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Bakht Khush Jameel Furqan Ali Zain Khan Wali Ullah Khan Imran Sardar SidhuGuftaar Ahmad Lee Jeong WooPower Allocation and User Assignment Scheme for beyond 5G Heterogeneous Networks

Published inWireless Communications and Mobile Computing

DOI10115520192472783

Published 01012019

Document VersionPublishers PDF also known as Version of record

Please cite the original versionBakht K Jameel F Ali Z Khan W U Khan I Sardar Sidhu G A amp Lee J W (2019) Power Allocationand User Assignment Scheme for beyond 5G Heterogeneous Networks Wireless Communications and MobileComputing 2019 [2472783] httpsdoiorg10115520192472783

Research ArticlePower Allocation and User Assignment Scheme forbeyond 5G Heterogeneous Networks

Khush Bakht1 Furqan Jameel2 Zain Ali3 Wali Ullah Khan 4 Imran Khan 5

Guftaar Ahmad Sardar Sidhu3 and Jeong Woo Lee 6

1Department of Electronic Engineering Fatima Jinnah Women University Rawalpindi 46000 Pakistan2Department of Communications and Networking Aalto University 02150 Espoo Finland3Department of Electrical Engineering COMSATS University Islamabad 45550 Pakistan4School of Information Science and Engineering Shandong University Qingdao 266237 China5University of Engineering and Technology Peshawar Peshawar Pakistan6School of Electrical and Electronics Engineering Chung-Ang University Seoul Republic of Korea

Correspondence should be addressed to Jeong Woo Lee jwlee2cauackr

Received 8 June 2019 Revised 31 August 2019 Accepted 25 October 2019 Published 16 November 2019

Academic Editor Olivier Berder

Copyright copy 2019 Khush Bakht et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e issue of spectrum scarcity in wireless networks is becoming prominent and critical with each passing year Although severalpromising solutions have been proposed to provide a solution to spectrum scarcity most of them have many associated tradeosIn this context one of the emerging ideas relates to the utilization of cognitive radios (CR) for future heterogeneous networks(HetNets) is paper provides a marriage of two promising candidates (ie CR and HetNets) for beyond fth generation (5G)wireless networks More specically a joint power allocation and user assignment solution for the multiuser underlay CR-basedHetNets has been proposed and evaluated To counter the limiting factors in these networks the individual power of transmittingnodes and interference temperature protection constraints of the primary networks have been considered An ecient solution isdesigned from the dual decomposition approach where the optimal user assignment is obtained for the optimized power al-location at each node e simulation results validate the superiority of the proposed optimization scheme against conventionalbaseline techniques

1 Introduction

In the last few years the wireless systems have evolved to thepoint where homogeneous cellular networks have achievednear optimal performance [1] ese advancements in thehomogeneous cellular networks though signicant may notbe enough to support beyond fth generation (5G) wirelessnetworks [2] To do so dynamic and exhaustive improve-ments in spectral eciency are needed One of the proposedsolutions is advanced networks which consist of multipletiers of high-powered base stations (BSs) overlaid with low-powered BS both having dierent coverage areas alsoknown as heterogeneous networks (HetNets) [3] Typicallya HetNet consists of main macro base stations (MBSs) a fewpico base stations (PBSs) and several femto base stations

(FBSs) An MBS in HetNets has a high transmission powerand greater coverage area which is then overlaid with low-powered PBS and FBS [4 5]

ere are dierent purposes of PBS and FBS in theHetNets Generally speaking the PBS are laid in a densetrac areas to improve coverage in hotspot areas while FBSare overlaid in a manner to remove the coverage issues inhomogeneous systems thereby improving the overall per-formance In conventional homogeneous cellular networkthe mobile terminal is associated with a BS on the basis ofdownlink signal-to-interference-and-noise ratio (SINR)Specically a user is associated with a particular BS whichoers SINR greater than the other BSs [6] In HetNets theSINR-based association principle leads to having load bal-ancing issue among MBS and PBS Hence the network

HindawiWireless Communications and Mobile ComputingVolume 2019 Article ID 2472783 11 pageshttpsdoiorg10115520192472783

resources need to be intelligently distributed among BSs forhigher spectral efficiency [5 6]

11 RelatedWorks By the end of 2020 it is anticipated thatup to 50 billion devices will exist in the world including staticand mobile platforms [7] Due to this reason there has beenan upsurge in the research on HetNets to provide efficientand long-term solutions [8] Zhang et al in [9] investigatedthe problem of user association in HetNets -ey consideredan optimization problem with different traffic capacitylimits quality of service (QoS) requirements and powerbudget constraints-e proposed scheme of user associationimproved the performance of the network A bit-rateadoption method was proposed by the authors of [10] forbody area HetNets -eir scheme showed the promise ofincreasing the efficiency of packet transmission regardless ofplacement of the node -e authors of [11] performed jointoptimization of resource allocation and user association inHetNets -ey presented and compared three allocationstrategies ie orthogonal deployment cochannel de-ployment and partially shared deployment Sequentialquadratic programming- (SQP-)based power optimizationapproach was presented in [12] to maximize the sum rate ofsmall-cell networks Similar works have been done by theauthors of [13] where a pricing-based approach for asso-ciating the user to a particular BS was proposed by theauthors of [14] and the idea of a distributed price updatestrategy was presented for achieving the user fairness

Liu et al presented a fair user association scheme in [15]-ey used the Nash bargaining solution (NBS) whereby theoptimization with fairness was achieved among competingBSs In [16] the HetNets using orthogonal frequency di-vision multiple access (OFDMA) network were presented bythe authors -e aim was to manage radio resource bymaximizing the throughput of user having minimum rateFrom the perspective of nonorthogonal multiple access(NOMA) the author of [17] provided a novel idea to useinterference-aided vehicular networks and highlighted somekey challenges Similarly the authors of [18] managed theradio resources by a scheduling algorithm implemented by acentral global resource controller (GRC) -e algorithmoptimized the attributes such as fairness among usersspectral efficiency and battery lifetime Relay-based HetNetswere considered by the authors of [19] wherein they pro-posed a method to suppress intercell and intracell in-terference-eir proposed scheme was shown to outperformexisting baseline methods in terms of sum-rate performanceAnother similar and much recent work [20] provided op-timal time switching and power splitting technique forimproving the performance of wireless network-e authorsin [21] jointly optimized power allocation and user asso-ciation in ultradense heterogeneous networks using non-cooperative game theory thus achieving the increase insystem throughput as well as optimal power allocation

Semov et al proved that in HetNets by taking thegeographical position into account the userrsquos throughputand fairness can be improved [22] A similar concept wasemployed in the form of relays by the authors of [23] to

ensure proportional fairness among user equipment bytaking into account backhaul links between the relays andBS On the other hand the authors of [24] considereddistributed antenna system (DAS) and compared cochannelresource allocation schemes for energy-efficient communi-cation In [25] the authors studied a single-cell HetNethaving one macro and one picocell for efficient resourceallocation such that the energy efficiency is maximized byproposing an iterative resource allocation algorithmHowever the same authors did not consider multicellularnetwork for the evaluation To achieve the spectral efficiencyof the system more advanced dynamic spectrum accesstechniques (DSA) should be employed -e cognitive radio(CR) is an efficient DSA technique [26] that allows sec-ondary (unlicensed) users (SU) to access the spectrum ofprimary (licensed) users (PU) in an opportunistic way [27]In CR spectrum sharing can be classified as spectrumoverlay and spectrum underlay In spectrum overlay the SUcan transmit simultaneously on the frequency band used bythe PU through adjusting its transmit power such that it doesnot cause much interference for the PU [27]

Of late the CR-based HetNets have been drawing a lot ofresearch attention nowadays due to their dynamic resourceallocation property -e power adjustment of BS and mobileusers can be achieved by dynamic resource allocation [28]In this regard the author in [29] maximized the energyefficiency subject to power and interference constraint inOFDMA-based CR networks -e authors applied theconvex optimization theory and proposed an iterative al-gorithm In [30] the authors considered a cognitive fem-tocell network using OFDMA and solved sum-utilitymaximization and dynamic resource allocation problemwith the help of the dual decomposition method In [31] theauthors considered the stochastic optimization model formaximizing the long term energy efficiency in time-varyingheterogeneous networks -e proposed problem is a mixedinteger problem and the Lagrange dual method has beenused to solve it In [32] the authors considered the resourceallocation problem for rate maximization in multiusercognitive heterogeneous networks -e authors consideredthe maximum transmit power of cognitive microcell basestation and cross-tier interference constraints simulta-neously -ey converted the nonconvex optimizationproblem into a geometric programming problem and solvedit in a distributed way using the Lagrange dual method

12 Motivation and Contributions Although the researchworks reported in recent years have considered the overallsystemrsquos performance maximization for CR-based HetNetsthe problem of fairness among different users has receivedlittle attention -e potential problem arises when theschemes proposed for the sum-rate maximization assignvery few or no resources to some of the users with higherfading conditions -is uneven distribution of resourcesresults in degrading the achievable performance for differentusers -is problem may become more serious for the CR-based HetNets due to the limiting factor of increased in-terference -us optimization of the transmission for user

2 Wireless Communications and Mobile Computing

fairness under more practical constraints becomes essentialTo the best of our knowledge the resource optimization anduser assignment techniques for fair rate allocation have notbeen jointly investigated in the literature due to the higherlevel of complexity involved in finding the optimal solution

To fill this gap in the literature and provide a compre-hensive solution to the user fairness problem in CR-basedHetNets we provide a joint strategy for power allocation anduser fairness In particular we consider the joint poweroptimization and user association problem in HetNets forachieving fairness among different users We first formulatea joint optimization problem subject to power and in-terference constraints -en to provide a less complex andan efficient solution we design an algorithm from the dualdecomposition strategy -e presented numerical resultsindicate the importance and utility of our scheme incomparison with the baseline methodologies

13 Organization -e remainder of this paper is organizedas follows -e system model and problem formulation arepresented in Section 2 In Section 3 the proposed solutionhas been described while Section 4 discusses the simulationresults Finally Section 5 presents some concluding remarksand future research directions In addition the list of ac-ronyms used throughout this paper has been provided inTable 1

2 System Model and Problem Formulation

-is section describes the considered system model andprovides the steps related to problem formulation

21 System Model A downlink CR transmission is con-sidered where the secondary HetNet system is reusing thespectrum of the primary network in an underlay mode asshown in Figure 1 -e HetNet consists of a single macrocelloverlaid with multiple pico BSs intended to transmit data tomultiple users It is assumed that all the devices are equippedwith a single antenna and experience independent andidentically distributed (iid) Rayleigh fading -e channelbandwidth is distributed among MBS and PBSs in such away that MBS and PBSs are nonorthogonal to each otherwhile each PBS is orthogonal to other PBSs in the HetNet

In this model BSs are denoted as BSb such thatB BSb | b 1 2 3 B where BS1 is modeled as macro

and the remaining BSs act as pico BSs Further there are Ausers and C channels allocated to BS -e signal-to-in-terference-and-noise ratio (SINR) of the ath user from thebth BS at the cth channel is given as in [11]

SINRacb Pacbgacb

Pacbprime facb + σ2a

(1)

where Pacb is the transmit power for the ath user connectedto the bth BS at cth channel αacb denotes the associationvariable which belongs to set 0 1 σ2a is the power spectraldensity (PSD) of the noise while gacb and facb are thechannel gains from intended BS to user and from interferingBS to the user respectively Furthermore Pacb

prime denotes thetransmit power of interfering BS

22 Problem Formulation One of our key objectives in thisarticle is fair maximization of the data rate of each user in thenetwork To facilitate mathematical analysis we introduce abinary variable αacb such that

αacb 1 when the ath user is associatedwith the bth BS through the cth channel

0 otherwise1113896 (2)

Based on the above expression the channel allocationfollows

1113944

C

c11113944

B

b1αacb 1 foralla 1 2 3 A (3)

To protect PUs from the interference and to improve theperformance of the system resource allocation at BSs needto be performed such that

1113944

A

a11113944

C

c11113944

B

b1αacbPacbhacb le Ith (4)

Table 1 List of acronyms

Acronym Definition5G Fifth generationAWGN Additive white Gaussian noiseBS Base stationCR Cognitive radioD2D Device to deviceDAS Distributed antenna systemDSA Dynamic spectrum accessFBS Femto base stationGRC Global resource controllerHetNet Heterogeneous networksKKT KarushndashKuhnndashTuckerMBS Macro base stationNBS Nash bargaining solutionOFDMA Orthogonal frequency division multiple accessPBS Pico base stationPSD Power spectral densityPU Primary userQoS Quality of serviceRRM Radio resource managementSQP Sequential quadratic programmingSU Secondary userSINR Signal-to-interference plus noise ratio

Wireless Communications and Mobile Computing 3

where Pacb is the power allocated by the bth BS to the cthchannel allocated to the ath user and hacb represents thegain of interference channel c from the bth BS to primary BSFinally to ensure that the power consumed by each BS is lessthan or equal to the power budget power allocation at eachBS needs to be ensured as

1113944A

a11113944

C

c1Pacbαacb lePb forallb 1 2 3 B (5)

To achieve fairness in rates of different users a max-min-based optimization framework is adopted such that theproblem is stated mathematically as

P1 maxPacbαacb

min log2 1 + SINRacb1113872 1113873

st (2) (3) and (4)

(6)

where Pb denotes total power available at the bth BS and Ithrepresents the sum interference threshold Here the firstconstraint ensures that the total power allocated by the bthBS must be within the available power budget Similarly thesecond constraint ensures that the primary system is wellprotected from interference due to the underlay commu-nication and the third constraint makes sure that the cthchannel is allocated to just one SU

3 Proposed Solution

To provide a viable solution to the aforementioned problemwe propose a solution based on duality theory As is evidentfrom the above expression the allocation of channels andpower loading are strongly coupled variables and thus ajoint optimization approach is needed -e problem P1 is amixed binary integer programming and requires an ex-haustive search Fortunately it has been shown by [33] thatfor a sufficiently large number of subchannels the gap

between the dual solution and the primal solution reduce tozero regardless of the nonconvexity of the original problem-us we exploit the duality theory to decompose and op-timally solve the coupled problem

To convert the complex max-min problem into standardoptimization form we introduce intermediate variable tsuch that

tle log2 1 + SINRacb1113872 1113873 foralla c b (7)

-e introduction of the intermediate variable transformsthe problem as

P2 maxPacbt

t

st (2) (3) (4) and (6)(8)

To obtain an immediate solution of auxiliary variablesfrom the structure of objective in (8) we utilize the fact thatfor any yge 0 minimizing y is equivalent to minimization ofy2 It is a known fact that maximizing y is equal to mini-mizing minus y Hence to transform the problem into a standardminimization problem we replace the objective in (8) withits negative After making these transformations P2 iswritten as

P3 minPacbt

minus t2

st (2) (3) (4) and (6)

(9)

Now substituting x minus t we obtainminPacbt

x2

st (2) (3) (4) and minus xle log2 1 + SINRacb1113872 1113873 foralla c b

(10)

-e dual function associated with (10) is given by

D λa ηb v( 1113857 minxPacbαacb

1113944

A

a1λa minus x minus 1113944

C

c11113944

B

b1αacblog2 1 + SINRacb1113872 1113873⎛⎝ ⎞⎠

+ x2

+ 1113944B

b1ηb 1113944

A

a11113944

C

c1αacbPacb minus P⎛⎝ ⎞⎠ + v 1113944

A

a11113944

C

c11113944

B

b1αacbPacbhacb minus Ith

⎛⎝ ⎞⎠

st 1113944C

c11113944

B

b1αacb 1 foralla 1 2 3 A

(11)

Secondary user network

Primary user network

Primary BS

UserMBS

PBS

Figure 1 Heterogeneous network using cognitive radio spectrum in underlay mode

4 Wireless Communications and Mobile Computing

-e expression in (11) can be written as

D λa ηb v( 1113857 minxPacbαacb

x2

+ 1113944A

a11113944

C

c11113944

B

b1αacb minus λalog2 1 + SINRacb1113872 1113873 + ηbPa c b + vPacbhacb1113872 1113873

minus 1113944A

a1λax minus 1113944

B

b1ηbPb minus vIth

st 1113944C

c11113944

B

b1αacb 1 foralla 1 2 3 A

(12)

For any given channel allocation dual decompositionguides to solve the following subproblems

P4 minx

x2

minus x 1113944A

a1λa

⎛⎝ ⎞⎠ (13)

P5 minPacb

minus λalog2 1 + SINRacb1113872 1113873 + ηbPacb + vPacbhacb1113872 1113873

(14)

Using the KKT conditions to find the solution of theproblems given in (13) and (14) we get

xlowast

12

1113944

A

a1λa

⎛⎝ ⎞⎠

+

Plowastacb

Φacb minus Pacbprime αacbfacb

gacb ηb + vhacb1113872 1113873⎛⎝ ⎞⎠

+

(15)

where Φacb λagacb + σ2aηb + vhacb for all a c b when(Ψ)+ max(0Ψ) -e detailed derivation steps have beenprovided in the Appendix

Now to find the optimum value of αacb the followingoptimization problem is considered

minαacb

1113944

A

a11113944

C

c11113944

B

b1αacb minus λalog2 1 + SINRacb1113872 1113873 + ηbP

lowastacb1113872

+ vPlowastacbhacb1113873

st 1113944C

c11113944

B

b1αacb 1 foralla 1 2 3 A

(16)

Evidently the optimal solution can be found such that

αlowastacb for a argmin

cminus λalog2 1 + SINRacb1113872 1113873 + ηbPlowastacb + vPlowastacbhacb1113872 1113873

otherwise

⎧⎪⎨

⎪⎩(17)

-e dual problem is convex hence subgradient methodcan be adopted to find the solution -e dual variables areupdated at each iteration as

λ(itr)a λ(itrminus 1)

a + δ(itrminus 1)times minus 1113944

B

b11113944

C

c1log2 1 + SINRacb1113872 1113873 minus x⎛⎝ ⎞⎠⎛⎝ ⎞⎠

η(itr)b η(itrminus 1)

b + δ(itrminus 1)1113944

A

a11113944

C

c1Pacb minus Pb

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

v(itr)

v(itrminus 1)

+ δ(itrminus 1)times 1113944

A

a11113944

C

c11113944

B

b1Pacbhacb minus Ith

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(18)

where δ is the step size To obtain the joint optimizationsolution power loading and channel allocation are updatedin each iteration

4 Simulation Results

In this section we present the performance of the proposedscheme -e simulation environment for the proposedschemes is MATLAB -e noise PSD is σ2a 01 For sim-ulation we have considered the maximum number of usersin the system which is A 30 the total channels available inthe system which is C 20 and the total BS available in thesystem which is B 5 For the sake of the evaluation wehave compared our proposed optimal allocation and optimalpower scheme (OAOP) with baseline schemes ie fixedchannel allocation and optimal power loading (FAOP) andfixed channel allocation and fixed power loading (FAFP)

In Figure 2 peak-to-average rate ratio (PR) has beenplotted against different parameters Different values of PRshow fairness among the users -e smaller values of PRindicate a more fair scheme Moreover the effect ofchanging picocell power PRp on PR has also been shown

Wireless Communications and Mobile Computing 5

-e picocell power has been varied from 025 to 2 Con-sidering Pp 025 the percentage gap between FAFP andFAOP is 4974 between FAFP and OAOP is 2259 andbetween FAOP and OAOP is 4542 It can be seen that thevalue of PR decreases with an increase in the power -isindicates that high picocell power causes high interferencewhich results in decreasing the PR Moreover a comparisonof the three schemes clearly indicates that the OAOP schemeis the fairest among all -is is because optimization onchannel allocation along with optimal power loading makesthe problem more flexible compared with the cases whenonly power loading is optimized and if channel allocationand power loading are fixed Also for Pp 05 the per-centage gap between FAFP and FAOP is 4983 betweenFAFP and OAOP is 2544 and between FAOP and OAOPis 5105 Finally increasing the value to Pp 2 the per-centage gap between FAFP and FAOP is 5699 betweenFAFP and OAOP is 3312 and between FAOP and OAOPis 5811

Figure 3 shows the PR with changing interferenceconstraint PRIth

-e interference constraint has been variedfrom 20 to 40 Considering Ith 20 the percentage gapbetween FAFP and FAOP is 5508 between FAFP andOAOP is 3183 and between FAOP and OAOP is 5779Also for Ith 25 the percentage gap between FAFP andFAOP is 5339 between FAFP and OAOP is 3057 andbetween FAOP and OAOP is 5727 Moreover if Ith 30then the percentage gap between FAFP and FAOP is 5241between FAFP and OAOP is 2995 and between FAOPand OAOP is 5714 Further increasing the value toIth 35 the percentage gap between FAFP and FAOP is5200 between FAFP and OAOP is 2962 and betweenFAOP and OAOP is 5711 Finally increasing the value toIth 40 the percentage gap between FAFP and FAOP is5186 between FAFP and OAOP is 2891 and betweenFAOP and OAOP is 5631 Comparison of schemes interms of the percentage gap shows that as the interferenceconstraint Ith increases the percentage gap is decreasingMoreover the average gap between the graphs of FAFP withFAOP and FAOP with OAOP scheme is near to each otherGraphs also show that OAOP outperforms all other schemesin terms of fairness -e reason behind this is the additionalflexibility provided by optimum channel allocation

