough

put o

f str

eam

ing

user

s (b

ps)

Us=8Us=6Us=4Us=2

Figure 7. Total throughput of streaming users versus interfer-ence threshold for two, four, six and eight streaming users, andunder minimum process gain = 10, Q� D 5 dB, For streaming

users, the minimum rate requirement is 128 kbps.

10−11

10−10

10−9

10−8

10−7

0

0.5

1

1.5

2

2.5x 10

6

Interference threshold

Thr

ough

put o

f ela

stic

use

rs (

bps)

Q*=5 dBQ*=10 dBQ*=15 dB

Figure 8. Total throughput of elastic users versus interferencethreshold for four streaming users (Us D 4), six elastic users,

Q� D 5;10;15 dB and under minimum process gain = 10.

For instant, by increasing the SINR constraint form Q� D

5 dB to Q� D 15 dB, for a given interference threshold.10�8/, the throughput degradation is about 1200 kbps. Inaddition, for a given interference threshold .10�11/, thethroughput degradation is about 720 kbps.

Figure 9 includes the plots for the total throughput ofelastic users versus the interference threshold under vari-ous PGmins, with Us D 4 for PF and max–min fairness.Throughput curves indicate that throughput of elastic usersdecreases when the minimum processing gain increases.This results from the fact that as the minimum processinggain increases, the maximum rate decrease; hence, totalthroughput of elastic users decreases. As mentioned pre-viously, It is also observed that the higher level of the

10−11

10−10

10−9

10−8

10−7

0

2

4

6

8

10x 10

6


Thr

ough

put o

f ela

stic

use

rs (

bps)

PGmin=2PGmin=5PGmin=10PGmin=27

Figure 9. Total throughput of elastic users versus interferencethreshold for four streaming users (Us D 4), six elastic users,Q� D 5 dB and under minimum process gain = 2,5,10 and 27.

interference threshold (more than 10�8) has negligibleeffects on the total throughput of elastic users.

5. CONCLUSION

The rate and power allocation problem for SUs withQoS requirements and interference threshold constraint inCDMD-based cognitive radio (CR) system has been for-mulated as an optimization problem. The geometric pro-gramming has been proposed for solving the problem.Based on the unique features of the harmonic mean, aneffective utility-based framework has been proposed forfairness analysis. Moreover, a comparison has been madebetween the proposed HRF method and the two otherconventional fairness methods: PF and max–min fairness.Simulation results have been provided, which showed thatthe proposed utility provides a better compromise betweenefficiency and fairness than the conventional fairness cri-teria. So that in comparison with max–min fairness, ourproposed criteria (HRF) can achieve higher throughput.Moreover, HRF has performed better than PF in termsof fairness. In addition, sum power of users in HRF isless than in PF. Furthermore, total interference imposedby SUs to PU in the HRF criteria is lower than one in thePF criteria.

Moreover, this work has been developed for a hetero-geneous traffic model (streaming and elastic). Then, aresource-allocation problem has been presented that allo-cated the available resources to the elastic users so that theQoS constraints of streaming users, which are measured bytheir delay, are satisfied. To solve the optimization prob-lem, the delay constraints are translated into physical-layerparameters. Simulations have been used for demonstratingthe effect of streaming traffic on the total throughput ofelastic users.



APPENDIX I

Equation (20) can be rewritten as follows:

�ksiE�X2ks

i

�6 �ks

i�E.Xks

i/�2C2�ks

i�ksi

�C2�ks

i

�E�Xks

i

��2(31)

because E�X2ksi

�>�E�Xks

i

��2. Consequently, Equation (31)

is given by

�ksi

�E�Xks

i

��2C �ks

i�E

�Xks

i

� �2C 2�ks

i�ksi

�> 0(32)

Because of the fact that �ksi> 0, the polynomial in the

left-hand side of (32) is greater than or equal to zero

if E�Xks

i

�6 E�

�Xks

i

�1

and E�Xks

i

�> E�

�Xks

i

�2

where E��Xks

i

�1;2

are roots of the polynomial in the

left-hand side of Equation (32), which are taken as follows:

E��Xks

i

�1;2D

�2C2�ks

i�ksi

�˙

r�2C2�ks

i�ksi

�2� 8�ks

i�ksi

2�ksi

(33)

It can be easily shown that E��Xks

i

�1

and E��Xks

i

�2

are

both positive. Because the system requires smaller servicetime, we choose the smaller root. Therefore, the inequality(32) satisfies if

E�Xks

i

�6

�2C 2�ks

i�ksi

��

r�2C 2�ks

i�ksi

�2� 8�ks

i�ksi

2�ksi

(34)If we denote the average packet size by z, then we have

E�Xks

i

�D

E.z/

Rksi

(35)

By using Equations (34) and (35), the delay constraintin Equation (14) transforms into the rate constraint inEquation (21).

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Harmonic mean rate fairness for cognitive radio networks with heterogeneous traffic