-e effect of changing the power of macro base stationPRm on the peak-to-average rate ratio is shown in Figure 4-emacro power has been varied from 10 to 30 ConsideringPRm 10 the percentage gap between FAFP and FAOP is5464 between FAFP and OAOP is 3183 and betweenFAOP and OAOP is 5826 Also for PRm 15 the per-centage gap between FAFP and FAOP is 5605 betweenFAFP and OAOP is 3446 and between FAOP and OAOPis 6148 Moreover if PRm 20 then the percentage gapbetween FAFP and FAOP is 5665 between FAFP andOAOP is 3627 and between FAOP and OAOP is 6403Further increasing the value to PRm 25 the percentage gapbetween FAFP and FAOP is 5709 between FAFP andOAOP is 3759 and between FAOP and OAOP is 6584Finally increasing the value to PRm 30 the percentage gapbetween FAFP and FAOP is 5749 between FAFP andOAOP is 3861 and between FAOP and OAOP is 6715Hence comparison of all the three schemes shows that theOAOP is the superior scheme

In the results presented here the sum throughput hasalso been maximized along with achieving fairness amongthe users Figure 5 shows the sum throughput versus picopower Pp -e Pp has been varied from 025 to 2 -epercentage gap between OAOP scheme and FAOP is 38between OAOP and FAFP is 7 and between FAOP andFAFP is 1842 where the value of Pp 025 -e resultsevidently show that the spectral efficiency of OAOP schemesis the most optimal among the other power allocationschemes -is is because the channels are allocated to usersin such a way that would maximize the total data rate ofusers -en increasing the value of Pp 05 the percentagegap between OAOP and FAOP is 4256 between OAOPand FAFP is 7825 and between FAOP and FAFP is1838 Further increasing the value to Pp 2 the per-centage gap between OAOP and FAOP is 4851 betweenOAOP and FAOP is 1012 and between FAOP and FAFPis 2085

Figure 6 shows the sum throughput versus Ith -epercentage gap between OAOP scheme and FAOP is4598 between OAOP and FAFP is 973 and betweenFAOP and FAFP is 2117 where the value of Ith 20Further increasing the value to Ith 25 the percentage gapbetween OAOP and FAOP is 4609 between OAOP and

FAFPFAOPOAOP

12345678

PRp

16 18 212 140806 104Pp

Figure 2 Peak-to-average ratio with changing picocell power

FAFPFAOPOAOP

115

225

335

445

PRI th

3836 4022 3426 282420 30 32Ith

Figure 3 Peak-to-average ratio with changing Ith

6 Wireless Communications and Mobile Computing

FAFP is 1022 and between FAOP and FAFP is 2217Moreover if Ith 30 percentage gap between OAOP andFAOP is 4621 between OAOP and FAFP is 1062 andbetween FAOP and FAFP is 23 Considering Ith 35 thepercentage gap between OAOP and FAOP is 4632 betweenOAOP and FAFP is 1098 and between FAOP and FAFP is2371 Further increasing the value to Ith 40 the percentagegap between OAOP and FAOP is 4647 between OAOP andFAOP is 1101 and between FAOP and FAFP is 2371Hence the sum throughput of OAOP scheme is maximum

-e effect of varying macro power Pm on the sumthroughput is shown in Figure 7 -e Pm has been variedfrom 10 to 30 -e percentage gap between OAOP schemeand FAOP is 4549 between OAOP and FAFP is 973and between FAOP and FAFP is 2140 where the value ofPm 10 Further increasing the value to Pm 15 the per-centage gap between OAOP and FAOP is 4669 betweenOAOP and FAFP is 965 and between FAOP and FAFP is2067 Moreover if Pm 20 the percentage gap betweenOAOP and FAOP is 4727 between OAOP and FAFP is961 and between FAOP and FAFP is 2032 ConsideringPm 25 the percentage gap between OAOP and FAOP is4774 between OAOP and FAFP is 958 and betweenFAOP and FAFP is 2006 Further increasing the value toPm 30 the percentage gap between OAOP and FAOP is4813 between OAOP and FAOP is 956 and betweenFAOP and FAFP is 1986 -e results show that OAOPscheme is the most optimal

Figure 8 shows the values of sum throughput with achanging number of users U From this plot we identifiedthat the performance of OAOP is far superior to otherschemes which is due to the fact that adding new users tothe system introduces new channel gains -is results inincreasing the flexibility of the problem In OAOP bothchannel allocation and power loading are optimizedtherefore this scheme is better at taking advantage of thenewly added users in the network In FAOP only powerloading is optimized so the addition of new users in-creases the data rate because of the reduced degree offreedom compared with OAOP A very slight increase isobserved in FAFP when new users are added as neitherchannel allocation nor power loading is optimized in thisscheme U has been varied from 20 to 100 ConsideringU 20 the percentage gap between OAOP scheme andFAOP is 6107 between OAOP and FAFP is 1482 andbetween FAOP and FAFP is 2427 Further increasingU 40 the percentage gap between OAOP and FAOP is5328 between OAOP and FAFP is 1099 and betweenFAOP and FAFP is 2063 Moreover if U 60 thepercentage gap between OAOP and FAOP is 4881between OAOP and FAFP is 1079 and between FAOPand FAFP is 2010

Figure 9 shows the convergence of dual variables λa ηand v with initial value as 06 η as 06 and v as 01 -esmaller the value of step size the finer the convergence iswhereas a bigger value leads to fast convergence

115

225

335

445

PRm

26 2818 3022 24201412 1610Pm

FAFPFAOPOAOP

Figure 4 Peak-to-average ratio with changing macro power

0

50

100

150

200

250

Rate

(bps

)

06 08 1 12 1404 18 216Pp

FAFPFAOPOAOP

Figure 5 Sum throughput with changing pico power

050

100150200250300

Rate

(bps

)

3224 26 28 3022 34 36 38 4020Ith

FAFPFAOPOAOP

Figure 6 Sum throughput with changing Ith

050

100150200250300

Rate

(bps

)

10 14 16 18 20 3012 26 2822 24Pm

FAFPFAOPOAOP

Figure 7 Sum throughput with changing macro power

Wireless Communications and Mobile Computing 7

5 Conclusion

-is paper optimized the power allocation and user asso-ciation to achieving fairness among different users in un-derlay CR-based HetNets Specifically we adopted a max-min-based fairness framework under various practicalconstraints Problem was first transformed into a standardmaximization optimization and then dual decomposition isused to solve the integer programming problem -e dualproblem was solved through the subgradient method -epower allocation at each node is obtained from the convexoptimization techniques for the known power allocation atall other nodes -e results are compared with the sub-optimal scenarios when the fixed channel allocation is as-sumed for the optimizednonoptimized power allocation-e results showed that the proposed scheme outperformsall other candidates and the spectral efficiency has beenimproved due to the simultaneous transmission with theprimary network

Appendix

-e Lagrangian L associated with the optimization problemP5 is

minPacb

minus λalog2 1 + SINRacb1113872 1113873 + ηbPacb + vPacbhacb1113872 1113873

(A1)In the above expression we have converted our problem

from a constrained problem to an unconstrained problem

minPacbge0

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A2)

By substituting the value of SINR we get

0100200300400500600700

Rate

(bps

)

20 70 10080 9060504030U

FAFPFAOPOAOP

Figure 8 Sum throughput with increasing users

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

Plot of v from recursion

Plot of λ from recursion

Plot of η from recursion

0055

005

0045

v

055

05

045

λ

η 1

0

Figure 9 Convergence of η λ and v with increasing iterations

8 Wireless Communications and Mobile Computing

z

zPacb

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A3)

Applying the KKT conditions for optimality and bysolving the first derivative test we obtain

minus λa

11 + Pacb + gacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

⎛⎝ ⎞⎠

timesz

zPacb

1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠ + ηb + vhacb

(A4)

Now solving the partial derivative as

ηb + vhacb minus λa

times1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

timesgacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠

(A5)

Taking LCM of previous step and finding the partialderivative of internal function we have

minus λa

gacb 1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

times1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠ + ηb + vhacb

(A6)

Further simplification results in

minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ ηb + vhacb

(A7)

By canceling the common terms and simplifying wehave

minus ηb minus vhacb minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

middot1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

ηb + vhacb

λacbgacb

(A8)

After some mathematical simplifications we get

1113944

A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ Pacbgacb λacbgacb

ηb + vhacb

(A9)

Inverting both sides of the equation as

Pacbgacb λacbgacb

ηb + vhacb

1113888 1113889

minus 1113944A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

(A10)

After cross multiplying we have

Pacb λacbgacb minus Ξηb + vhacb1113872 1113873

ηb + vhacb

(A11)

where Ξ (1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a)

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Wireless Communications and Mobile Computing 9

Acknowledgments

-is research was supported by the Chung-Ang Universityresearch grant in 2017 and the Basic Science ResearchProgram through the National Research Foundation ofKorea funded by the Ministry of Education (NRF-2019R1F1A1042599)

References

[1] F Jameel Z Hamid F Jabeen S Zeadally and M A JavedldquoA survey of device-to-device communications research is-sues and challengesrdquo IEEE Communications Surveys amp Tu-torials vol 20 no 3 pp 2133ndash2168 2018

[2] B Wang F Gao S Jin H Lin and G Y Li ldquoSpatial- andfrequency-wideband effects in millimeter-wave massiveMIMO systemsrdquo IEEE Transactions on Signal Processingvol 66 no 13 pp 3393ndash3406 2018

[3] X Wang E Turgut and M Cenk Gursoy ldquoCoverage indownlink heterogeneous mmWave cellular networks withuser-centric small cell deploymentrdquo 2017 httpsarxivorgabs170707035

[4] Y Xu Y Hu and G Li ldquoRobust rate maximization forheterogeneous wireless networks under channel un-certaintiesrdquo Sensors vol 18 no 2 p 639 2018

[5] Y Xu and S Mao ldquoUser association in massive MIMOHetNetsrdquo IEEE Systems Journal vol 11 no 1 pp 7ndash19 2017

[6] A Khandekar N Bhushan J Tingfang and V Vanghi ldquoLTE-advanced heterogeneous networksrdquo in Proceedings of the2010 EuropeanWireless Conference (EW) pp 978ndash982 LuccaItaly April 2010

[7] F Jameel S Kumar Z Chang T Hamalainan andT Ristaniemi ldquoOperator revenue analysis for device-to-devicecommunications overlaying cellular networkrdquo in Proceedings ofthe 2018 IEEE Conference on Standards for Communicationsand Networking (CSCN) pp 1ndash6 Paris France October 2018

[8] Y Sun W Xia S Zhang Y Wu T Wang and Y FangldquoEnergy efficient pico cell range expansion and density jointoptimization for heterogeneous networks with eICICrdquo Sen-sors vol 18 no 3 p 762 2018

[9] H Zhang H Ji and Y Wang ldquoUser association scheme inheterogeneous networks considering multiple real-worldpoliciesrdquo in Proceedings of the National Doctoral AcademicForum on Information and Communications Technology 2013Beijing China 2013

[10] K Cwalina S Ambroziak P Rajchowski J Sadowski andJ Stefanski ldquoA novel bitrate adaptation method for hetero-geneous wireless body area networksrdquo Applied Sciences vol 8no 7 p 1209 2018

[11] D Fooladivanda and C Rosenberg ldquoJoint resource allocationand user association for heterogeneous wireless cellularnetworksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 1 pp 248ndash257 2013

[12] W U Khan Z Ali M Waqas and G A S Sidhu ldquoEfficientpower allocation with individual QoS guarantees in futuresmall-cell networksrdquoAEU-International Journal of Electronicsand Communications vol 105 pp 36ndash41 2019

[13] K Shen and W Yu ldquoDownlink cell association optimizationfor heterogeneous networks via dual coordinate descentrdquo inProceedings of the 2013 IEEE International Conference onAcoustics Speech and Signal Processing pp 4779ndash4783 IEEEVancouver Canada May 2013

[14] K Shen and W Yu ldquoDistributed pricing-based user associ-ation for downlink heterogeneous cellular networksrdquo IEEE

Journal on Selected Areas in Communications vol 32 no 6pp 1100ndash1113 2014

[15] D Liu Y Chen K K Chai and T Zhang ldquoNash bargainingsolution based user association optimization in HetNetsrdquo inProceedings of the 2014 IEEE 11th Consumer Communicationsand Networking Conference (CCNC) pp 587ndash592 IEEE LasVegas NV USA January 2014

[16] P Xue P Gong J H Park D Park and D K Kim ldquoMax-minfairness based radio resource management in fourth gener-ation heterogeneous networksrdquo in Proceedings of the 2009 9thInternational Symposium on Communications and In-formation Technology pp 208ndash213 IEEE Icheon SouthKorea September 2009

[17] W U Khan F Jameel T Ristaniemi B M Elhalawany andJ Liu ldquoEfficient power allocation for multi-cell uplink NOMAnetworkrdquo in Proceedings of the 2019 IEEE 89th VehicularTechnology Conference (VTC2019-Spring) pp 1ndash5 IEEEKuala Lumpur Malaysia April-May 2019

[18] R Amin J Martin J Deaton L A DaSilva A Hussien andA Eltawil ldquoBalancing spectral efficiency energy consump-tion and fairness in future heterogeneous wireless systemswith reconfigurable devicesrdquo IEEE Journal on Selected Areasin Communications vol 31 no 5 pp 969ndash980 2013

[19] C Qin C Wang D Pan W Wang and Y Zhang ldquoA crosstime slot partial interference alignment scheme in two-cellrelay heterogeneous networksrdquo Applied Sciences vol 9 no 4p 652 2019

[20] T Jabeen Z Ali W U Khan et al ldquoJoint power allocationand link selection for multi-carrier buffer aided relay net-workrdquo Electronics vol 8 no 6 p 686 2019

[21] A Khodmi S B R Chaouch N Agoulmine and Z ChoukairldquoA joint power allocation and user association based on non-cooperative game theory in an heterogeneous ultra-densenetworkrdquo IEEE Access vol 7 pp 111790ndash111800 2019

[22] P T Semov V Poulkov A Mihovska and R Prasad ldquoIn-creasing throughput and fairness for users in heterogeneoussemi coordinated deploymentsrdquo in Proceedings of the 2014IEEE Wireless Communications and Networking ConferenceWorkshops (WCNCW) pp 40ndash45 IEEE Istanbul TurkeyApril 2014

[23] Q Li R Q Hu Y Qian and G Wu ldquoA proportional fairradio resource allocation for heterogeneous cellular networkswith relaysrdquo in Proceedings of the 2012 IEEE Global Com-munications Conference (GLOBECOM) pp 5457ndash5463 IEEEAnaheim CA USA December 2012

[24] J Jiang W Wang M Peng and Y Huang ldquoEnergy efficiencycomparison between orthogonal and co-channel resourceallocation schemes in distributed antenna systemsrdquo in Pro-ceedings of the 2012 IEEE 23rd International Symposium onPersonal Indoor and Mobile Radio Communications-(PIMRC) pp 541ndash545 IEEE Sydney Australia September2012

[25] J Jiang M Peng K Zhang and L Li ldquoEnergy-efficient re-source allocation in heterogeneous network with cross-tierinterference constraintrdquo in Proceedings of the 2013 IEEE 24thInternational Symposium on Personal Indoor and MobileRadio Communications (PIMRC Workshops) pp 168ndash172IEEE London UK September 2013

[26] X Wang M Jia Q Guo I W-H Ho and J Wu ldquoJointpower original bandwidth and detected hole bandwidthallocation for multi-homing heterogeneous networks basedon cognitive radiordquo IEEE Transactions on Vehicular Tech-nology vol 68 no 3 pp 2777ndash2790 2019

10 Wireless Communications and Mobile Computing

fairness under more practical constraints becomes essentialTo the best of our knowledge the resource optimization anduser assignment techniques for fair rate allocation have notbeen jointly investigated in the literature due to the higherlevel of complexity involved in finding the optimal solution

To fill this gap in the literature and provide a compre-hensive solution to the user fairness problem in CR-basedHetNets we provide a joint strategy for power allocation anduser fairness In particular we consider the joint poweroptimization and user association problem in HetNets forachieving fairness among different users We first formulatea joint optimization problem subject to power and in-terference constraints -en to provide a less complex andan efficient solution we design an algorithm from the dualdecomposition strategy -e presented numerical resultsindicate the importance and utility of our scheme incomparison with the baseline methodologies

13 Organization -e remainder of this paper is organizedas follows -e system model and problem formulation arepresented in Section 2 In Section 3 the proposed solutionhas been described while Section 4 discusses the simulationresults Finally Section 5 presents some concluding remarksand future research directions In addition the list of ac-ronyms used throughout this paper has been provided inTable 1

2 System Model and Problem Formulation

-is section describes the considered system model andprovides the steps related to problem formulation

21 System Model A downlink CR transmission is con-sidered where the secondary HetNet system is reusing thespectrum of the primary network in an underlay mode asshown in Figure 1 -e HetNet consists of a single macrocelloverlaid with multiple pico BSs intended to transmit data tomultiple users It is assumed that all the devices are equippedwith a single antenna and experience independent andidentically distributed (iid) Rayleigh fading -e channelbandwidth is distributed among MBS and PBSs in such away that MBS and PBSs are nonorthogonal to each otherwhile each PBS is orthogonal to other PBSs in the HetNet

In this model BSs are denoted as BSb such thatB BSb | b 1 2 3 B where BS1 is modeled as macro

and the remaining BSs act as pico BSs Further there are Ausers and C channels allocated to BS -e signal-to-in-terference-and-noise ratio (SINR) of the ath user from thebth BS at the cth channel is given as in [11]

SINRacb Pacbgacb

Pacbprime facb + σ2a

(1)

where Pacb is the transmit power for the ath user connectedto the bth BS at cth channel αacb denotes the associationvariable which belongs to set 0 1 σ2a is the power spectraldensity (PSD) of the noise while gacb and facb are thechannel gains from intended BS to user and from interferingBS to the user respectively Furthermore Pacb

prime denotes thetransmit power of interfering BS

22 Problem Formulation One of our key objectives in thisarticle is fair maximization of the data rate of each user in thenetwork To facilitate mathematical analysis we introduce abinary variable αacb such that

αacb 1 when the ath user is associatedwith the bth BS through the cth channel

0 otherwise1113896 (2)

Based on the above expression the channel allocationfollows

1113944

C

c11113944

B

b1αacb 1 foralla 1 2 3 A (3)

To protect PUs from the interference and to improve theperformance of the system resource allocation at BSs needto be performed such that

1113944

A

a11113944

C

c11113944

B

b1αacbPacbhacb le Ith (4)

Table 1 List of acronyms

Acronym Definition5G Fifth generationAWGN Additive white Gaussian noiseBS Base stationCR Cognitive radioD2D Device to deviceDAS Distributed antenna systemDSA Dynamic spectrum accessFBS Femto base stationGRC Global resource controllerHetNet Heterogeneous networksKKT KarushndashKuhnndashTuckerMBS Macro base stationNBS Nash bargaining solutionOFDMA Orthogonal frequency division multiple accessPBS Pico base stationPSD Power spectral densityPU Primary userQoS Quality of serviceRRM Radio resource managementSQP Sequential quadratic programmingSU Secondary userSINR Signal-to-interference plus noise ratio

Wireless Communications and Mobile Computing 3

where Pacb is the power allocated by the bth BS to the cthchannel allocated to the ath user and hacb represents thegain of interference channel c from the bth BS to primary BSFinally to ensure that the power consumed by each BS is lessthan or equal to the power budget power allocation at eachBS needs to be ensured as

1113944A

a11113944

C

c1Pacbαacb lePb forallb 1 2 3 B (5)

To achieve fairness in rates of different users a max-min-based optimization framework is adopted such that theproblem is stated mathematically as

P1 maxPacbαacb

min log2 1 + SINRacb1113872 1113873

st (2) (3) and (4)

(6)

where Pb denotes total power available at the bth BS and Ithrepresents the sum interference threshold Here the firstconstraint ensures that the total power allocated by the bthBS must be within the available power budget Similarly thesecond constraint ensures that the primary system is wellprotected from interference due to the underlay commu-nication and the third constraint makes sure that the cthchannel is allocated to just one SU

3 Proposed Solution

To provide a viable solution to the aforementioned problemwe propose a solution based on duality theory As is evidentfrom the above expression the allocation of channels andpower loading are strongly coupled variables and thus ajoint optimization approach is needed -e problem P1 is amixed binary integer programming and requires an ex-haustive search Fortunately it has been shown by [33] thatfor a sufficiently large number of subchannels the gap

between the dual solution and the primal solution reduce tozero regardless of the nonconvexity of the original problem-us we exploit the duality theory to decompose and op-timally solve the coupled problem

To convert the complex max-min problem into standardoptimization form we introduce intermediate variable tsuch that

tle log2 1 + SINRacb1113872 1113873 foralla c b (7)

-e introduction of the intermediate variable transformsthe problem as

P2 maxPacbt

t

st (2) (3) (4) and (6)(8)

To obtain an immediate solution of auxiliary variablesfrom the structure of objective in (8) we utilize the fact thatfor any yge 0 minimizing y is equivalent to minimization ofy2 It is a known fact that maximizing y is equal to mini-mizing minus y Hence to transform the problem into a standardminimization problem we replace the objective in (8) withits negative After making these transformations P2 iswritten as

P3 minPacbt

minus t2

st (2) (3) (4) and (6)

(9)

Now substituting x minus t we obtainminPacbt

x2

st (2) (3) (4) and minus xle log2 1 + SINRacb1113872 1113873 foralla c b

(10)

-e dual function associated with (10) is given by

D λa ηb v( 1113857 minxPacbαacb

1113944

A

a1λa minus x minus 1113944

C

c11113944

B

b1αacblog2 1 + SINRacb1113872 1113873⎛⎝ ⎞⎠

+ x2

+ 1113944B

b1ηb 1113944

A

a11113944

C

c1αacbPacb minus P⎛⎝ ⎞⎠ + v 1113944

A

a11113944

C

c11113944

B

b1αacbPacbhacb minus Ith

⎛⎝ ⎞⎠

st 1113944C

c11113944

B

b1αacb 1 foralla 1 2 3 A

(11)

Secondary user network

Primary user network

Primary BS

UserMBS

PBS

Figure 1 Heterogeneous network using cognitive radio spectrum in underlay mode

4 Wireless Communications and Mobile Computing

-e expression in (11) can be written as

D λa ηb v( 1113857 minxPacbαacb

x2

+ 1113944A

a11113944

C

c11113944

B

b1αacb minus λalog2 1 + SINRacb1113872 1113873 + ηbPa c b + vPacbhacb1113872 1113873

minus 1113944A

a1λax minus 1113944

B

b1ηbPb minus vIth

st 1113944C

c11113944

B

b1αacb 1 foralla 1 2 3 A

(12)

For any given channel allocation dual decompositionguides to solve the following subproblems

P4 minx

x2

minus x 1113944A

a1λa

⎛⎝ ⎞⎠ (13)

P5 minPacb

minus λalog2 1 + SINRacb1113872 1113873 + ηbPacb + vPacbhacb1113872 1113873

(14)

Using the KKT conditions to find the solution of theproblems given in (13) and (14) we get

xlowast

12

1113944

A

a1λa

⎛⎝ ⎞⎠

+

Plowastacb

Φacb minus Pacbprime αacbfacb

gacb ηb + vhacb1113872 1113873⎛⎝ ⎞⎠

+

(15)

where Φacb λagacb + σ2aηb + vhacb for all a c b when(Ψ)+ max(0Ψ) -e detailed derivation steps have beenprovided in the Appendix

Now to find the optimum value of αacb the followingoptimization problem is considered

minαacb

1113944

A

a11113944

C

c11113944

B

b1αacb minus λalog2 1 + SINRacb1113872 1113873 + ηbP

lowastacb1113872

+ vPlowastacbhacb1113873

st 1113944C

c11113944

B

b1αacb 1 foralla 1 2 3 A

(16)

Evidently the optimal solution can be found such that

αlowastacb for a argmin

cminus λalog2 1 + SINRacb1113872 1113873 + ηbPlowastacb + vPlowastacbhacb1113872 1113873

otherwise

⎧⎪⎨

⎪⎩(17)

-e dual problem is convex hence subgradient methodcan be adopted to find the solution -e dual variables areupdated at each iteration as

λ(itr)a λ(itrminus 1)

a + δ(itrminus 1)times minus 1113944

B

b11113944

C

c1log2 1 + SINRacb1113872 1113873 minus x⎛⎝ ⎞⎠⎛⎝ ⎞⎠

η(itr)b η(itrminus 1)

b + δ(itrminus 1)1113944

A

a11113944

C

c1Pacb minus Pb

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

v(itr)

v(itrminus 1)

+ δ(itrminus 1)times 1113944

A

a11113944

C

c11113944

B

b1Pacbhacb minus Ith

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(18)

where δ is the step size To obtain the joint optimizationsolution power loading and channel allocation are updatedin each iteration

4 Simulation Results

In this section we present the performance of the proposedscheme -e simulation environment for the proposedschemes is MATLAB -e noise PSD is σ2a 01 For sim-ulation we have considered the maximum number of usersin the system which is A 30 the total channels available inthe system which is C 20 and the total BS available in thesystem which is B 5 For the sake of the evaluation wehave compared our proposed optimal allocation and optimalpower scheme (OAOP) with baseline schemes ie fixedchannel allocation and optimal power loading (FAOP) andfixed channel allocation and fixed power loading (FAFP)

In Figure 2 peak-to-average rate ratio (PR) has beenplotted against different parameters Different values of PRshow fairness among the users -e smaller values of PRindicate a more fair scheme Moreover the effect ofchanging picocell power PRp on PR has also been shown

Wireless Communications and Mobile Computing 5

-e picocell power has been varied from 025 to 2 Con-sidering Pp 025 the percentage gap between FAFP andFAOP is 4974 between FAFP and OAOP is 2259 andbetween FAOP and OAOP is 4542 It can be seen that thevalue of PR decreases with an increase in the power -isindicates that high picocell power causes high interferencewhich results in decreasing the PR Moreover a comparisonof the three schemes clearly indicates that the OAOP schemeis the fairest among all -is is because optimization onchannel allocation along with optimal power loading makesthe problem more flexible compared with the cases whenonly power loading is optimized and if channel allocationand power loading are fixed Also for Pp 05 the per-centage gap between FAFP and FAOP is 4983 betweenFAFP and OAOP is 2544 and between FAOP and OAOPis 5105 Finally increasing the value to Pp 2 the per-centage gap between FAFP and FAOP is 5699 betweenFAFP and OAOP is 3312 and between FAOP and OAOPis 5811

Figure 3 shows the PR with changing interferenceconstraint PRIth

-e interference constraint has been variedfrom 20 to 40 Considering Ith 20 the percentage gapbetween FAFP and FAOP is 5508 between FAFP andOAOP is 3183 and between FAOP and OAOP is 5779Also for Ith 25 the percentage gap between FAFP andFAOP is 5339 between FAFP and OAOP is 3057 andbetween FAOP and OAOP is 5727 Moreover if Ith 30then the percentage gap between FAFP and FAOP is 5241between FAFP and OAOP is 2995 and between FAOPand OAOP is 5714 Further increasing the value toIth 35 the percentage gap between FAFP and FAOP is5200 between FAFP and OAOP is 2962 and betweenFAOP and OAOP is 5711 Finally increasing the value toIth 40 the percentage gap between FAFP and FAOP is5186 between FAFP and OAOP is 2891 and betweenFAOP and OAOP is 5631 Comparison of schemes interms of the percentage gap shows that as the interferenceconstraint Ith increases the percentage gap is decreasingMoreover the average gap between the graphs of FAFP withFAOP and FAOP with OAOP scheme is near to each otherGraphs also show that OAOP outperforms all other schemesin terms of fairness -e reason behind this is the additionalflexibility provided by optimum channel allocation

-e effect of changing the power of macro base stationPRm on the peak-to-average rate ratio is shown in Figure 4-emacro power has been varied from 10 to 30 ConsideringPRm 10 the percentage gap between FAFP and FAOP is5464 between FAFP and OAOP is 3183 and betweenFAOP and OAOP is 5826 Also for PRm 15 the per-centage gap between FAFP and FAOP is 5605 betweenFAFP and OAOP is 3446 and between FAOP and OAOPis 6148 Moreover if PRm 20 then the percentage gapbetween FAFP and FAOP is 5665 between FAFP andOAOP is 3627 and between FAOP and OAOP is 6403Further increasing the value to PRm 25 the percentage gapbetween FAFP and FAOP is 5709 between FAFP andOAOP is 3759 and between FAOP and OAOP is 6584Finally increasing the value to PRm 30 the percentage gapbetween FAFP and FAOP is 5749 between FAFP andOAOP is 3861 and between FAOP and OAOP is 6715Hence comparison of all the three schemes shows that theOAOP is the superior scheme

In the results presented here the sum throughput hasalso been maximized along with achieving fairness amongthe users Figure 5 shows the sum throughput versus picopower Pp -e Pp has been varied from 025 to 2 -epercentage gap between OAOP scheme and FAOP is 38between OAOP and FAFP is 7 and between FAOP andFAFP is 1842 where the value of Pp 025 -e resultsevidently show that the spectral efficiency of OAOP schemesis the most optimal among the other power allocationschemes -is is because the channels are allocated to usersin such a way that would maximize the total data rate ofusers -en increasing the value of Pp 05 the percentagegap between OAOP and FAOP is 4256 between OAOPand FAFP is 7825 and between FAOP and FAFP is1838 Further increasing the value to Pp 2 the per-centage gap between OAOP and FAOP is 4851 betweenOAOP and FAOP is 1012 and between FAOP and FAFPis 2085

Figure 6 shows the sum throughput versus Ith -epercentage gap between OAOP scheme and FAOP is4598 between OAOP and FAFP is 973 and betweenFAOP and FAFP is 2117 where the value of Ith 20Further increasing the value to Ith 25 the percentage gapbetween OAOP and FAOP is 4609 between OAOP and

FAFPFAOPOAOP

12345678

PRp

16 18 212 140806 104Pp

Figure 2 Peak-to-average ratio with changing picocell power

FAFPFAOPOAOP

115

225

335

445

PRI th

3836 4022 3426 282420 30 32Ith

Figure 3 Peak-to-average ratio with changing Ith

6 Wireless Communications and Mobile Computing

FAFP is 1022 and between FAOP and FAFP is 2217Moreover if Ith 30 percentage gap between OAOP andFAOP is 4621 between OAOP and FAFP is 1062 andbetween FAOP and FAFP is 23 Considering Ith 35 thepercentage gap between OAOP and FAOP is 4632 betweenOAOP and FAFP is 1098 and between FAOP and FAFP is2371 Further increasing the value to Ith 40 the percentagegap between OAOP and FAOP is 4647 between OAOP andFAOP is 1101 and between FAOP and FAFP is 2371Hence the sum throughput of OAOP scheme is maximum

-e effect of varying macro power Pm on the sumthroughput is shown in Figure 7 -e Pm has been variedfrom 10 to 30 -e percentage gap between OAOP schemeand FAOP is 4549 between OAOP and FAFP is 973and between FAOP and FAFP is 2140 where the value ofPm 10 Further increasing the value to Pm 15 the per-centage gap between OAOP and FAOP is 4669 betweenOAOP and FAFP is 965 and between FAOP and FAFP is2067 Moreover if Pm 20 the percentage gap betweenOAOP and FAOP is 4727 between OAOP and FAFP is961 and between FAOP and FAFP is 2032 ConsideringPm 25 the percentage gap between OAOP and FAOP is4774 between OAOP and FAFP is 958 and betweenFAOP and FAFP is 2006 Further increasing the value toPm 30 the percentage gap between OAOP and FAOP is4813 between OAOP and FAOP is 956 and betweenFAOP and FAFP is 1986 -e results show that OAOPscheme is the most optimal

Figure 8 shows the values of sum throughput with achanging number of users U From this plot we identifiedthat the performance of OAOP is far superior to otherschemes which is due to the fact that adding new users tothe system introduces new channel gains -is results inincreasing the flexibility of the problem In OAOP bothchannel allocation and power loading are optimizedtherefore this scheme is better at taking advantage of thenewly added users in the network In FAOP only powerloading is optimized so the addition of new users in-creases the data rate because of the reduced degree offreedom compared with OAOP A very slight increase isobserved in FAFP when new users are added as neitherchannel allocation nor power loading is optimized in thisscheme U has been varied from 20 to 100 ConsideringU 20 the percentage gap between OAOP scheme andFAOP is 6107 between OAOP and FAFP is 1482 andbetween FAOP and FAFP is 2427 Further increasingU 40 the percentage gap between OAOP and FAOP is5328 between OAOP and FAFP is 1099 and betweenFAOP and FAFP is 2063 Moreover if U 60 thepercentage gap between OAOP and FAOP is 4881between OAOP and FAFP is 1079 and between FAOPand FAFP is 2010

Figure 9 shows the convergence of dual variables λa ηand v with initial value as 06 η as 06 and v as 01 -esmaller the value of step size the finer the convergence iswhereas a bigger value leads to fast convergence

115

225

335

445

PRm

26 2818 3022 24201412 1610Pm

FAFPFAOPOAOP

Figure 4 Peak-to-average ratio with changing macro power

0

50

100

150

200

250

Rate

(bps

)

06 08 1 12 1404 18 216Pp

FAFPFAOPOAOP

Figure 5 Sum throughput with changing pico power

050

100150200250300

Rate

(bps

)

3224 26 28 3022 34 36 38 4020Ith

FAFPFAOPOAOP

Figure 6 Sum throughput with changing Ith

050

100150200250300

Rate

(bps

)

10 14 16 18 20 3012 26 2822 24Pm

FAFPFAOPOAOP

Figure 7 Sum throughput with changing macro power

Wireless Communications and Mobile Computing 7

5 Conclusion

-is paper optimized the power allocation and user asso-ciation to achieving fairness among different users in un-derlay CR-based HetNets Specifically we adopted a max-min-based fairness framework under various practicalconstraints Problem was first transformed into a standardmaximization optimization and then dual decomposition isused to solve the integer programming problem -e dualproblem was solved through the subgradient method -epower allocation at each node is obtained from the convexoptimization techniques for the known power allocation atall other nodes -e results are compared with the sub-optimal scenarios when the fixed channel allocation is as-sumed for the optimizednonoptimized power allocation-e results showed that the proposed scheme outperformsall other candidates and the spectral efficiency has beenimproved due to the simultaneous transmission with theprimary network

Appendix

-e Lagrangian L associated with the optimization problemP5 is

minPacb

minus λalog2 1 + SINRacb1113872 1113873 + ηbPacb + vPacbhacb1113872 1113873

(A1)In the above expression we have converted our problem

from a constrained problem to an unconstrained problem

minPacbge0

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A2)

By substituting the value of SINR we get

0100200300400500600700

Rate

(bps

)

20 70 10080 9060504030U

FAFPFAOPOAOP

Figure 8 Sum throughput with increasing users

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

Plot of v from recursion

Plot of λ from recursion

Plot of η from recursion

0055

005

0045

v

055

05

045

λ

η 1

0

Figure 9 Convergence of η λ and v with increasing iterations

8 Wireless Communications and Mobile Computing

z

zPacb

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A3)

Applying the KKT conditions for optimality and bysolving the first derivative test we obtain

minus λa

11 + Pacb + gacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

⎛⎝ ⎞⎠

timesz

zPacb

1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠ + ηb + vhacb

(A4)

Now solving the partial derivative as

ηb + vhacb minus λa

times1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

timesgacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠

(A5)

Taking LCM of previous step and finding the partialderivative of internal function we have

minus λa

gacb 1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

times1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠ + ηb + vhacb

(A6)

Further simplification results in

minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ ηb + vhacb

(A7)

By canceling the common terms and simplifying wehave

minus ηb minus vhacb minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

middot1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

ηb + vhacb

λacbgacb

(A8)

After some mathematical simplifications we get

1113944

A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ Pacbgacb λacbgacb

ηb + vhacb

(A9)

Inverting both sides of the equation as

Pacbgacb λacbgacb

ηb + vhacb

1113888 1113889

minus 1113944A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

(A10)

After cross multiplying we have

Pacb λacbgacb minus Ξηb + vhacb1113872 1113873

ηb + vhacb

(A11)

where Ξ (1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a)

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Wireless Communications and Mobile Computing 9

Acknowledgments

-is research was supported by the Chung-Ang Universityresearch grant in 2017 and the Basic Science ResearchProgram through the National Research Foundation ofKorea funded by the Ministry of Education (NRF-2019R1F1A1042599)

References

[1] F Jameel Z Hamid F Jabeen S Zeadally and M A JavedldquoA survey of device-to-device communications research is-sues and challengesrdquo IEEE Communications Surveys amp Tu-torials vol 20 no 3 pp 2133ndash2168 2018

[2] B Wang F Gao S Jin H Lin and G Y Li ldquoSpatial- andfrequency-wideband effects in millimeter-wave massiveMIMO systemsrdquo IEEE Transactions on Signal Processingvol 66 no 13 pp 3393ndash3406 2018

[3] X Wang E Turgut and M Cenk Gursoy ldquoCoverage indownlink heterogeneous mmWave cellular networks withuser-centric small cell deploymentrdquo 2017 httpsarxivorgabs170707035

[4] Y Xu Y Hu and G Li ldquoRobust rate maximization forheterogeneous wireless networks under channel un-certaintiesrdquo Sensors vol 18 no 2 p 639 2018

[5] Y Xu and S Mao ldquoUser association in massive MIMOHetNetsrdquo IEEE Systems Journal vol 11 no 1 pp 7ndash19 2017

[6] A Khandekar N Bhushan J Tingfang and V Vanghi ldquoLTE-advanced heterogeneous networksrdquo in Proceedings of the2010 EuropeanWireless Conference (EW) pp 978ndash982 LuccaItaly April 2010

[7] F Jameel S Kumar Z Chang T Hamalainan andT Ristaniemi ldquoOperator revenue analysis for device-to-devicecommunications overlaying cellular networkrdquo in Proceedings ofthe 2018 IEEE Conference on Standards for Communicationsand Networking (CSCN) pp 1ndash6 Paris France October 2018

[8] Y Sun W Xia S Zhang Y Wu T Wang and Y FangldquoEnergy efficient pico cell range expansion and density jointoptimization for heterogeneous networks with eICICrdquo Sen-sors vol 18 no 3 p 762 2018

[9] H Zhang H Ji and Y Wang ldquoUser association scheme inheterogeneous networks considering multiple real-worldpoliciesrdquo in Proceedings of the National Doctoral AcademicForum on Information and Communications Technology 2013Beijing China 2013

[10] K Cwalina S Ambroziak P Rajchowski J Sadowski andJ Stefanski ldquoA novel bitrate adaptation method for hetero-geneous wireless body area networksrdquo Applied Sciences vol 8no 7 p 1209 2018

[11] D Fooladivanda and C Rosenberg ldquoJoint resource allocationand user association for heterogeneous wireless cellularnetworksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 1 pp 248ndash257 2013

[12] W U Khan Z Ali M Waqas and G A S Sidhu ldquoEfficientpower allocation with individual QoS guarantees in futuresmall-cell networksrdquoAEU-International Journal of Electronicsand Communications vol 105 pp 36ndash41 2019

[13] K Shen and W Yu ldquoDownlink cell association optimizationfor heterogeneous networks via dual coordinate descentrdquo inProceedings of the 2013 IEEE International Conference onAcoustics Speech and Signal Processing pp 4779ndash4783 IEEEVancouver Canada May 2013

[14] K Shen and W Yu ldquoDistributed pricing-based user associ-ation for downlink heterogeneous cellular networksrdquo IEEE

Journal on Selected Areas in Communications vol 32 no 6pp 1100ndash1113 2014

[15] D Liu Y Chen K K Chai and T Zhang ldquoNash bargainingsolution based user association optimization in HetNetsrdquo inProceedings of the 2014 IEEE 11th Consumer Communicationsand Networking Conference (CCNC) pp 587ndash592 IEEE LasVegas NV USA January 2014

[16] P Xue P Gong J H Park D Park and D K Kim ldquoMax-minfairness based radio resource management in fourth gener-ation heterogeneous networksrdquo in Proceedings of the 2009 9thInternational Symposium on Communications and In-formation Technology pp 208ndash213 IEEE Icheon SouthKorea September 2009

[17] W U Khan F Jameel T Ristaniemi B M Elhalawany andJ Liu ldquoEfficient power allocation for multi-cell uplink NOMAnetworkrdquo in Proceedings of the 2019 IEEE 89th VehicularTechnology Conference (VTC2019-Spring) pp 1ndash5 IEEEKuala Lumpur Malaysia April-May 2019

[18] R Amin J Martin J Deaton L A DaSilva A Hussien andA Eltawil ldquoBalancing spectral efficiency energy consump-tion and fairness in future heterogeneous wireless systemswith reconfigurable devicesrdquo IEEE Journal on Selected Areasin Communications vol 31 no 5 pp 969ndash980 2013

[19] C Qin C Wang D Pan W Wang and Y Zhang ldquoA crosstime slot partial interference alignment scheme in two-cellrelay heterogeneous networksrdquo Applied Sciences vol 9 no 4p 652 2019

[20] T Jabeen Z Ali W U Khan et al ldquoJoint power allocationand link selection for multi-carrier buffer aided relay net-workrdquo Electronics vol 8 no 6 p 686 2019

[21] A Khodmi S B R Chaouch N Agoulmine and Z ChoukairldquoA joint power allocation and user association based on non-cooperative game theory in an heterogeneous ultra-densenetworkrdquo IEEE Access vol 7 pp 111790ndash111800 2019

[22] P T Semov V Poulkov A Mihovska and R Prasad ldquoIn-creasing throughput and fairness for users in heterogeneoussemi coordinated deploymentsrdquo in Proceedings of the 2014IEEE Wireless Communications and Networking ConferenceWorkshops (WCNCW) pp 40ndash45 IEEE Istanbul TurkeyApril 2014

[23] Q Li R Q Hu Y Qian and G Wu ldquoA proportional fairradio resource allocation for heterogeneous cellular networkswith relaysrdquo in Proceedings of the 2012 IEEE Global Com-munications Conference (GLOBECOM) pp 5457ndash5463 IEEEAnaheim CA USA December 2012

[24] J Jiang W Wang M Peng and Y Huang ldquoEnergy efficiencycomparison between orthogonal and co-channel resourceallocation schemes in distributed antenna systemsrdquo in Pro-ceedings of the 2012 IEEE 23rd International Symposium onPersonal Indoor and Mobile Radio Communications-(PIMRC) pp 541ndash545 IEEE Sydney Australia September2012

[25] J Jiang M Peng K Zhang and L Li ldquoEnergy-efficient re-source allocation in heterogeneous network with cross-tierinterference constraintrdquo in Proceedings of the 2013 IEEE 24thInternational Symposium on Personal Indoor and MobileRadio Communications (PIMRC Workshops) pp 168ndash172IEEE London UK September 2013

[26] X Wang M Jia Q Guo I W-H Ho and J Wu ldquoJointpower original bandwidth and detected hole bandwidthallocation for multi-homing heterogeneous networks basedon cognitive radiordquo IEEE Transactions on Vehicular Tech-nology vol 68 no 3 pp 2777ndash2790 2019

10 Wireless Communications and Mobile Computing

where Pacb is the power allocated by the bth BS to the cthchannel allocated to the ath user and hacb represents thegain of interference channel c from the bth BS to primary BSFinally to ensure that the power consumed by each BS is lessthan or equal to the power budget power allocation at eachBS needs to be ensured as

1113944A

a11113944

C

c1Pacbαacb lePb forallb 1 2 3 B (5)

To achieve fairness in rates of different users a max-min-based optimization framework is adopted such that theproblem is stated mathematically as

P1 maxPacbαacb

min log2 1 + SINRacb1113872 1113873

st (2) (3) and (4)

(6)

where Pb denotes total power available at the bth BS and Ithrepresents the sum interference threshold Here the firstconstraint ensures that the total power allocated by the bthBS must be within the available power budget Similarly thesecond constraint ensures that the primary system is wellprotected from interference due to the underlay commu-nication and the third constraint makes sure that the cthchannel is allocated to just one SU

3 Proposed Solution

To provide a viable solution to the aforementioned problemwe propose a solution based on duality theory As is evidentfrom the above expression the allocation of channels andpower loading are strongly coupled variables and thus ajoint optimization approach is needed -e problem P1 is amixed binary integer programming and requires an ex-haustive search Fortunately it has been shown by [33] thatfor a sufficiently large number of subchannels the gap

between the dual solution and the primal solution reduce tozero regardless of the nonconvexity of the original problem-us we exploit the duality theory to decompose and op-timally solve the coupled problem

To convert the complex max-min problem into standardoptimization form we introduce intermediate variable tsuch that

tle log2 1 + SINRacb1113872 1113873 foralla c b (7)

-e introduction of the intermediate variable transformsthe problem as

P2 maxPacbt

t

st (2) (3) (4) and (6)(8)

To obtain an immediate solution of auxiliary variablesfrom the structure of objective in (8) we utilize the fact thatfor any yge 0 minimizing y is equivalent to minimization ofy2 It is a known fact that maximizing y is equal to mini-mizing minus y Hence to transform the problem into a standardminimization problem we replace the objective in (8) withits negative After making these transformations P2 iswritten as

P3 minPacbt

minus t2

st (2) (3) (4) and (6)

(9)

Now substituting x minus t we obtainminPacbt

x2

st (2) (3) (4) and minus xle log2 1 + SINRacb1113872 1113873 foralla c b

(10)

-e dual function associated with (10) is given by

D λa ηb v( 1113857 minxPacbαacb

1113944

A

a1λa minus x minus 1113944

C

c11113944

B

b1αacblog2 1 + SINRacb1113872 1113873⎛⎝ ⎞⎠

+ x2

+ 1113944B

b1ηb 1113944

A

a11113944

C

c1αacbPacb minus P⎛⎝ ⎞⎠ + v 1113944

A

a11113944

C

c11113944

B

b1αacbPacbhacb minus Ith

⎛⎝ ⎞⎠

st 1113944C

c11113944

B

b1αacb 1 foralla 1 2 3 A

(11)

Secondary user network

Primary user network

Primary BS

UserMBS

PBS

Figure 1 Heterogeneous network using cognitive radio spectrum in underlay mode

4 Wireless Communications and Mobile Computing

-e expression in (11) can be written as

D λa ηb v( 1113857 minxPacbαacb

x2

+ 1113944A

a11113944

C

c11113944

B

b1αacb minus λalog2 1 + SINRacb1113872 1113873 + ηbPa c b + vPacbhacb1113872 1113873

minus 1113944A

a1λax minus 1113944

B

b1ηbPb minus vIth

st 1113944C

c11113944

B

b1αacb 1 foralla 1 2 3 A

(12)

For any given channel allocation dual decompositionguides to solve the following subproblems

P4 minx

x2

minus x 1113944A

a1λa

⎛⎝ ⎞⎠ (13)

P5 minPacb

minus λalog2 1 + SINRacb1113872 1113873 + ηbPacb + vPacbhacb1113872 1113873

(14)

Using the KKT conditions to find the solution of theproblems given in (13) and (14) we get

xlowast

12

1113944

A

a1λa

⎛⎝ ⎞⎠

+

Plowastacb

Φacb minus Pacbprime αacbfacb

gacb ηb + vhacb1113872 1113873⎛⎝ ⎞⎠

+

(15)

where Φacb λagacb + σ2aηb + vhacb for all a c b when(Ψ)+ max(0Ψ) -e detailed derivation steps have beenprovided in the Appendix

Now to find the optimum value of αacb the followingoptimization problem is considered

minαacb

1113944

A

a11113944

C

c11113944

B

b1αacb minus λalog2 1 + SINRacb1113872 1113873 + ηbP

lowastacb1113872

+ vPlowastacbhacb1113873

st 1113944C

c11113944

B

b1αacb 1 foralla 1 2 3 A

(16)

Evidently the optimal solution can be found such that

αlowastacb for a argmin

cminus λalog2 1 + SINRacb1113872 1113873 + ηbPlowastacb + vPlowastacbhacb1113872 1113873

otherwise

⎧⎪⎨

⎪⎩(17)

-e dual problem is convex hence subgradient methodcan be adopted to find the solution -e dual variables areupdated at each iteration as

λ(itr)a λ(itrminus 1)

a + δ(itrminus 1)times minus 1113944

B

b11113944

C

c1log2 1 + SINRacb1113872 1113873 minus x⎛⎝ ⎞⎠⎛⎝ ⎞⎠

η(itr)b η(itrminus 1)

b + δ(itrminus 1)1113944

A

a11113944

C

c1Pacb minus Pb

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

v(itr)

v(itrminus 1)

+ δ(itrminus 1)times 1113944

A

a11113944

C

c11113944

B

b1Pacbhacb minus Ith

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(18)

where δ is the step size To obtain the joint optimizationsolution power loading and channel allocation are updatedin each iteration

4 Simulation Results

In this section we present the performance of the proposedscheme -e simulation environment for the proposedschemes is MATLAB -e noise PSD is σ2a 01 For sim-ulation we have considered the maximum number of usersin the system which is A 30 the total channels available inthe system which is C 20 and the total BS available in thesystem which is B 5 For the sake of the evaluation wehave compared our proposed optimal allocation and optimalpower scheme (OAOP) with baseline schemes ie fixedchannel allocation and optimal power loading (FAOP) andfixed channel allocation and fixed power loading (FAFP)

In Figure 2 peak-to-average rate ratio (PR) has beenplotted against different parameters Different values of PRshow fairness among the users -e smaller values of PRindicate a more fair scheme Moreover the effect ofchanging picocell power PRp on PR has also been shown

Wireless Communications and Mobile Computing 5

-e picocell power has been varied from 025 to 2 Con-sidering Pp 025 the percentage gap between FAFP andFAOP is 4974 between FAFP and OAOP is 2259 andbetween FAOP and OAOP is 4542 It can be seen that thevalue of PR decreases with an increase in the power -isindicates that high picocell power causes high interferencewhich results in decreasing the PR Moreover a comparisonof the three schemes clearly indicates that the OAOP schemeis the fairest among all -is is because optimization onchannel allocation along with optimal power loading makesthe problem more flexible compared with the cases whenonly power loading is optimized and if channel allocationand power loading are fixed Also for Pp 05 the per-centage gap between FAFP and FAOP is 4983 betweenFAFP and OAOP is 2544 and between FAOP and OAOPis 5105 Finally increasing the value to Pp 2 the per-centage gap between FAFP and FAOP is 5699 betweenFAFP and OAOP is 3312 and between FAOP and OAOPis 5811

Figure 3 shows the PR with changing interferenceconstraint PRIth

-e interference constraint has been variedfrom 20 to 40 Considering Ith 20 the percentage gapbetween FAFP and FAOP is 5508 between FAFP andOAOP is 3183 and between FAOP and OAOP is 5779Also for Ith 25 the percentage gap between FAFP andFAOP is 5339 between FAFP and OAOP is 3057 andbetween FAOP and OAOP is 5727 Moreover if Ith 30then the percentage gap between FAFP and FAOP is 5241between FAFP and OAOP is 2995 and between FAOPand OAOP is 5714 Further increasing the value toIth 35 the percentage gap between FAFP and FAOP is5200 between FAFP and OAOP is 2962 and betweenFAOP and OAOP is 5711 Finally increasing the value toIth 40 the percentage gap between FAFP and FAOP is5186 between FAFP and OAOP is 2891 and betweenFAOP and OAOP is 5631 Comparison of schemes interms of the percentage gap shows that as the interferenceconstraint Ith increases the percentage gap is decreasingMoreover the average gap between the graphs of FAFP withFAOP and FAOP with OAOP scheme is near to each otherGraphs also show that OAOP outperforms all other schemesin terms of fairness -e reason behind this is the additionalflexibility provided by optimum channel allocation

-e effect of changing the power of macro base stationPRm on the peak-to-average rate ratio is shown in Figure 4-emacro power has been varied from 10 to 30 ConsideringPRm 10 the percentage gap between FAFP and FAOP is5464 between FAFP and OAOP is 3183 and betweenFAOP and OAOP is 5826 Also for PRm 15 the per-centage gap between FAFP and FAOP is 5605 betweenFAFP and OAOP is 3446 and between FAOP and OAOPis 6148 Moreover if PRm 20 then the percentage gapbetween FAFP and FAOP is 5665 between FAFP andOAOP is 3627 and between FAOP and OAOP is 6403Further increasing the value to PRm 25 the percentage gapbetween FAFP and FAOP is 5709 between FAFP andOAOP is 3759 and between FAOP and OAOP is 6584Finally increasing the value to PRm 30 the percentage gapbetween FAFP and FAOP is 5749 between FAFP andOAOP is 3861 and between FAOP and OAOP is 6715Hence comparison of all the three schemes shows that theOAOP is the superior scheme

In the results presented here the sum throughput hasalso been maximized along with achieving fairness amongthe users Figure 5 shows the sum throughput versus picopower Pp -e Pp has been varied from 025 to 2 -epercentage gap between OAOP scheme and FAOP is 38between OAOP and FAFP is 7 and between FAOP andFAFP is 1842 where the value of Pp 025 -e resultsevidently show that the spectral efficiency of OAOP schemesis the most optimal among the other power allocationschemes -is is because the channels are allocated to usersin such a way that would maximize the total data rate ofusers -en increasing the value of Pp 05 the percentagegap between OAOP and FAOP is 4256 between OAOPand FAFP is 7825 and between FAOP and FAFP is1838 Further increasing the value to Pp 2 the per-centage gap between OAOP and FAOP is 4851 betweenOAOP and FAOP is 1012 and between FAOP and FAFPis 2085

Figure 6 shows the sum throughput versus Ith -epercentage gap between OAOP scheme and FAOP is4598 between OAOP and FAFP is 973 and betweenFAOP and FAFP is 2117 where the value of Ith 20Further increasing the value to Ith 25 the percentage gapbetween OAOP and FAOP is 4609 between OAOP and

FAFPFAOPOAOP

12345678

PRp

16 18 212 140806 104Pp

Figure 2 Peak-to-average ratio with changing picocell power

FAFPFAOPOAOP

115

225

335

445

PRI th

3836 4022 3426 282420 30 32Ith

Figure 3 Peak-to-average ratio with changing Ith

6 Wireless Communications and Mobile Computing

FAFP is 1022 and between FAOP and FAFP is 2217Moreover if Ith 30 percentage gap between OAOP andFAOP is 4621 between OAOP and FAFP is 1062 andbetween FAOP and FAFP is 23 Considering Ith 35 thepercentage gap between OAOP and FAOP is 4632 betweenOAOP and FAFP is 1098 and between FAOP and FAFP is2371 Further increasing the value to Ith 40 the percentagegap between OAOP and FAOP is 4647 between OAOP andFAOP is 1101 and between FAOP and FAFP is 2371Hence the sum throughput of OAOP scheme is maximum

-e effect of varying macro power Pm on the sumthroughput is shown in Figure 7 -e Pm has been variedfrom 10 to 30 -e percentage gap between OAOP schemeand FAOP is 4549 between OAOP and FAFP is 973and between FAOP and FAFP is 2140 where the value ofPm 10 Further increasing the value to Pm 15 the per-centage gap between OAOP and FAOP is 4669 betweenOAOP and FAFP is 965 and between FAOP and FAFP is2067 Moreover if Pm 20 the percentage gap betweenOAOP and FAOP is 4727 between OAOP and FAFP is961 and between FAOP and FAFP is 2032 ConsideringPm 25 the percentage gap between OAOP and FAOP is4774 between OAOP and FAFP is 958 and betweenFAOP and FAFP is 2006 Further increasing the value toPm 30 the percentage gap between OAOP and FAOP is4813 between OAOP and FAOP is 956 and betweenFAOP and FAFP is 1986 -e results show that OAOPscheme is the most optimal

Figure 8 shows the values of sum throughput with achanging number of users U From this plot we identifiedthat the performance of OAOP is far superior to otherschemes which is due to the fact that adding new users tothe system introduces new channel gains -is results inincreasing the flexibility of the problem In OAOP bothchannel allocation and power loading are optimizedtherefore this scheme is better at taking advantage of thenewly added users in the network In FAOP only powerloading is optimized so the addition of new users in-creases the data rate because of the reduced degree offreedom compared with OAOP A very slight increase isobserved in FAFP when new users are added as neitherchannel allocation nor power loading is optimized in thisscheme U has been varied from 20 to 100 ConsideringU 20 the percentage gap between OAOP scheme andFAOP is 6107 between OAOP and FAFP is 1482 andbetween FAOP and FAFP is 2427 Further increasingU 40 the percentage gap between OAOP and FAOP is5328 between OAOP and FAFP is 1099 and betweenFAOP and FAFP is 2063 Moreover if U 60 thepercentage gap between OAOP and FAOP is 4881between OAOP and FAFP is 1079 and between FAOPand FAFP is 2010

Figure 9 shows the convergence of dual variables λa ηand v with initial value as 06 η as 06 and v as 01 -esmaller the value of step size the finer the convergence iswhereas a bigger value leads to fast convergence

115

225

335

445

PRm

26 2818 3022 24201412 1610Pm

FAFPFAOPOAOP

Figure 4 Peak-to-average ratio with changing macro power

0

50

100

150

200

250

Rate

(bps

)

06 08 1 12 1404 18 216Pp

FAFPFAOPOAOP

Figure 5 Sum throughput with changing pico power

050

100150200250300

Rate

(bps

)

3224 26 28 3022 34 36 38 4020Ith

FAFPFAOPOAOP

Figure 6 Sum throughput with changing Ith

050

100150200250300

Rate

(bps

)

10 14 16 18 20 3012 26 2822 24Pm

FAFPFAOPOAOP

Figure 7 Sum throughput with changing macro power

Wireless Communications and Mobile Computing 7

5 Conclusion

-is paper optimized the power allocation and user asso-ciation to achieving fairness among different users in un-derlay CR-based HetNets Specifically we adopted a max-min-based fairness framework under various practicalconstraints Problem was first transformed into a standardmaximization optimization and then dual decomposition isused to solve the integer programming problem -e dualproblem was solved through the subgradient method -epower allocation at each node is obtained from the convexoptimization techniques for the known power allocation atall other nodes -e results are compared with the sub-optimal scenarios when the fixed channel allocation is as-sumed for the optimizednonoptimized power allocation-e results showed that the proposed scheme outperformsall other candidates and the spectral efficiency has beenimproved due to the simultaneous transmission with theprimary network

Appendix

-e Lagrangian L associated with the optimization problemP5 is

minPacb

minus λalog2 1 + SINRacb1113872 1113873 + ηbPacb + vPacbhacb1113872 1113873

(A1)In the above expression we have converted our problem

from a constrained problem to an unconstrained problem

minPacbge0

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A2)

By substituting the value of SINR we get

0100200300400500600700

Rate

(bps

)

20 70 10080 9060504030U

FAFPFAOPOAOP

Figure 8 Sum throughput with increasing users

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

Plot of v from recursion

Plot of λ from recursion

Plot of η from recursion

0055

005

0045

v

055

05

045

λ

η 1

0

Figure 9 Convergence of η λ and v with increasing iterations

8 Wireless Communications and Mobile Computing

z

zPacb

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A3)

Applying the KKT conditions for optimality and bysolving the first derivative test we obtain

minus λa

11 + Pacb + gacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

⎛⎝ ⎞⎠

timesz

zPacb

1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠ + ηb + vhacb

(A4)

Now solving the partial derivative as

ηb + vhacb minus λa

times1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

timesgacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠

(A5)

Taking LCM of previous step and finding the partialderivative of internal function we have

minus λa

gacb 1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

times1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠ + ηb + vhacb

(A6)

Further simplification results in

minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ ηb + vhacb

(A7)

By canceling the common terms and simplifying wehave

minus ηb minus vhacb minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

middot1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

ηb + vhacb

λacbgacb

(A8)

After some mathematical simplifications we get

1113944

A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ Pacbgacb λacbgacb

ηb + vhacb

(A9)

Inverting both sides of the equation as

Pacbgacb λacbgacb

ηb + vhacb

1113888 1113889

minus 1113944A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

(A10)

After cross multiplying we have

Pacb λacbgacb minus Ξηb + vhacb1113872 1113873

ηb + vhacb

(A11)

where Ξ (1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a)

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Wireless Communications and Mobile Computing 9

Acknowledgments

-is research was supported by the Chung-Ang Universityresearch grant in 2017 and the Basic Science ResearchProgram through the National Research Foundation ofKorea funded by the Ministry of Education (NRF-2019R1F1A1042599)

References

[1] F Jameel Z Hamid F Jabeen S Zeadally and M A JavedldquoA survey of device-to-device communications research is-sues and challengesrdquo IEEE Communications Surveys amp Tu-torials vol 20 no 3 pp 2133ndash2168 2018

[2] B Wang F Gao S Jin H Lin and G Y Li ldquoSpatial- andfrequency-wideband effects in millimeter-wave massiveMIMO systemsrdquo IEEE Transactions on Signal Processingvol 66 no 13 pp 3393ndash3406 2018

[3] X Wang E Turgut and M Cenk Gursoy ldquoCoverage indownlink heterogeneous mmWave cellular networks withuser-centric small cell deploymentrdquo 2017 httpsarxivorgabs170707035

[4] Y Xu Y Hu and G Li ldquoRobust rate maximization forheterogeneous wireless networks under channel un-certaintiesrdquo Sensors vol 18 no 2 p 639 2018

[5] Y Xu and S Mao ldquoUser association in massive MIMOHetNetsrdquo IEEE Systems Journal vol 11 no 1 pp 7ndash19 2017

[6] A Khandekar N Bhushan J Tingfang and V Vanghi ldquoLTE-advanced heterogeneous networksrdquo in Proceedings of the2010 EuropeanWireless Conference (EW) pp 978ndash982 LuccaItaly April 2010

[7] F Jameel S Kumar Z Chang T Hamalainan andT Ristaniemi ldquoOperator revenue analysis for device-to-devicecommunications overlaying cellular networkrdquo in Proceedings ofthe 2018 IEEE Conference on Standards for Communicationsand Networking (CSCN) pp 1ndash6 Paris France October 2018

[8] Y Sun W Xia S Zhang Y Wu T Wang and Y FangldquoEnergy efficient pico cell range expansion and density jointoptimization for heterogeneous networks with eICICrdquo Sen-sors vol 18 no 3 p 762 2018

[9] H Zhang H Ji and Y Wang ldquoUser association scheme inheterogeneous networks considering multiple real-worldpoliciesrdquo in Proceedings of the National Doctoral AcademicForum on Information and Communications Technology 2013Beijing China 2013

[10] K Cwalina S Ambroziak P Rajchowski J Sadowski andJ Stefanski ldquoA novel bitrate adaptation method for hetero-geneous wireless body area networksrdquo Applied Sciences vol 8no 7 p 1209 2018

[11] D Fooladivanda and C Rosenberg ldquoJoint resource allocationand user association for heterogeneous wireless cellularnetworksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 1 pp 248ndash257 2013

[12] W U Khan Z Ali M Waqas and G A S Sidhu ldquoEfficientpower allocation with individual QoS guarantees in futuresmall-cell networksrdquoAEU-International Journal of Electronicsand Communications vol 105 pp 36ndash41 2019

[13] K Shen and W Yu ldquoDownlink cell association optimizationfor heterogeneous networks via dual coordinate descentrdquo inProceedings of the 2013 IEEE International Conference onAcoustics Speech and Signal Processing pp 4779ndash4783 IEEEVancouver Canada May 2013

[14] K Shen and W Yu ldquoDistributed pricing-based user associ-ation for downlink heterogeneous cellular networksrdquo IEEE

Journal on Selected Areas in Communications vol 32 no 6pp 1100ndash1113 2014

[15] D Liu Y Chen K K Chai and T Zhang ldquoNash bargainingsolution based user association optimization in HetNetsrdquo inProceedings of the 2014 IEEE 11th Consumer Communicationsand Networking Conference (CCNC) pp 587ndash592 IEEE LasVegas NV USA January 2014

[16] P Xue P Gong J H Park D Park and D K Kim ldquoMax-minfairness based radio resource management in fourth gener-ation heterogeneous networksrdquo in Proceedings of the 2009 9thInternational Symposium on Communications and In-formation Technology pp 208ndash213 IEEE Icheon SouthKorea September 2009

[17] W U Khan F Jameel T Ristaniemi B M Elhalawany andJ Liu ldquoEfficient power allocation for multi-cell uplink NOMAnetworkrdquo in Proceedings of the 2019 IEEE 89th VehicularTechnology Conference (VTC2019-Spring) pp 1ndash5 IEEEKuala Lumpur Malaysia April-May 2019

[18] R Amin J Martin J Deaton L A DaSilva A Hussien andA Eltawil ldquoBalancing spectral efficiency energy consump-tion and fairness in future heterogeneous wireless systemswith reconfigurable devicesrdquo IEEE Journal on Selected Areasin Communications vol 31 no 5 pp 969ndash980 2013

[19] C Qin C Wang D Pan W Wang and Y Zhang ldquoA crosstime slot partial interference alignment scheme in two-cellrelay heterogeneous networksrdquo Applied Sciences vol 9 no 4p 652 2019

[20] T Jabeen Z Ali W U Khan et al ldquoJoint power allocationand link selection for multi-carrier buffer aided relay net-workrdquo Electronics vol 8 no 6 p 686 2019

[21] A Khodmi S B R Chaouch N Agoulmine and Z ChoukairldquoA joint power allocation and user association based on non-cooperative game theory in an heterogeneous ultra-densenetworkrdquo IEEE Access vol 7 pp 111790ndash111800 2019

[22] P T Semov V Poulkov A Mihovska and R Prasad ldquoIn-creasing throughput and fairness for users in heterogeneoussemi coordinated deploymentsrdquo in Proceedings of the 2014IEEE Wireless Communications and Networking ConferenceWorkshops (WCNCW) pp 40ndash45 IEEE Istanbul TurkeyApril 2014

[23] Q Li R Q Hu Y Qian and G Wu ldquoA proportional fairradio resource allocation for heterogeneous cellular networkswith relaysrdquo in Proceedings of the 2012 IEEE Global Com-munications Conference (GLOBECOM) pp 5457ndash5463 IEEEAnaheim CA USA December 2012

[24] J Jiang W Wang M Peng and Y Huang ldquoEnergy efficiencycomparison between orthogonal and co-channel resourceallocation schemes in distributed antenna systemsrdquo in Pro-ceedings of the 2012 IEEE 23rd International Symposium onPersonal Indoor and Mobile Radio Communications-(PIMRC) pp 541ndash545 IEEE Sydney Australia September2012

[25] J Jiang M Peng K Zhang and L Li ldquoEnergy-efficient re-source allocation in heterogeneous network with cross-tierinterference constraintrdquo in Proceedings of the 2013 IEEE 24thInternational Symposium on Personal Indoor and MobileRadio Communications (PIMRC Workshops) pp 168ndash172IEEE London UK September 2013

[26] X Wang M Jia Q Guo I W-H Ho and J Wu ldquoJointpower original bandwidth and detected hole bandwidthallocation for multi-homing heterogeneous networks basedon cognitive radiordquo IEEE Transactions on Vehicular Tech-nology vol 68 no 3 pp 2777ndash2790 2019

10 Wireless Communications and Mobile Computing

-e expression in (11) can be written as

D λa ηb v( 1113857 minxPacbαacb

x2

+ 1113944A

a11113944

C

c11113944

B

b1αacb minus λalog2 1 + SINRacb1113872 1113873 + ηbPa c b + vPacbhacb1113872 1113873

minus 1113944A

a1λax minus 1113944

B

b1ηbPb minus vIth

st 1113944C

c11113944

B

b1αacb 1 foralla 1 2 3 A

(12)

For any given channel allocation dual decompositionguides to solve the following subproblems

P4 minx

x2

minus x 1113944A

a1λa

⎛⎝ ⎞⎠ (13)

P5 minPacb

minus λalog2 1 + SINRacb1113872 1113873 + ηbPacb + vPacbhacb1113872 1113873

(14)

Using the KKT conditions to find the solution of theproblems given in (13) and (14) we get

xlowast

12

1113944

A

a1λa

⎛⎝ ⎞⎠

+

Plowastacb

Φacb minus Pacbprime αacbfacb

gacb ηb + vhacb1113872 1113873⎛⎝ ⎞⎠

+

(15)

where Φacb λagacb + σ2aηb + vhacb for all a c b when(Ψ)+ max(0Ψ) -e detailed derivation steps have beenprovided in the Appendix

Now to find the optimum value of αacb the followingoptimization problem is considered

minαacb

1113944

A

a11113944

C

c11113944

B

b1αacb minus λalog2 1 + SINRacb1113872 1113873 + ηbP

lowastacb1113872

+ vPlowastacbhacb1113873

st 1113944C

c11113944

B

b1αacb 1 foralla 1 2 3 A

(16)

Evidently the optimal solution can be found such that

αlowastacb for a argmin

cminus λalog2 1 + SINRacb1113872 1113873 + ηbPlowastacb + vPlowastacbhacb1113872 1113873

otherwise

⎧⎪⎨

⎪⎩(17)

-e dual problem is convex hence subgradient methodcan be adopted to find the solution -e dual variables areupdated at each iteration as

λ(itr)a λ(itrminus 1)

a + δ(itrminus 1)times minus 1113944

B

b11113944

C

c1log2 1 + SINRacb1113872 1113873 minus x⎛⎝ ⎞⎠⎛⎝ ⎞⎠

η(itr)b η(itrminus 1)

b + δ(itrminus 1)1113944

A

a11113944

C

c1Pacb minus Pb

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

v(itr)

v(itrminus 1)

+ δ(itrminus 1)times 1113944

A

a11113944

C

c11113944

B

b1Pacbhacb minus Ith

⎛⎝ ⎞⎠⎛⎝ ⎞⎠

(18)

where δ is the step size To obtain the joint optimizationsolution power loading and channel allocation are updatedin each iteration

4 Simulation Results

In this section we present the performance of the proposedscheme -e simulation environment for the proposedschemes is MATLAB -e noise PSD is σ2a 01 For sim-ulation we have considered the maximum number of usersin the system which is A 30 the total channels available inthe system which is C 20 and the total BS available in thesystem which is B 5 For the sake of the evaluation wehave compared our proposed optimal allocation and optimalpower scheme (OAOP) with baseline schemes ie fixedchannel allocation and optimal power loading (FAOP) andfixed channel allocation and fixed power loading (FAFP)

In Figure 2 peak-to-average rate ratio (PR) has beenplotted against different parameters Different values of PRshow fairness among the users -e smaller values of PRindicate a more fair scheme Moreover the effect ofchanging picocell power PRp on PR has also been shown

Wireless Communications and Mobile Computing 5

-e picocell power has been varied from 025 to 2 Con-sidering Pp 025 the percentage gap between FAFP andFAOP is 4974 between FAFP and OAOP is 2259 andbetween FAOP and OAOP is 4542 It can be seen that thevalue of PR decreases with an increase in the power -isindicates that high picocell power causes high interferencewhich results in decreasing the PR Moreover a comparisonof the three schemes clearly indicates that the OAOP schemeis the fairest among all -is is because optimization onchannel allocation along with optimal power loading makesthe problem more flexible compared with the cases whenonly power loading is optimized and if channel allocationand power loading are fixed Also for Pp 05 the per-centage gap between FAFP and FAOP is 4983 betweenFAFP and OAOP is 2544 and between FAOP and OAOPis 5105 Finally increasing the value to Pp 2 the per-centage gap between FAFP and FAOP is 5699 betweenFAFP and OAOP is 3312 and between FAOP and OAOPis 5811

Figure 3 shows the PR with changing interferenceconstraint PRIth

-e interference constraint has been variedfrom 20 to 40 Considering Ith 20 the percentage gapbetween FAFP and FAOP is 5508 between FAFP andOAOP is 3183 and between FAOP and OAOP is 5779Also for Ith 25 the percentage gap between FAFP andFAOP is 5339 between FAFP and OAOP is 3057 andbetween FAOP and OAOP is 5727 Moreover if Ith 30then the percentage gap between FAFP and FAOP is 5241between FAFP and OAOP is 2995 and between FAOPand OAOP is 5714 Further increasing the value toIth 35 the percentage gap between FAFP and FAOP is5200 between FAFP and OAOP is 2962 and betweenFAOP and OAOP is 5711 Finally increasing the value toIth 40 the percentage gap between FAFP and FAOP is5186 between FAFP and OAOP is 2891 and betweenFAOP and OAOP is 5631 Comparison of schemes interms of the percentage gap shows that as the interferenceconstraint Ith increases the percentage gap is decreasingMoreover the average gap between the graphs of FAFP withFAOP and FAOP with OAOP scheme is near to each otherGraphs also show that OAOP outperforms all other schemesin terms of fairness -e reason behind this is the additionalflexibility provided by optimum channel allocation

-e effect of changing the power of macro base stationPRm on the peak-to-average rate ratio is shown in Figure 4-emacro power has been varied from 10 to 30 ConsideringPRm 10 the percentage gap between FAFP and FAOP is5464 between FAFP and OAOP is 3183 and betweenFAOP and OAOP is 5826 Also for PRm 15 the per-centage gap between FAFP and FAOP is 5605 betweenFAFP and OAOP is 3446 and between FAOP and OAOPis 6148 Moreover if PRm 20 then the percentage gapbetween FAFP and FAOP is 5665 between FAFP andOAOP is 3627 and between FAOP and OAOP is 6403Further increasing the value to PRm 25 the percentage gapbetween FAFP and FAOP is 5709 between FAFP andOAOP is 3759 and between FAOP and OAOP is 6584Finally increasing the value to PRm 30 the percentage gapbetween FAFP and FAOP is 5749 between FAFP andOAOP is 3861 and between FAOP and OAOP is 6715Hence comparison of all the three schemes shows that theOAOP is the superior scheme

In the results presented here the sum throughput hasalso been maximized along with achieving fairness amongthe users Figure 5 shows the sum throughput versus picopower Pp -e Pp has been varied from 025 to 2 -epercentage gap between OAOP scheme and FAOP is 38between OAOP and FAFP is 7 and between FAOP andFAFP is 1842 where the value of Pp 025 -e resultsevidently show that the spectral efficiency of OAOP schemesis the most optimal among the other power allocationschemes -is is because the channels are allocated to usersin such a way that would maximize the total data rate ofusers -en increasing the value of Pp 05 the percentagegap between OAOP and FAOP is 4256 between OAOPand FAFP is 7825 and between FAOP and FAFP is1838 Further increasing the value to Pp 2 the per-centage gap between OAOP and FAOP is 4851 betweenOAOP and FAOP is 1012 and between FAOP and FAFPis 2085

Figure 6 shows the sum throughput versus Ith -epercentage gap between OAOP scheme and FAOP is4598 between OAOP and FAFP is 973 and betweenFAOP and FAFP is 2117 where the value of Ith 20Further increasing the value to Ith 25 the percentage gapbetween OAOP and FAOP is 4609 between OAOP and

FAFPFAOPOAOP

12345678

PRp

16 18 212 140806 104Pp

Figure 2 Peak-to-average ratio with changing picocell power

FAFPFAOPOAOP

115

225

335

445

PRI th

3836 4022 3426 282420 30 32Ith

Figure 3 Peak-to-average ratio with changing Ith

6 Wireless Communications and Mobile Computing

FAFP is 1022 and between FAOP and FAFP is 2217Moreover if Ith 30 percentage gap between OAOP andFAOP is 4621 between OAOP and FAFP is 1062 andbetween FAOP and FAFP is 23 Considering Ith 35 thepercentage gap between OAOP and FAOP is 4632 betweenOAOP and FAFP is 1098 and between FAOP and FAFP is2371 Further increasing the value to Ith 40 the percentagegap between OAOP and FAOP is 4647 between OAOP andFAOP is 1101 and between FAOP and FAFP is 2371Hence the sum throughput of OAOP scheme is maximum

-e effect of varying macro power Pm on the sumthroughput is shown in Figure 7 -e Pm has been variedfrom 10 to 30 -e percentage gap between OAOP schemeand FAOP is 4549 between OAOP and FAFP is 973and between FAOP and FAFP is 2140 where the value ofPm 10 Further increasing the value to Pm 15 the per-centage gap between OAOP and FAOP is 4669 betweenOAOP and FAFP is 965 and between FAOP and FAFP is2067 Moreover if Pm 20 the percentage gap betweenOAOP and FAOP is 4727 between OAOP and FAFP is961 and between FAOP and FAFP is 2032 ConsideringPm 25 the percentage gap between OAOP and FAOP is4774 between OAOP and FAFP is 958 and betweenFAOP and FAFP is 2006 Further increasing the value toPm 30 the percentage gap between OAOP and FAOP is4813 between OAOP and FAOP is 956 and betweenFAOP and FAFP is 1986 -e results show that OAOPscheme is the most optimal

Figure 8 shows the values of sum throughput with achanging number of users U From this plot we identifiedthat the performance of OAOP is far superior to otherschemes which is due to the fact that adding new users tothe system introduces new channel gains -is results inincreasing the flexibility of the problem In OAOP bothchannel allocation and power loading are optimizedtherefore this scheme is better at taking advantage of thenewly added users in the network In FAOP only powerloading is optimized so the addition of new users in-creases the data rate because of the reduced degree offreedom compared with OAOP A very slight increase isobserved in FAFP when new users are added as neitherchannel allocation nor power loading is optimized in thisscheme U has been varied from 20 to 100 ConsideringU 20 the percentage gap between OAOP scheme andFAOP is 6107 between OAOP and FAFP is 1482 andbetween FAOP and FAFP is 2427 Further increasingU 40 the percentage gap between OAOP and FAOP is5328 between OAOP and FAFP is 1099 and betweenFAOP and FAFP is 2063 Moreover if U 60 thepercentage gap between OAOP and FAOP is 4881between OAOP and FAFP is 1079 and between FAOPand FAFP is 2010

Figure 9 shows the convergence of dual variables λa ηand v with initial value as 06 η as 06 and v as 01 -esmaller the value of step size the finer the convergence iswhereas a bigger value leads to fast convergence

115

225

335

445

PRm

26 2818 3022 24201412 1610Pm

FAFPFAOPOAOP

Figure 4 Peak-to-average ratio with changing macro power

0

50

100

150

200

250

Rate

(bps

)

06 08 1 12 1404 18 216Pp

FAFPFAOPOAOP

Figure 5 Sum throughput with changing pico power

050

100150200250300

Rate

(bps

)

3224 26 28 3022 34 36 38 4020Ith

FAFPFAOPOAOP

Figure 6 Sum throughput with changing Ith

050

100150200250300

Rate

(bps

)

10 14 16 18 20 3012 26 2822 24Pm

FAFPFAOPOAOP

Figure 7 Sum throughput with changing macro power

Wireless Communications and Mobile Computing 7

5 Conclusion

-is paper optimized the power allocation and user asso-ciation to achieving fairness among different users in un-derlay CR-based HetNets Specifically we adopted a max-min-based fairness framework under various practicalconstraints Problem was first transformed into a standardmaximization optimization and then dual decomposition isused to solve the integer programming problem -e dualproblem was solved through the subgradient method -epower allocation at each node is obtained from the convexoptimization techniques for the known power allocation atall other nodes -e results are compared with the sub-optimal scenarios when the fixed channel allocation is as-sumed for the optimizednonoptimized power allocation-e results showed that the proposed scheme outperformsall other candidates and the spectral efficiency has beenimproved due to the simultaneous transmission with theprimary network

Appendix

-e Lagrangian L associated with the optimization problemP5 is

minPacb

minus λalog2 1 + SINRacb1113872 1113873 + ηbPacb + vPacbhacb1113872 1113873

(A1)In the above expression we have converted our problem

from a constrained problem to an unconstrained problem

minPacbge0

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A2)

By substituting the value of SINR we get

0100200300400500600700

Rate

(bps

)

20 70 10080 9060504030U

FAFPFAOPOAOP

Figure 8 Sum throughput with increasing users

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

Plot of v from recursion

Plot of λ from recursion

Plot of η from recursion

0055

005

0045

v

055

05

045

λ

η 1

0

Figure 9 Convergence of η λ and v with increasing iterations

8 Wireless Communications and Mobile Computing

z

zPacb

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A3)

Applying the KKT conditions for optimality and bysolving the first derivative test we obtain

minus λa

11 + Pacb + gacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

⎛⎝ ⎞⎠

timesz

zPacb

1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠ + ηb + vhacb

(A4)

Now solving the partial derivative as

ηb + vhacb minus λa

times1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

timesgacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠

(A5)

Taking LCM of previous step and finding the partialderivative of internal function we have

minus λa

gacb 1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

times1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠ + ηb + vhacb

(A6)

Further simplification results in

minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ ηb + vhacb

(A7)

By canceling the common terms and simplifying wehave

minus ηb minus vhacb minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

middot1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

ηb + vhacb

λacbgacb

(A8)

After some mathematical simplifications we get

1113944

A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ Pacbgacb λacbgacb

ηb + vhacb

(A9)

Inverting both sides of the equation as

Pacbgacb λacbgacb

ηb + vhacb

1113888 1113889

minus 1113944A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

(A10)

After cross multiplying we have

Pacb λacbgacb minus Ξηb + vhacb1113872 1113873

ηb + vhacb

(A11)

where Ξ (1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a)

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Wireless Communications and Mobile Computing 9

Acknowledgments

-is research was supported by the Chung-Ang Universityresearch grant in 2017 and the Basic Science ResearchProgram through the National Research Foundation ofKorea funded by the Ministry of Education (NRF-2019R1F1A1042599)

References

[1] F Jameel Z Hamid F Jabeen S Zeadally and M A JavedldquoA survey of device-to-device communications research is-sues and challengesrdquo IEEE Communications Surveys amp Tu-torials vol 20 no 3 pp 2133ndash2168 2018

[2] B Wang F Gao S Jin H Lin and G Y Li ldquoSpatial- andfrequency-wideband effects in millimeter-wave massiveMIMO systemsrdquo IEEE Transactions on Signal Processingvol 66 no 13 pp 3393ndash3406 2018

[3] X Wang E Turgut and M Cenk Gursoy ldquoCoverage indownlink heterogeneous mmWave cellular networks withuser-centric small cell deploymentrdquo 2017 httpsarxivorgabs170707035

[4] Y Xu Y Hu and G Li ldquoRobust rate maximization forheterogeneous wireless networks under channel un-certaintiesrdquo Sensors vol 18 no 2 p 639 2018

[5] Y Xu and S Mao ldquoUser association in massive MIMOHetNetsrdquo IEEE Systems Journal vol 11 no 1 pp 7ndash19 2017

[6] A Khandekar N Bhushan J Tingfang and V Vanghi ldquoLTE-advanced heterogeneous networksrdquo in Proceedings of the2010 EuropeanWireless Conference (EW) pp 978ndash982 LuccaItaly April 2010

[7] F Jameel S Kumar Z Chang T Hamalainan andT Ristaniemi ldquoOperator revenue analysis for device-to-devicecommunications overlaying cellular networkrdquo in Proceedings ofthe 2018 IEEE Conference on Standards for Communicationsand Networking (CSCN) pp 1ndash6 Paris France October 2018

[8] Y Sun W Xia S Zhang Y Wu T Wang and Y FangldquoEnergy efficient pico cell range expansion and density jointoptimization for heterogeneous networks with eICICrdquo Sen-sors vol 18 no 3 p 762 2018

[9] H Zhang H Ji and Y Wang ldquoUser association scheme inheterogeneous networks considering multiple real-worldpoliciesrdquo in Proceedings of the National Doctoral AcademicForum on Information and Communications Technology 2013Beijing China 2013

[10] K Cwalina S Ambroziak P Rajchowski J Sadowski andJ Stefanski ldquoA novel bitrate adaptation method for hetero-geneous wireless body area networksrdquo Applied Sciences vol 8no 7 p 1209 2018

[11] D Fooladivanda and C Rosenberg ldquoJoint resource allocationand user association for heterogeneous wireless cellularnetworksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 1 pp 248ndash257 2013

[12] W U Khan Z Ali M Waqas and G A S Sidhu ldquoEfficientpower allocation with individual QoS guarantees in futuresmall-cell networksrdquoAEU-International Journal of Electronicsand Communications vol 105 pp 36ndash41 2019

[13] K Shen and W Yu ldquoDownlink cell association optimizationfor heterogeneous networks via dual coordinate descentrdquo inProceedings of the 2013 IEEE International Conference onAcoustics Speech and Signal Processing pp 4779ndash4783 IEEEVancouver Canada May 2013

[14] K Shen and W Yu ldquoDistributed pricing-based user associ-ation for downlink heterogeneous cellular networksrdquo IEEE

Journal on Selected Areas in Communications vol 32 no 6pp 1100ndash1113 2014

[15] D Liu Y Chen K K Chai and T Zhang ldquoNash bargainingsolution based user association optimization in HetNetsrdquo inProceedings of the 2014 IEEE 11th Consumer Communicationsand Networking Conference (CCNC) pp 587ndash592 IEEE LasVegas NV USA January 2014

[16] P Xue P Gong J H Park D Park and D K Kim ldquoMax-minfairness based radio resource management in fourth gener-ation heterogeneous networksrdquo in Proceedings of the 2009 9thInternational Symposium on Communications and In-formation Technology pp 208ndash213 IEEE Icheon SouthKorea September 2009

[17] W U Khan F Jameel T Ristaniemi B M Elhalawany andJ Liu ldquoEfficient power allocation for multi-cell uplink NOMAnetworkrdquo in Proceedings of the 2019 IEEE 89th VehicularTechnology Conference (VTC2019-Spring) pp 1ndash5 IEEEKuala Lumpur Malaysia April-May 2019

[18] R Amin J Martin J Deaton L A DaSilva A Hussien andA Eltawil ldquoBalancing spectral efficiency energy consump-tion and fairness in future heterogeneous wireless systemswith reconfigurable devicesrdquo IEEE Journal on Selected Areasin Communications vol 31 no 5 pp 969ndash980 2013

[19] C Qin C Wang D Pan W Wang and Y Zhang ldquoA crosstime slot partial interference alignment scheme in two-cellrelay heterogeneous networksrdquo Applied Sciences vol 9 no 4p 652 2019

[20] T Jabeen Z Ali W U Khan et al ldquoJoint power allocationand link selection for multi-carrier buffer aided relay net-workrdquo Electronics vol 8 no 6 p 686 2019

[21] A Khodmi S B R Chaouch N Agoulmine and Z ChoukairldquoA joint power allocation and user association based on non-cooperative game theory in an heterogeneous ultra-densenetworkrdquo IEEE Access vol 7 pp 111790ndash111800 2019

[22] P T Semov V Poulkov A Mihovska and R Prasad ldquoIn-creasing throughput and fairness for users in heterogeneoussemi coordinated deploymentsrdquo in Proceedings of the 2014IEEE Wireless Communications and Networking ConferenceWorkshops (WCNCW) pp 40ndash45 IEEE Istanbul TurkeyApril 2014

[23] Q Li R Q Hu Y Qian and G Wu ldquoA proportional fairradio resource allocation for heterogeneous cellular networkswith relaysrdquo in Proceedings of the 2012 IEEE Global Com-munications Conference (GLOBECOM) pp 5457ndash5463 IEEEAnaheim CA USA December 2012

[24] J Jiang W Wang M Peng and Y Huang ldquoEnergy efficiencycomparison between orthogonal and co-channel resourceallocation schemes in distributed antenna systemsrdquo in Pro-ceedings of the 2012 IEEE 23rd International Symposium onPersonal Indoor and Mobile Radio Communications-(PIMRC) pp 541ndash545 IEEE Sydney Australia September2012

[25] J Jiang M Peng K Zhang and L Li ldquoEnergy-efficient re-source allocation in heterogeneous network with cross-tierinterference constraintrdquo in Proceedings of the 2013 IEEE 24thInternational Symposium on Personal Indoor and MobileRadio Communications (PIMRC Workshops) pp 168ndash172IEEE London UK September 2013

[26] X Wang M Jia Q Guo I W-H Ho and J Wu ldquoJointpower original bandwidth and detected hole bandwidthallocation for multi-homing heterogeneous networks basedon cognitive radiordquo IEEE Transactions on Vehicular Tech-nology vol 68 no 3 pp 2777ndash2790 2019

10 Wireless Communications and Mobile Computing

-e picocell power has been varied from 025 to 2 Con-sidering Pp 025 the percentage gap between FAFP andFAOP is 4974 between FAFP and OAOP is 2259 andbetween FAOP and OAOP is 4542 It can be seen that thevalue of PR decreases with an increase in the power -isindicates that high picocell power causes high interferencewhich results in decreasing the PR Moreover a comparisonof the three schemes clearly indicates that the OAOP schemeis the fairest among all -is is because optimization onchannel allocation along with optimal power loading makesthe problem more flexible compared with the cases whenonly power loading is optimized and if channel allocationand power loading are fixed Also for Pp 05 the per-centage gap between FAFP and FAOP is 4983 betweenFAFP and OAOP is 2544 and between FAOP and OAOPis 5105 Finally increasing the value to Pp 2 the per-centage gap between FAFP and FAOP is 5699 betweenFAFP and OAOP is 3312 and between FAOP and OAOPis 5811

Figure 3 shows the PR with changing interferenceconstraint PRIth

-e interference constraint has been variedfrom 20 to 40 Considering Ith 20 the percentage gapbetween FAFP and FAOP is 5508 between FAFP andOAOP is 3183 and between FAOP and OAOP is 5779Also for Ith 25 the percentage gap between FAFP andFAOP is 5339 between FAFP and OAOP is 3057 andbetween FAOP and OAOP is 5727 Moreover if Ith 30then the percentage gap between FAFP and FAOP is 5241between FAFP and OAOP is 2995 and between FAOPand OAOP is 5714 Further increasing the value toIth 35 the percentage gap between FAFP and FAOP is5200 between FAFP and OAOP is 2962 and betweenFAOP and OAOP is 5711 Finally increasing the value toIth 40 the percentage gap between FAFP and FAOP is5186 between FAFP and OAOP is 2891 and betweenFAOP and OAOP is 5631 Comparison of schemes interms of the percentage gap shows that as the interferenceconstraint Ith increases the percentage gap is decreasingMoreover the average gap between the graphs of FAFP withFAOP and FAOP with OAOP scheme is near to each otherGraphs also show that OAOP outperforms all other schemesin terms of fairness -e reason behind this is the additionalflexibility provided by optimum channel allocation

-e effect of changing the power of macro base stationPRm on the peak-to-average rate ratio is shown in Figure 4-emacro power has been varied from 10 to 30 ConsideringPRm 10 the percentage gap between FAFP and FAOP is5464 between FAFP and OAOP is 3183 and betweenFAOP and OAOP is 5826 Also for PRm 15 the per-centage gap between FAFP and FAOP is 5605 betweenFAFP and OAOP is 3446 and between FAOP and OAOPis 6148 Moreover if PRm 20 then the percentage gapbetween FAFP and FAOP is 5665 between FAFP andOAOP is 3627 and between FAOP and OAOP is 6403Further increasing the value to PRm 25 the percentage gapbetween FAFP and FAOP is 5709 between FAFP andOAOP is 3759 and between FAOP and OAOP is 6584Finally increasing the value to PRm 30 the percentage gapbetween FAFP and FAOP is 5749 between FAFP andOAOP is 3861 and between FAOP and OAOP is 6715Hence comparison of all the three schemes shows that theOAOP is the superior scheme

In the results presented here the sum throughput hasalso been maximized along with achieving fairness amongthe users Figure 5 shows the sum throughput versus picopower Pp -e Pp has been varied from 025 to 2 -epercentage gap between OAOP scheme and FAOP is 38between OAOP and FAFP is 7 and between FAOP andFAFP is 1842 where the value of Pp 025 -e resultsevidently show that the spectral efficiency of OAOP schemesis the most optimal among the other power allocationschemes -is is because the channels are allocated to usersin such a way that would maximize the total data rate ofusers -en increasing the value of Pp 05 the percentagegap between OAOP and FAOP is 4256 between OAOPand FAFP is 7825 and between FAOP and FAFP is1838 Further increasing the value to Pp 2 the per-centage gap between OAOP and FAOP is 4851 betweenOAOP and FAOP is 1012 and between FAOP and FAFPis 2085

Figure 6 shows the sum throughput versus Ith -epercentage gap between OAOP scheme and FAOP is4598 between OAOP and FAFP is 973 and betweenFAOP and FAFP is 2117 where the value of Ith 20Further increasing the value to Ith 25 the percentage gapbetween OAOP and FAOP is 4609 between OAOP and

FAFPFAOPOAOP

12345678

PRp

16 18 212 140806 104Pp

Figure 2 Peak-to-average ratio with changing picocell power

FAFPFAOPOAOP

115

225

335

445

PRI th

3836 4022 3426 282420 30 32Ith

Figure 3 Peak-to-average ratio with changing Ith

6 Wireless Communications and Mobile Computing

FAFP is 1022 and between FAOP and FAFP is 2217Moreover if Ith 30 percentage gap between OAOP andFAOP is 4621 between OAOP and FAFP is 1062 andbetween FAOP and FAFP is 23 Considering Ith 35 thepercentage gap between OAOP and FAOP is 4632 betweenOAOP and FAFP is 1098 and between FAOP and FAFP is2371 Further increasing the value to Ith 40 the percentagegap between OAOP and FAOP is 4647 between OAOP andFAOP is 1101 and between FAOP and FAFP is 2371Hence the sum throughput of OAOP scheme is maximum

-e effect of varying macro power Pm on the sumthroughput is shown in Figure 7 -e Pm has been variedfrom 10 to 30 -e percentage gap between OAOP schemeand FAOP is 4549 between OAOP and FAFP is 973and between FAOP and FAFP is 2140 where the value ofPm 10 Further increasing the value to Pm 15 the per-centage gap between OAOP and FAOP is 4669 betweenOAOP and FAFP is 965 and between FAOP and FAFP is2067 Moreover if Pm 20 the percentage gap betweenOAOP and FAOP is 4727 between OAOP and FAFP is961 and between FAOP and FAFP is 2032 ConsideringPm 25 the percentage gap between OAOP and FAOP is4774 between OAOP and FAFP is 958 and betweenFAOP and FAFP is 2006 Further increasing the value toPm 30 the percentage gap between OAOP and FAOP is4813 between OAOP and FAOP is 956 and betweenFAOP and FAFP is 1986 -e results show that OAOPscheme is the most optimal

Figure 8 shows the values of sum throughput with achanging number of users U From this plot we identifiedthat the performance of OAOP is far superior to otherschemes which is due to the fact that adding new users tothe system introduces new channel gains -is results inincreasing the flexibility of the problem In OAOP bothchannel allocation and power loading are optimizedtherefore this scheme is better at taking advantage of thenewly added users in the network In FAOP only powerloading is optimized so the addition of new users in-creases the data rate because of the reduced degree offreedom compared with OAOP A very slight increase isobserved in FAFP when new users are added as neitherchannel allocation nor power loading is optimized in thisscheme U has been varied from 20 to 100 ConsideringU 20 the percentage gap between OAOP scheme andFAOP is 6107 between OAOP and FAFP is 1482 andbetween FAOP and FAFP is 2427 Further increasingU 40 the percentage gap between OAOP and FAOP is5328 between OAOP and FAFP is 1099 and betweenFAOP and FAFP is 2063 Moreover if U 60 thepercentage gap between OAOP and FAOP is 4881between OAOP and FAFP is 1079 and between FAOPand FAFP is 2010

Figure 9 shows the convergence of dual variables λa ηand v with initial value as 06 η as 06 and v as 01 -esmaller the value of step size the finer the convergence iswhereas a bigger value leads to fast convergence

115

225

335

445

PRm

26 2818 3022 24201412 1610Pm

FAFPFAOPOAOP

Figure 4 Peak-to-average ratio with changing macro power

0

50

100

150

200

250

Rate

(bps

)

06 08 1 12 1404 18 216Pp

FAFPFAOPOAOP

Figure 5 Sum throughput with changing pico power

050

100150200250300

Rate

(bps

)

3224 26 28 3022 34 36 38 4020Ith

FAFPFAOPOAOP

Figure 6 Sum throughput with changing Ith

050

100150200250300

Rate

(bps

)

10 14 16 18 20 3012 26 2822 24Pm

FAFPFAOPOAOP

Figure 7 Sum throughput with changing macro power

Wireless Communications and Mobile Computing 7

5 Conclusion

-is paper optimized the power allocation and user asso-ciation to achieving fairness among different users in un-derlay CR-based HetNets Specifically we adopted a max-min-based fairness framework under various practicalconstraints Problem was first transformed into a standardmaximization optimization and then dual decomposition isused to solve the integer programming problem -e dualproblem was solved through the subgradient method -epower allocation at each node is obtained from the convexoptimization techniques for the known power allocation atall other nodes -e results are compared with the sub-optimal scenarios when the fixed channel allocation is as-sumed for the optimizednonoptimized power allocation-e results showed that the proposed scheme outperformsall other candidates and the spectral efficiency has beenimproved due to the simultaneous transmission with theprimary network

Appendix

-e Lagrangian L associated with the optimization problemP5 is

minPacb

minus λalog2 1 + SINRacb1113872 1113873 + ηbPacb + vPacbhacb1113872 1113873

(A1)In the above expression we have converted our problem

from a constrained problem to an unconstrained problem

minPacbge0

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A2)

By substituting the value of SINR we get

0100200300400500600700

Rate

(bps

)

20 70 10080 9060504030U

FAFPFAOPOAOP

Figure 8 Sum throughput with increasing users

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

Plot of v from recursion

Plot of λ from recursion

Plot of η from recursion

0055

005

0045

v

055

05

045

λ

η 1

0

Figure 9 Convergence of η λ and v with increasing iterations

8 Wireless Communications and Mobile Computing

z

zPacb

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A3)

Applying the KKT conditions for optimality and bysolving the first derivative test we obtain

minus λa

11 + Pacb + gacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

⎛⎝ ⎞⎠

timesz

zPacb

1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠ + ηb + vhacb

(A4)

Now solving the partial derivative as

ηb + vhacb minus λa

times1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

timesgacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠

(A5)

Taking LCM of previous step and finding the partialderivative of internal function we have

minus λa

gacb 1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

times1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠ + ηb + vhacb

(A6)

Further simplification results in

minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ ηb + vhacb

(A7)

By canceling the common terms and simplifying wehave

minus ηb minus vhacb minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

middot1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

ηb + vhacb

λacbgacb

(A8)

After some mathematical simplifications we get

1113944

A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ Pacbgacb λacbgacb

ηb + vhacb

(A9)

Inverting both sides of the equation as

Pacbgacb λacbgacb

ηb + vhacb

1113888 1113889

minus 1113944A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

(A10)

After cross multiplying we have

Pacb λacbgacb minus Ξηb + vhacb1113872 1113873

ηb + vhacb

(A11)

where Ξ (1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a)

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Wireless Communications and Mobile Computing 9

Acknowledgments

-is research was supported by the Chung-Ang Universityresearch grant in 2017 and the Basic Science ResearchProgram through the National Research Foundation ofKorea funded by the Ministry of Education (NRF-2019R1F1A1042599)

References

[1] F Jameel Z Hamid F Jabeen S Zeadally and M A JavedldquoA survey of device-to-device communications research is-sues and challengesrdquo IEEE Communications Surveys amp Tu-torials vol 20 no 3 pp 2133ndash2168 2018

[2] B Wang F Gao S Jin H Lin and G Y Li ldquoSpatial- andfrequency-wideband effects in millimeter-wave massiveMIMO systemsrdquo IEEE Transactions on Signal Processingvol 66 no 13 pp 3393ndash3406 2018

[3] X Wang E Turgut and M Cenk Gursoy ldquoCoverage indownlink heterogeneous mmWave cellular networks withuser-centric small cell deploymentrdquo 2017 httpsarxivorgabs170707035

[4] Y Xu Y Hu and G Li ldquoRobust rate maximization forheterogeneous wireless networks under channel un-certaintiesrdquo Sensors vol 18 no 2 p 639 2018

[5] Y Xu and S Mao ldquoUser association in massive MIMOHetNetsrdquo IEEE Systems Journal vol 11 no 1 pp 7ndash19 2017

[6] A Khandekar N Bhushan J Tingfang and V Vanghi ldquoLTE-advanced heterogeneous networksrdquo in Proceedings of the2010 EuropeanWireless Conference (EW) pp 978ndash982 LuccaItaly April 2010

[7] F Jameel S Kumar Z Chang T Hamalainan andT Ristaniemi ldquoOperator revenue analysis for device-to-devicecommunications overlaying cellular networkrdquo in Proceedings ofthe 2018 IEEE Conference on Standards for Communicationsand Networking (CSCN) pp 1ndash6 Paris France October 2018

[8] Y Sun W Xia S Zhang Y Wu T Wang and Y FangldquoEnergy efficient pico cell range expansion and density jointoptimization for heterogeneous networks with eICICrdquo Sen-sors vol 18 no 3 p 762 2018

[9] H Zhang H Ji and Y Wang ldquoUser association scheme inheterogeneous networks considering multiple real-worldpoliciesrdquo in Proceedings of the National Doctoral AcademicForum on Information and Communications Technology 2013Beijing China 2013

[10] K Cwalina S Ambroziak P Rajchowski J Sadowski andJ Stefanski ldquoA novel bitrate adaptation method for hetero-geneous wireless body area networksrdquo Applied Sciences vol 8no 7 p 1209 2018

[11] D Fooladivanda and C Rosenberg ldquoJoint resource allocationand user association for heterogeneous wireless cellularnetworksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 1 pp 248ndash257 2013

[12] W U Khan Z Ali M Waqas and G A S Sidhu ldquoEfficientpower allocation with individual QoS guarantees in futuresmall-cell networksrdquoAEU-International Journal of Electronicsand Communications vol 105 pp 36ndash41 2019

[13] K Shen and W Yu ldquoDownlink cell association optimizationfor heterogeneous networks via dual coordinate descentrdquo inProceedings of the 2013 IEEE International Conference onAcoustics Speech and Signal Processing pp 4779ndash4783 IEEEVancouver Canada May 2013

[14] K Shen and W Yu ldquoDistributed pricing-based user associ-ation for downlink heterogeneous cellular networksrdquo IEEE

Journal on Selected Areas in Communications vol 32 no 6pp 1100ndash1113 2014

[15] D Liu Y Chen K K Chai and T Zhang ldquoNash bargainingsolution based user association optimization in HetNetsrdquo inProceedings of the 2014 IEEE 11th Consumer Communicationsand Networking Conference (CCNC) pp 587ndash592 IEEE LasVegas NV USA January 2014

[16] P Xue P Gong J H Park D Park and D K Kim ldquoMax-minfairness based radio resource management in fourth gener-ation heterogeneous networksrdquo in Proceedings of the 2009 9thInternational Symposium on Communications and In-formation Technology pp 208ndash213 IEEE Icheon SouthKorea September 2009

[17] W U Khan F Jameel T Ristaniemi B M Elhalawany andJ Liu ldquoEfficient power allocation for multi-cell uplink NOMAnetworkrdquo in Proceedings of the 2019 IEEE 89th VehicularTechnology Conference (VTC2019-Spring) pp 1ndash5 IEEEKuala Lumpur Malaysia April-May 2019

[18] R Amin J Martin J Deaton L A DaSilva A Hussien andA Eltawil ldquoBalancing spectral efficiency energy consump-tion and fairness in future heterogeneous wireless systemswith reconfigurable devicesrdquo IEEE Journal on Selected Areasin Communications vol 31 no 5 pp 969ndash980 2013

[19] C Qin C Wang D Pan W Wang and Y Zhang ldquoA crosstime slot partial interference alignment scheme in two-cellrelay heterogeneous networksrdquo Applied Sciences vol 9 no 4p 652 2019

[20] T Jabeen Z Ali W U Khan et al ldquoJoint power allocationand link selection for multi-carrier buffer aided relay net-workrdquo Electronics vol 8 no 6 p 686 2019

[21] A Khodmi S B R Chaouch N Agoulmine and Z ChoukairldquoA joint power allocation and user association based on non-cooperative game theory in an heterogeneous ultra-densenetworkrdquo IEEE Access vol 7 pp 111790ndash111800 2019

[22] P T Semov V Poulkov A Mihovska and R Prasad ldquoIn-creasing throughput and fairness for users in heterogeneoussemi coordinated deploymentsrdquo in Proceedings of the 2014IEEE Wireless Communications and Networking ConferenceWorkshops (WCNCW) pp 40ndash45 IEEE Istanbul TurkeyApril 2014

[23] Q Li R Q Hu Y Qian and G Wu ldquoA proportional fairradio resource allocation for heterogeneous cellular networkswith relaysrdquo in Proceedings of the 2012 IEEE Global Com-munications Conference (GLOBECOM) pp 5457ndash5463 IEEEAnaheim CA USA December 2012

[24] J Jiang W Wang M Peng and Y Huang ldquoEnergy efficiencycomparison between orthogonal and co-channel resourceallocation schemes in distributed antenna systemsrdquo in Pro-ceedings of the 2012 IEEE 23rd International Symposium onPersonal Indoor and Mobile Radio Communications-(PIMRC) pp 541ndash545 IEEE Sydney Australia September2012

[25] J Jiang M Peng K Zhang and L Li ldquoEnergy-efficient re-source allocation in heterogeneous network with cross-tierinterference constraintrdquo in Proceedings of the 2013 IEEE 24thInternational Symposium on Personal Indoor and MobileRadio Communications (PIMRC Workshops) pp 168ndash172IEEE London UK September 2013

[26] X Wang M Jia Q Guo I W-H Ho and J Wu ldquoJointpower original bandwidth and detected hole bandwidthallocation for multi-homing heterogeneous networks basedon cognitive radiordquo IEEE Transactions on Vehicular Tech-nology vol 68 no 3 pp 2777ndash2790 2019

10 Wireless Communications and Mobile Computing

FAFP is 1022 and between FAOP and FAFP is 2217Moreover if Ith 30 percentage gap between OAOP andFAOP is 4621 between OAOP and FAFP is 1062 andbetween FAOP and FAFP is 23 Considering Ith 35 thepercentage gap between OAOP and FAOP is 4632 betweenOAOP and FAFP is 1098 and between FAOP and FAFP is2371 Further increasing the value to Ith 40 the percentagegap between OAOP and FAOP is 4647 between OAOP andFAOP is 1101 and between FAOP and FAFP is 2371Hence the sum throughput of OAOP scheme is maximum

-e effect of varying macro power Pm on the sumthroughput is shown in Figure 7 -e Pm has been variedfrom 10 to 30 -e percentage gap between OAOP schemeand FAOP is 4549 between OAOP and FAFP is 973and between FAOP and FAFP is 2140 where the value ofPm 10 Further increasing the value to Pm 15 the per-centage gap between OAOP and FAOP is 4669 betweenOAOP and FAFP is 965 and between FAOP and FAFP is2067 Moreover if Pm 20 the percentage gap betweenOAOP and FAOP is 4727 between OAOP and FAFP is961 and between FAOP and FAFP is 2032 ConsideringPm 25 the percentage gap between OAOP and FAOP is4774 between OAOP and FAFP is 958 and betweenFAOP and FAFP is 2006 Further increasing the value toPm 30 the percentage gap between OAOP and FAOP is4813 between OAOP and FAOP is 956 and betweenFAOP and FAFP is 1986 -e results show that OAOPscheme is the most optimal

Figure 8 shows the values of sum throughput with achanging number of users U From this plot we identifiedthat the performance of OAOP is far superior to otherschemes which is due to the fact that adding new users tothe system introduces new channel gains -is results inincreasing the flexibility of the problem In OAOP bothchannel allocation and power loading are optimizedtherefore this scheme is better at taking advantage of thenewly added users in the network In FAOP only powerloading is optimized so the addition of new users in-creases the data rate because of the reduced degree offreedom compared with OAOP A very slight increase isobserved in FAFP when new users are added as neitherchannel allocation nor power loading is optimized in thisscheme U has been varied from 20 to 100 ConsideringU 20 the percentage gap between OAOP scheme andFAOP is 6107 between OAOP and FAFP is 1482 andbetween FAOP and FAFP is 2427 Further increasingU 40 the percentage gap between OAOP and FAOP is5328 between OAOP and FAFP is 1099 and betweenFAOP and FAFP is 2063 Moreover if U 60 thepercentage gap between OAOP and FAOP is 4881between OAOP and FAFP is 1079 and between FAOPand FAFP is 2010

Figure 9 shows the convergence of dual variables λa ηand v with initial value as 06 η as 06 and v as 01 -esmaller the value of step size the finer the convergence iswhereas a bigger value leads to fast convergence

115

225

335

445

PRm

26 2818 3022 24201412 1610Pm

FAFPFAOPOAOP

Figure 4 Peak-to-average ratio with changing macro power

0

50

100

150

200

250

Rate

(bps

)

06 08 1 12 1404 18 216Pp

FAFPFAOPOAOP

Figure 5 Sum throughput with changing pico power

050

100150200250300

Rate

(bps

)

3224 26 28 3022 34 36 38 4020Ith

FAFPFAOPOAOP

Figure 6 Sum throughput with changing Ith

050

100150200250300

Rate

(bps

)

10 14 16 18 20 3012 26 2822 24Pm

FAFPFAOPOAOP

Figure 7 Sum throughput with changing macro power

Wireless Communications and Mobile Computing 7

5 Conclusion

-is paper optimized the power allocation and user asso-ciation to achieving fairness among different users in un-derlay CR-based HetNets Specifically we adopted a max-min-based fairness framework under various practicalconstraints Problem was first transformed into a standardmaximization optimization and then dual decomposition isused to solve the integer programming problem -e dualproblem was solved through the subgradient method -epower allocation at each node is obtained from the convexoptimization techniques for the known power allocation atall other nodes -e results are compared with the sub-optimal scenarios when the fixed channel allocation is as-sumed for the optimizednonoptimized power allocation-e results showed that the proposed scheme outperformsall other candidates and the spectral efficiency has beenimproved due to the simultaneous transmission with theprimary network

Appendix

-e Lagrangian L associated with the optimization problemP5 is

minPacb

minus λalog2 1 + SINRacb1113872 1113873 + ηbPacb + vPacbhacb1113872 1113873

(A1)In the above expression we have converted our problem

from a constrained problem to an unconstrained problem

minPacbge0

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A2)

By substituting the value of SINR we get

0100200300400500600700

Rate

(bps

)

20 70 10080 9060504030U

FAFPFAOPOAOP

Figure 8 Sum throughput with increasing users

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

Plot of v from recursion

Plot of λ from recursion

Plot of η from recursion

0055

005

0045

v

055

05

045

λ

η 1

0

Figure 9 Convergence of η λ and v with increasing iterations

8 Wireless Communications and Mobile Computing

z

zPacb

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A3)

Applying the KKT conditions for optimality and bysolving the first derivative test we obtain

minus λa

11 + Pacb + gacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

⎛⎝ ⎞⎠

timesz

zPacb

1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠ + ηb + vhacb

(A4)

Now solving the partial derivative as

ηb + vhacb minus λa

times1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

timesgacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠

(A5)

Taking LCM of previous step and finding the partialderivative of internal function we have

minus λa

gacb 1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

times1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠ + ηb + vhacb

(A6)

Further simplification results in

minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ ηb + vhacb

(A7)

By canceling the common terms and simplifying wehave

minus ηb minus vhacb minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

middot1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

ηb + vhacb

λacbgacb

(A8)

After some mathematical simplifications we get

1113944

A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ Pacbgacb λacbgacb

ηb + vhacb

(A9)

Inverting both sides of the equation as

Pacbgacb λacbgacb

ηb + vhacb

1113888 1113889

minus 1113944A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

(A10)

After cross multiplying we have

Pacb λacbgacb minus Ξηb + vhacb1113872 1113873

ηb + vhacb

(A11)

where Ξ (1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a)

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Wireless Communications and Mobile Computing 9

Acknowledgments

-is research was supported by the Chung-Ang Universityresearch grant in 2017 and the Basic Science ResearchProgram through the National Research Foundation ofKorea funded by the Ministry of Education (NRF-2019R1F1A1042599)

References

[1] F Jameel Z Hamid F Jabeen S Zeadally and M A JavedldquoA survey of device-to-device communications research is-sues and challengesrdquo IEEE Communications Surveys amp Tu-torials vol 20 no 3 pp 2133ndash2168 2018

[2] B Wang F Gao S Jin H Lin and G Y Li ldquoSpatial- andfrequency-wideband effects in millimeter-wave massiveMIMO systemsrdquo IEEE Transactions on Signal Processingvol 66 no 13 pp 3393ndash3406 2018

[3] X Wang E Turgut and M Cenk Gursoy ldquoCoverage indownlink heterogeneous mmWave cellular networks withuser-centric small cell deploymentrdquo 2017 httpsarxivorgabs170707035

[4] Y Xu Y Hu and G Li ldquoRobust rate maximization forheterogeneous wireless networks under channel un-certaintiesrdquo Sensors vol 18 no 2 p 639 2018

[5] Y Xu and S Mao ldquoUser association in massive MIMOHetNetsrdquo IEEE Systems Journal vol 11 no 1 pp 7ndash19 2017

[6] A Khandekar N Bhushan J Tingfang and V Vanghi ldquoLTE-advanced heterogeneous networksrdquo in Proceedings of the2010 EuropeanWireless Conference (EW) pp 978ndash982 LuccaItaly April 2010

[7] F Jameel S Kumar Z Chang T Hamalainan andT Ristaniemi ldquoOperator revenue analysis for device-to-devicecommunications overlaying cellular networkrdquo in Proceedings ofthe 2018 IEEE Conference on Standards for Communicationsand Networking (CSCN) pp 1ndash6 Paris France October 2018

[8] Y Sun W Xia S Zhang Y Wu T Wang and Y FangldquoEnergy efficient pico cell range expansion and density jointoptimization for heterogeneous networks with eICICrdquo Sen-sors vol 18 no 3 p 762 2018

[9] H Zhang H Ji and Y Wang ldquoUser association scheme inheterogeneous networks considering multiple real-worldpoliciesrdquo in Proceedings of the National Doctoral AcademicForum on Information and Communications Technology 2013Beijing China 2013

[10] K Cwalina S Ambroziak P Rajchowski J Sadowski andJ Stefanski ldquoA novel bitrate adaptation method for hetero-geneous wireless body area networksrdquo Applied Sciences vol 8no 7 p 1209 2018

[11] D Fooladivanda and C Rosenberg ldquoJoint resource allocationand user association for heterogeneous wireless cellularnetworksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 1 pp 248ndash257 2013

[12] W U Khan Z Ali M Waqas and G A S Sidhu ldquoEfficientpower allocation with individual QoS guarantees in futuresmall-cell networksrdquoAEU-International Journal of Electronicsand Communications vol 105 pp 36ndash41 2019

[13] K Shen and W Yu ldquoDownlink cell association optimizationfor heterogeneous networks via dual coordinate descentrdquo inProceedings of the 2013 IEEE International Conference onAcoustics Speech and Signal Processing pp 4779ndash4783 IEEEVancouver Canada May 2013

[14] K Shen and W Yu ldquoDistributed pricing-based user associ-ation for downlink heterogeneous cellular networksrdquo IEEE

Journal on Selected Areas in Communications vol 32 no 6pp 1100ndash1113 2014

[15] D Liu Y Chen K K Chai and T Zhang ldquoNash bargainingsolution based user association optimization in HetNetsrdquo inProceedings of the 2014 IEEE 11th Consumer Communicationsand Networking Conference (CCNC) pp 587ndash592 IEEE LasVegas NV USA January 2014

[16] P Xue P Gong J H Park D Park and D K Kim ldquoMax-minfairness based radio resource management in fourth gener-ation heterogeneous networksrdquo in Proceedings of the 2009 9thInternational Symposium on Communications and In-formation Technology pp 208ndash213 IEEE Icheon SouthKorea September 2009

[17] W U Khan F Jameel T Ristaniemi B M Elhalawany andJ Liu ldquoEfficient power allocation for multi-cell uplink NOMAnetworkrdquo in Proceedings of the 2019 IEEE 89th VehicularTechnology Conference (VTC2019-Spring) pp 1ndash5 IEEEKuala Lumpur Malaysia April-May 2019

[18] R Amin J Martin J Deaton L A DaSilva A Hussien andA Eltawil ldquoBalancing spectral efficiency energy consump-tion and fairness in future heterogeneous wireless systemswith reconfigurable devicesrdquo IEEE Journal on Selected Areasin Communications vol 31 no 5 pp 969ndash980 2013

[19] C Qin C Wang D Pan W Wang and Y Zhang ldquoA crosstime slot partial interference alignment scheme in two-cellrelay heterogeneous networksrdquo Applied Sciences vol 9 no 4p 652 2019

[20] T Jabeen Z Ali W U Khan et al ldquoJoint power allocationand link selection for multi-carrier buffer aided relay net-workrdquo Electronics vol 8 no 6 p 686 2019

[21] A Khodmi S B R Chaouch N Agoulmine and Z ChoukairldquoA joint power allocation and user association based on non-cooperative game theory in an heterogeneous ultra-densenetworkrdquo IEEE Access vol 7 pp 111790ndash111800 2019

[22] P T Semov V Poulkov A Mihovska and R Prasad ldquoIn-creasing throughput and fairness for users in heterogeneoussemi coordinated deploymentsrdquo in Proceedings of the 2014IEEE Wireless Communications and Networking ConferenceWorkshops (WCNCW) pp 40ndash45 IEEE Istanbul TurkeyApril 2014

[23] Q Li R Q Hu Y Qian and G Wu ldquoA proportional fairradio resource allocation for heterogeneous cellular networkswith relaysrdquo in Proceedings of the 2012 IEEE Global Com-munications Conference (GLOBECOM) pp 5457ndash5463 IEEEAnaheim CA USA December 2012

[24] J Jiang W Wang M Peng and Y Huang ldquoEnergy efficiencycomparison between orthogonal and co-channel resourceallocation schemes in distributed antenna systemsrdquo in Pro-ceedings of the 2012 IEEE 23rd International Symposium onPersonal Indoor and Mobile Radio Communications-(PIMRC) pp 541ndash545 IEEE Sydney Australia September2012

[25] J Jiang M Peng K Zhang and L Li ldquoEnergy-efficient re-source allocation in heterogeneous network with cross-tierinterference constraintrdquo in Proceedings of the 2013 IEEE 24thInternational Symposium on Personal Indoor and MobileRadio Communications (PIMRC Workshops) pp 168ndash172IEEE London UK September 2013

[26] X Wang M Jia Q Guo I W-H Ho and J Wu ldquoJointpower original bandwidth and detected hole bandwidthallocation for multi-homing heterogeneous networks basedon cognitive radiordquo IEEE Transactions on Vehicular Tech-nology vol 68 no 3 pp 2777ndash2790 2019

10 Wireless Communications and Mobile Computing

5 Conclusion

-is paper optimized the power allocation and user asso-ciation to achieving fairness among different users in un-derlay CR-based HetNets Specifically we adopted a max-min-based fairness framework under various practicalconstraints Problem was first transformed into a standardmaximization optimization and then dual decomposition isused to solve the integer programming problem -e dualproblem was solved through the subgradient method -epower allocation at each node is obtained from the convexoptimization techniques for the known power allocation atall other nodes -e results are compared with the sub-optimal scenarios when the fixed channel allocation is as-sumed for the optimizednonoptimized power allocation-e results showed that the proposed scheme outperformsall other candidates and the spectral efficiency has beenimproved due to the simultaneous transmission with theprimary network

Appendix

-e Lagrangian L associated with the optimization problemP5 is

minPacb

minus λalog2 1 + SINRacb1113872 1113873 + ηbPacb + vPacbhacb1113872 1113873

(A1)In the above expression we have converted our problem

from a constrained problem to an unconstrained problem

minPacbge0

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A2)

By substituting the value of SINR we get

0100200300400500600700

Rate

(bps

)

20 70 10080 9060504030U

FAFPFAOPOAOP

Figure 8 Sum throughput with increasing users

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

0 50 100 150 200 250 300 350 400 450 500Iterations

Plot of v from recursion

Plot of λ from recursion

Plot of η from recursion

0055

005

0045

v

055

05

045

λ

η 1

0

Figure 9 Convergence of η λ and v with increasing iterations

8 Wireless Communications and Mobile Computing

z

zPacb

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A3)

Applying the KKT conditions for optimality and bysolving the first derivative test we obtain

minus λa

11 + Pacb + gacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

⎛⎝ ⎞⎠

timesz

zPacb

1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠ + ηb + vhacb

(A4)

Now solving the partial derivative as

ηb + vhacb minus λa

times1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

timesgacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠

(A5)

Taking LCM of previous step and finding the partialderivative of internal function we have

minus λa

gacb 1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

times1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠ + ηb + vhacb

(A6)

Further simplification results in

minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ ηb + vhacb

(A7)

By canceling the common terms and simplifying wehave

minus ηb minus vhacb minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

middot1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

ηb + vhacb

λacbgacb

(A8)

After some mathematical simplifications we get

1113944

A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ Pacbgacb λacbgacb

ηb + vhacb

(A9)

Inverting both sides of the equation as

Pacbgacb λacbgacb

ηb + vhacb

1113888 1113889

minus 1113944A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

(A10)

After cross multiplying we have

Pacb λacbgacb minus Ξηb + vhacb1113872 1113873

ηb + vhacb

(A11)

where Ξ (1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a)

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Wireless Communications and Mobile Computing 9

Acknowledgments

-is research was supported by the Chung-Ang Universityresearch grant in 2017 and the Basic Science ResearchProgram through the National Research Foundation ofKorea funded by the Ministry of Education (NRF-2019R1F1A1042599)

References

[1] F Jameel Z Hamid F Jabeen S Zeadally and M A JavedldquoA survey of device-to-device communications research is-sues and challengesrdquo IEEE Communications Surveys amp Tu-torials vol 20 no 3 pp 2133ndash2168 2018

[2] B Wang F Gao S Jin H Lin and G Y Li ldquoSpatial- andfrequency-wideband effects in millimeter-wave massiveMIMO systemsrdquo IEEE Transactions on Signal Processingvol 66 no 13 pp 3393ndash3406 2018

[3] X Wang E Turgut and M Cenk Gursoy ldquoCoverage indownlink heterogeneous mmWave cellular networks withuser-centric small cell deploymentrdquo 2017 httpsarxivorgabs170707035

[4] Y Xu Y Hu and G Li ldquoRobust rate maximization forheterogeneous wireless networks under channel un-certaintiesrdquo Sensors vol 18 no 2 p 639 2018

[5] Y Xu and S Mao ldquoUser association in massive MIMOHetNetsrdquo IEEE Systems Journal vol 11 no 1 pp 7ndash19 2017

[6] A Khandekar N Bhushan J Tingfang and V Vanghi ldquoLTE-advanced heterogeneous networksrdquo in Proceedings of the2010 EuropeanWireless Conference (EW) pp 978ndash982 LuccaItaly April 2010

[7] F Jameel S Kumar Z Chang T Hamalainan andT Ristaniemi ldquoOperator revenue analysis for device-to-devicecommunications overlaying cellular networkrdquo in Proceedings ofthe 2018 IEEE Conference on Standards for Communicationsand Networking (CSCN) pp 1ndash6 Paris France October 2018

[8] Y Sun W Xia S Zhang Y Wu T Wang and Y FangldquoEnergy efficient pico cell range expansion and density jointoptimization for heterogeneous networks with eICICrdquo Sen-sors vol 18 no 3 p 762 2018

[9] H Zhang H Ji and Y Wang ldquoUser association scheme inheterogeneous networks considering multiple real-worldpoliciesrdquo in Proceedings of the National Doctoral AcademicForum on Information and Communications Technology 2013Beijing China 2013

[10] K Cwalina S Ambroziak P Rajchowski J Sadowski andJ Stefanski ldquoA novel bitrate adaptation method for hetero-geneous wireless body area networksrdquo Applied Sciences vol 8no 7 p 1209 2018

[11] D Fooladivanda and C Rosenberg ldquoJoint resource allocationand user association for heterogeneous wireless cellularnetworksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 1 pp 248ndash257 2013

[12] W U Khan Z Ali M Waqas and G A S Sidhu ldquoEfficientpower allocation with individual QoS guarantees in futuresmall-cell networksrdquoAEU-International Journal of Electronicsand Communications vol 105 pp 36ndash41 2019

[13] K Shen and W Yu ldquoDownlink cell association optimizationfor heterogeneous networks via dual coordinate descentrdquo inProceedings of the 2013 IEEE International Conference onAcoustics Speech and Signal Processing pp 4779ndash4783 IEEEVancouver Canada May 2013

[14] K Shen and W Yu ldquoDistributed pricing-based user associ-ation for downlink heterogeneous cellular networksrdquo IEEE

Journal on Selected Areas in Communications vol 32 no 6pp 1100ndash1113 2014

[15] D Liu Y Chen K K Chai and T Zhang ldquoNash bargainingsolution based user association optimization in HetNetsrdquo inProceedings of the 2014 IEEE 11th Consumer Communicationsand Networking Conference (CCNC) pp 587ndash592 IEEE LasVegas NV USA January 2014

[16] P Xue P Gong J H Park D Park and D K Kim ldquoMax-minfairness based radio resource management in fourth gener-ation heterogeneous networksrdquo in Proceedings of the 2009 9thInternational Symposium on Communications and In-formation Technology pp 208ndash213 IEEE Icheon SouthKorea September 2009

[17] W U Khan F Jameel T Ristaniemi B M Elhalawany andJ Liu ldquoEfficient power allocation for multi-cell uplink NOMAnetworkrdquo in Proceedings of the 2019 IEEE 89th VehicularTechnology Conference (VTC2019-Spring) pp 1ndash5 IEEEKuala Lumpur Malaysia April-May 2019

[18] R Amin J Martin J Deaton L A DaSilva A Hussien andA Eltawil ldquoBalancing spectral efficiency energy consump-tion and fairness in future heterogeneous wireless systemswith reconfigurable devicesrdquo IEEE Journal on Selected Areasin Communications vol 31 no 5 pp 969ndash980 2013

[19] C Qin C Wang D Pan W Wang and Y Zhang ldquoA crosstime slot partial interference alignment scheme in two-cellrelay heterogeneous networksrdquo Applied Sciences vol 9 no 4p 652 2019

[20] T Jabeen Z Ali W U Khan et al ldquoJoint power allocationand link selection for multi-carrier buffer aided relay net-workrdquo Electronics vol 8 no 6 p 686 2019

[21] A Khodmi S B R Chaouch N Agoulmine and Z ChoukairldquoA joint power allocation and user association based on non-cooperative game theory in an heterogeneous ultra-densenetworkrdquo IEEE Access vol 7 pp 111790ndash111800 2019

[22] P T Semov V Poulkov A Mihovska and R Prasad ldquoIn-creasing throughput and fairness for users in heterogeneoussemi coordinated deploymentsrdquo in Proceedings of the 2014IEEE Wireless Communications and Networking ConferenceWorkshops (WCNCW) pp 40ndash45 IEEE Istanbul TurkeyApril 2014

[23] Q Li R Q Hu Y Qian and G Wu ldquoA proportional fairradio resource allocation for heterogeneous cellular networkswith relaysrdquo in Proceedings of the 2012 IEEE Global Com-munications Conference (GLOBECOM) pp 5457ndash5463 IEEEAnaheim CA USA December 2012

[24] J Jiang W Wang M Peng and Y Huang ldquoEnergy efficiencycomparison between orthogonal and co-channel resourceallocation schemes in distributed antenna systemsrdquo in Pro-ceedings of the 2012 IEEE 23rd International Symposium onPersonal Indoor and Mobile Radio Communications-(PIMRC) pp 541ndash545 IEEE Sydney Australia September2012

[25] J Jiang M Peng K Zhang and L Li ldquoEnergy-efficient re-source allocation in heterogeneous network with cross-tierinterference constraintrdquo in Proceedings of the 2013 IEEE 24thInternational Symposium on Personal Indoor and MobileRadio Communications (PIMRC Workshops) pp 168ndash172IEEE London UK September 2013

[26] X Wang M Jia Q Guo I W-H Ho and J Wu ldquoJointpower original bandwidth and detected hole bandwidthallocation for multi-homing heterogeneous networks basedon cognitive radiordquo IEEE Transactions on Vehicular Tech-nology vol 68 no 3 pp 2777ndash2790 2019

10 Wireless Communications and Mobile Computing

z

zPacb

⎛⎝ minus λalog2 1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠

+ ηbPacb + vPacbhacb⎞⎠

(A3)

Applying the KKT conditions for optimality and bysolving the first derivative test we obtain

minus λa

11 + Pacb + gacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

⎛⎝ ⎞⎠

timesz

zPacb

1 +Pacbgacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

⎛⎝ ⎞⎠ + ηb + vhacb

(A4)

Now solving the partial derivative as

ηb + vhacb minus λa

times1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

timesgacb1113936

Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠

(A5)

Taking LCM of previous step and finding the partialderivative of internal function we have

minus λa

gacb 1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

times1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a1113872 1113873

2⎛⎜⎝ ⎞⎟⎠ + ηb + vhacb

(A6)

Further simplification results in

minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ ηb + vhacb

(A7)

By canceling the common terms and simplifying wehave

minus ηb minus vhacb minus λa

gacb

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

⎛⎝ ⎞⎠

middot1

1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a + Pacbgacb1113872 1113873 + Pacbgacb

ηb + vhacb

λacbgacb

(A8)

After some mathematical simplifications we get

1113944

A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

+ Pacbgacb λacbgacb

ηb + vhacb

(A9)

Inverting both sides of the equation as

Pacbgacb λacbgacb

ηb + vhacb

1113888 1113889

minus 1113944A

a11113944

C

c11113944

B

b1Pacbprime facb + σ2a + Pacbgacb

⎛⎝ ⎞⎠

(A10)

After cross multiplying we have

Pacb λacbgacb minus Ξηb + vhacb1113872 1113873

ηb + vhacb

(A11)

where Ξ (1113936Aa11113936

Cc11113936

Bb1Pacbprime facb + σ2a)

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Wireless Communications and Mobile Computing 9

Acknowledgments

-is research was supported by the Chung-Ang Universityresearch grant in 2017 and the Basic Science ResearchProgram through the National Research Foundation ofKorea funded by the Ministry of Education (NRF-2019R1F1A1042599)

References

[1] F Jameel Z Hamid F Jabeen S Zeadally and M A JavedldquoA survey of device-to-device communications research is-sues and challengesrdquo IEEE Communications Surveys amp Tu-torials vol 20 no 3 pp 2133ndash2168 2018

[2] B Wang F Gao S Jin H Lin and G Y Li ldquoSpatial- andfrequency-wideband effects in millimeter-wave massiveMIMO systemsrdquo IEEE Transactions on Signal Processingvol 66 no 13 pp 3393ndash3406 2018

[3] X Wang E Turgut and M Cenk Gursoy ldquoCoverage indownlink heterogeneous mmWave cellular networks withuser-centric small cell deploymentrdquo 2017 httpsarxivorgabs170707035

[4] Y Xu Y Hu and G Li ldquoRobust rate maximization forheterogeneous wireless networks under channel un-certaintiesrdquo Sensors vol 18 no 2 p 639 2018

[5] Y Xu and S Mao ldquoUser association in massive MIMOHetNetsrdquo IEEE Systems Journal vol 11 no 1 pp 7ndash19 2017

[6] A Khandekar N Bhushan J Tingfang and V Vanghi ldquoLTE-advanced heterogeneous networksrdquo in Proceedings of the2010 EuropeanWireless Conference (EW) pp 978ndash982 LuccaItaly April 2010

[7] F Jameel S Kumar Z Chang T Hamalainan andT Ristaniemi ldquoOperator revenue analysis for device-to-devicecommunications overlaying cellular networkrdquo in Proceedings ofthe 2018 IEEE Conference on Standards for Communicationsand Networking (CSCN) pp 1ndash6 Paris France October 2018

[8] Y Sun W Xia S Zhang Y Wu T Wang and Y FangldquoEnergy efficient pico cell range expansion and density jointoptimization for heterogeneous networks with eICICrdquo Sen-sors vol 18 no 3 p 762 2018

[9] H Zhang H Ji and Y Wang ldquoUser association scheme inheterogeneous networks considering multiple real-worldpoliciesrdquo in Proceedings of the National Doctoral AcademicForum on Information and Communications Technology 2013Beijing China 2013

[10] K Cwalina S Ambroziak P Rajchowski J Sadowski andJ Stefanski ldquoA novel bitrate adaptation method for hetero-geneous wireless body area networksrdquo Applied Sciences vol 8no 7 p 1209 2018

[11] D Fooladivanda and C Rosenberg ldquoJoint resource allocationand user association for heterogeneous wireless cellularnetworksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 1 pp 248ndash257 2013

[12] W U Khan Z Ali M Waqas and G A S Sidhu ldquoEfficientpower allocation with individual QoS guarantees in futuresmall-cell networksrdquoAEU-International Journal of Electronicsand Communications vol 105 pp 36ndash41 2019

[13] K Shen and W Yu ldquoDownlink cell association optimizationfor heterogeneous networks via dual coordinate descentrdquo inProceedings of the 2013 IEEE International Conference onAcoustics Speech and Signal Processing pp 4779ndash4783 IEEEVancouver Canada May 2013

[14] K Shen and W Yu ldquoDistributed pricing-based user associ-ation for downlink heterogeneous cellular networksrdquo IEEE

Journal on Selected Areas in Communications vol 32 no 6pp 1100ndash1113 2014

[15] D Liu Y Chen K K Chai and T Zhang ldquoNash bargainingsolution based user association optimization in HetNetsrdquo inProceedings of the 2014 IEEE 11th Consumer Communicationsand Networking Conference (CCNC) pp 587ndash592 IEEE LasVegas NV USA January 2014

[16] P Xue P Gong J H Park D Park and D K Kim ldquoMax-minfairness based radio resource management in fourth gener-ation heterogeneous networksrdquo in Proceedings of the 2009 9thInternational Symposium on Communications and In-formation Technology pp 208ndash213 IEEE Icheon SouthKorea September 2009

[17] W U Khan F Jameel T Ristaniemi B M Elhalawany andJ Liu ldquoEfficient power allocation for multi-cell uplink NOMAnetworkrdquo in Proceedings of the 2019 IEEE 89th VehicularTechnology Conference (VTC2019-Spring) pp 1ndash5 IEEEKuala Lumpur Malaysia April-May 2019

[18] R Amin J Martin J Deaton L A DaSilva A Hussien andA Eltawil ldquoBalancing spectral efficiency energy consump-tion and fairness in future heterogeneous wireless systemswith reconfigurable devicesrdquo IEEE Journal on Selected Areasin Communications vol 31 no 5 pp 969ndash980 2013

[19] C Qin C Wang D Pan W Wang and Y Zhang ldquoA crosstime slot partial interference alignment scheme in two-cellrelay heterogeneous networksrdquo Applied Sciences vol 9 no 4p 652 2019

[20] T Jabeen Z Ali W U Khan et al ldquoJoint power allocationand link selection for multi-carrier buffer aided relay net-workrdquo Electronics vol 8 no 6 p 686 2019

[21] A Khodmi S B R Chaouch N Agoulmine and Z ChoukairldquoA joint power allocation and user association based on non-cooperative game theory in an heterogeneous ultra-densenetworkrdquo IEEE Access vol 7 pp 111790ndash111800 2019

[22] P T Semov V Poulkov A Mihovska and R Prasad ldquoIn-creasing throughput and fairness for users in heterogeneoussemi coordinated deploymentsrdquo in Proceedings of the 2014IEEE Wireless Communications and Networking ConferenceWorkshops (WCNCW) pp 40ndash45 IEEE Istanbul TurkeyApril 2014

[23] Q Li R Q Hu Y Qian and G Wu ldquoA proportional fairradio resource allocation for heterogeneous cellular networkswith relaysrdquo in Proceedings of the 2012 IEEE Global Com-munications Conference (GLOBECOM) pp 5457ndash5463 IEEEAnaheim CA USA December 2012

[24] J Jiang W Wang M Peng and Y Huang ldquoEnergy efficiencycomparison between orthogonal and co-channel resourceallocation schemes in distributed antenna systemsrdquo in Pro-ceedings of the 2012 IEEE 23rd International Symposium onPersonal Indoor and Mobile Radio Communications-(PIMRC) pp 541ndash545 IEEE Sydney Australia September2012

[25] J Jiang M Peng K Zhang and L Li ldquoEnergy-efficient re-source allocation in heterogeneous network with cross-tierinterference constraintrdquo in Proceedings of the 2013 IEEE 24thInternational Symposium on Personal Indoor and MobileRadio Communications (PIMRC Workshops) pp 168ndash172IEEE London UK September 2013

[26] X Wang M Jia Q Guo I W-H Ho and J Wu ldquoJointpower original bandwidth and detected hole bandwidthallocation for multi-homing heterogeneous networks basedon cognitive radiordquo IEEE Transactions on Vehicular Tech-nology vol 68 no 3 pp 2777ndash2790 2019

10 Wireless Communications and Mobile Computing

Acknowledgments

-is research was supported by the Chung-Ang Universityresearch grant in 2017 and the Basic Science ResearchProgram through the National Research Foundation ofKorea funded by the Ministry of Education (NRF-2019R1F1A1042599)

References

[1] F Jameel Z Hamid F Jabeen S Zeadally and M A JavedldquoA survey of device-to-device communications research is-sues and challengesrdquo IEEE Communications Surveys amp Tu-torials vol 20 no 3 pp 2133ndash2168 2018

[2] B Wang F Gao S Jin H Lin and G Y Li ldquoSpatial- andfrequency-wideband effects in millimeter-wave massiveMIMO systemsrdquo IEEE Transactions on Signal Processingvol 66 no 13 pp 3393ndash3406 2018

[3] X Wang E Turgut and M Cenk Gursoy ldquoCoverage indownlink heterogeneous mmWave cellular networks withuser-centric small cell deploymentrdquo 2017 httpsarxivorgabs170707035

[4] Y Xu Y Hu and G Li ldquoRobust rate maximization forheterogeneous wireless networks under channel un-certaintiesrdquo Sensors vol 18 no 2 p 639 2018

[5] Y Xu and S Mao ldquoUser association in massive MIMOHetNetsrdquo IEEE Systems Journal vol 11 no 1 pp 7ndash19 2017

[6] A Khandekar N Bhushan J Tingfang and V Vanghi ldquoLTE-advanced heterogeneous networksrdquo in Proceedings of the2010 EuropeanWireless Conference (EW) pp 978ndash982 LuccaItaly April 2010

[7] F Jameel S Kumar Z Chang T Hamalainan andT Ristaniemi ldquoOperator revenue analysis for device-to-devicecommunications overlaying cellular networkrdquo in Proceedings ofthe 2018 IEEE Conference on Standards for Communicationsand Networking (CSCN) pp 1ndash6 Paris France October 2018

[8] Y Sun W Xia S Zhang Y Wu T Wang and Y FangldquoEnergy efficient pico cell range expansion and density jointoptimization for heterogeneous networks with eICICrdquo Sen-sors vol 18 no 3 p 762 2018

[9] H Zhang H Ji and Y Wang ldquoUser association scheme inheterogeneous networks considering multiple real-worldpoliciesrdquo in Proceedings of the National Doctoral AcademicForum on Information and Communications Technology 2013Beijing China 2013

[10] K Cwalina S Ambroziak P Rajchowski J Sadowski andJ Stefanski ldquoA novel bitrate adaptation method for hetero-geneous wireless body area networksrdquo Applied Sciences vol 8no 7 p 1209 2018

[11] D Fooladivanda and C Rosenberg ldquoJoint resource allocationand user association for heterogeneous wireless cellularnetworksrdquo IEEE Transactions on Wireless Communicationsvol 12 no 1 pp 248ndash257 2013

[12] W U Khan Z Ali M Waqas and G A S Sidhu ldquoEfficientpower allocation with individual QoS guarantees in futuresmall-cell networksrdquoAEU-International Journal of Electronicsand Communications vol 105 pp 36ndash41 2019

[13] K Shen and W Yu ldquoDownlink cell association optimizationfor heterogeneous networks via dual coordinate descentrdquo inProceedings of the 2013 IEEE International Conference onAcoustics Speech and Signal Processing pp 4779ndash4783 IEEEVancouver Canada May 2013

[14] K Shen and W Yu ldquoDistributed pricing-based user associ-ation for downlink heterogeneous cellular networksrdquo IEEE

Journal on Selected Areas in Communications vol 32 no 6pp 1100ndash1113 2014

[15] D Liu Y Chen K K Chai and T Zhang ldquoNash bargainingsolution based user association optimization in HetNetsrdquo inProceedings of the 2014 IEEE 11th Consumer Communicationsand Networking Conference (CCNC) pp 587ndash592 IEEE LasVegas NV USA January 2014

[16] P Xue P Gong J H Park D Park and D K Kim ldquoMax-minfairness based radio resource management in fourth gener-ation heterogeneous networksrdquo in Proceedings of the 2009 9thInternational Symposium on Communications and In-formation Technology pp 208ndash213 IEEE Icheon SouthKorea September 2009

[17] W U Khan F Jameel T Ristaniemi B M Elhalawany andJ Liu ldquoEfficient power allocation for multi-cell uplink NOMAnetworkrdquo in Proceedings of the 2019 IEEE 89th VehicularTechnology Conference (VTC2019-Spring) pp 1ndash5 IEEEKuala Lumpur Malaysia April-May 2019

[18] R Amin J Martin J Deaton L A DaSilva A Hussien andA Eltawil ldquoBalancing spectral efficiency energy consump-tion and fairness in future heterogeneous wireless systemswith reconfigurable devicesrdquo IEEE Journal on Selected Areasin Communications vol 31 no 5 pp 969ndash980 2013

[19] C Qin C Wang D Pan W Wang and Y Zhang ldquoA crosstime slot partial interference alignment scheme in two-cellrelay heterogeneous networksrdquo Applied Sciences vol 9 no 4p 652 2019

[20] T Jabeen Z Ali W U Khan et al ldquoJoint power allocationand link selection for multi-carrier buffer aided relay net-workrdquo Electronics vol 8 no 6 p 686 2019

[21] A Khodmi S B R Chaouch N Agoulmine and Z ChoukairldquoA joint power allocation and user association based on non-cooperative game theory in an heterogeneous ultra-densenetworkrdquo IEEE Access vol 7 pp 111790ndash111800 2019

[22] P T Semov V Poulkov A Mihovska and R Prasad ldquoIn-creasing throughput and fairness for users in heterogeneoussemi coordinated deploymentsrdquo in Proceedings of the 2014IEEE Wireless Communications and Networking ConferenceWorkshops (WCNCW) pp 40ndash45 IEEE Istanbul TurkeyApril 2014

[23] Q Li R Q Hu Y Qian and G Wu ldquoA proportional fairradio resource allocation for heterogeneous cellular networkswith relaysrdquo in Proceedings of the 2012 IEEE Global Com-munications Conference (GLOBECOM) pp 5457ndash5463 IEEEAnaheim CA USA December 2012

[24] J Jiang W Wang M Peng and Y Huang ldquoEnergy efficiencycomparison between orthogonal and co-channel resourceallocation schemes in distributed antenna systemsrdquo in Pro-ceedings of the 2012 IEEE 23rd International Symposium onPersonal Indoor and Mobile Radio Communications-(PIMRC) pp 541ndash545 IEEE Sydney Australia September2012

[25] J Jiang M Peng K Zhang and L Li ldquoEnergy-efficient re-source allocation in heterogeneous network with cross-tierinterference constraintrdquo in Proceedings of the 2013 IEEE 24thInternational Symposium on Personal Indoor and MobileRadio Communications (PIMRC Workshops) pp 168ndash172IEEE London UK September 2013

[26] X Wang M Jia Q Guo I W-H Ho and J Wu ldquoJointpower original bandwidth and detected hole bandwidthallocation for multi-homing heterogeneous networks basedon cognitive radiordquo IEEE Transactions on Vehicular Tech-nology vol 68 no 3 pp 2777ndash2790 2019

10 Wireless Communications and Mobile Computing

[27] X Liu Y Zhang Y Li Z Zhang and K Long ldquoA survey ofcognitive radio technologies and their optimization ap-proachesrdquo in Proceedings of the 2013 8th InternationalConference on Communications and Networking in China(CHINACOM) pp 973ndash978 IEEE Guilin China August2013

[28] H O Kpojime and G A Safdar ldquoInterference mitigation incognitive-radio-based femtocellsrdquo IEEE CommunicationsSurveys amp Tutorials vol 17 no 3 pp 1511ndash1534 2015

[29] Y Wang W Xu K Yang and J Lin ldquoOptimal energy-ef-ficient power allocation for OFDM-based cognitive radionetworksrdquo IEEE Communications Letters vol 16 no 9pp 1420ndash1423 2012

[30] L Zhang T Jiang and K Luo ldquoDynamic spectrum allocationfor the downlink of OFDMA-based hybrid-access cognitivefemtocell networksrdquo IEEE Transactions on Vehicular Tech-nology vol 65 no 3 pp 1772ndash1781 2016

[31] R Liu M Sheng and W Wu ldquoEnergy-efficient resourceallocation for heterogeneous wireless network with multi-homed user equipmentsrdquo IEEE Access vol 6 pp 14591ndash14601 2018

[32] Y Xu Y Hu Q Chen R Chai and G Li ldquoDistributed re-source allocation for cognitive HetNets with cross-tier in-terference constraintrdquo in Proceedings of the 2017 IEEEWireless Communications and Networking Conference(WCNC) pp 1ndash6 San Francisco CA USA March 2017

[33] W Yu and R Lui ldquoDual methods for nonconvex spectrumoptimization of multicarrier systemsrdquo IEEE Transactions onCommunications vol 54 no 7 pp 1310ndash1322 2006

Wireless Communications and Mobile Computing 11

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom