A Price Selective Centralized Algorithm for Resource Allocation with Carrier Aggregation in LTE Cellular Networks
aa r X i v : . [ c s . N I] A ug A Price Selective Centralized Algorithm forResource Allocation with Carrier Aggregation inLTE Cellular Networks
Haya Shajaiah, Ahmed Abdelhadi and Charles ClancyBradley Department of Electrical and Computer EngineeringHume Center, Virginia Tech, Arlington, VA, 22203, USA { hayajs, aabdelhadi, tcc } @vt.edu Abstract —In this paper, we consider a resource allocation withcarrier aggregation optimization problem in long term evolution(LTE) cellular networks. In our proposed model, users arerunning elastic or inelastic traffic. Each user equipment (UE)is assigned an application utility function based on the typeof its application. Our objective is to allocate multiple carriersresources optimally among users in their coverage area whilegiving the user the ability to select one of the carriers to be itsprimary carrier and the others to be its secondary carriers. TheUE’s decision is based on the carrier price per unit bandwidth.We present a price selective centralized resource allocationwith carrier aggregation algorithm to allocate multiple carriersresources optimally among users while providing a minimumprice for the allocated resources. In addition, we analyze theconvergence of the algorithm with different carriers rates. Finally,we present simulation results for the performance of the proposedalgorithm.
Index Terms —Resource Allocation with Carrier Aggregation,Elastic Traffic, Inelastic Traffic
I. I
NTRODUCTION
In recent years, the number of mobile subscribers andtheir traffic have increased rapidly. Network providers arenow offering multiple services such as multimedia telephonyand mobile-TV [1]. More spectrum is required to meet thesedemands. Therefore, allowing mobile users to employ multiplecarriers by aggregating their different frequency bands isneeded [2]. The carrier aggregation (CA) feature was addedto the 3GPP LTE standard in Release 10 [3]. This featureallows the LTE Advanced to meet the International Mo-bile Telecommunications (IMT) requirements for the fourth-generation standards defined by the International Telecommu-nications Union (ITU) [4]. A resource allocation with carrieraggregation optimization problem is formulated in [5]. The au-thors proposed two-stage distributed resource allocation (RA)algorithm that allocates the primary and secondary carriersresources optimally among users.In this paper, we formulate the RA with CA probleminto a convex optimization framework. We use logarithmicutility functions to represent delay-tolerant applications andsigmoidal-like utility functions to represent real-time applica-tions running on the UEs subscribing for a mobile service.The primary and secondary carriers optimization problemsassign part of the bandwidth from the multiple carriers to eachuser. A minimum QoS is guaranteed for each user by using a proportional fairness approach. Our centralized RA with CAalgorithm provides a minimum price per unit bandwidth byallowing users under the coverage area of multiple evolvednode Bs (eNodeB)s to select the carrier with the lowestprice to be their primary carrier and the others to be theirsecondary carriers. This mechanism allows users to improvetheir allocated rates by using the CA feature while maintainingthe lowest possible price for their allocated aggregated rates.Additionally, our centralized algorithm is performed mostly inthe eNodeBs which reduces the transmission overhead createdby the distributed algorithm introduced in [5].
A. Related Work
In [6], the authors introduced bandwidth proportional fairresource allocation with logarithmic utilities. The algorithmsat the links are based on Lagrange multiplier methods ofoptimization theory. In [7], the authors used sigmoidal-likeutility functions to represent real-time applications. In [8], theauthors proposed weighted aggregated utility functions for theelastic and inelastic traffic. An optimal resource allocationalgorithm is presented in [9] and [10] to allocate a singlecarrier resources optimally among mobile users. In [11], two-stage resource allocation algorithm is proposed to allocate theeNodeB resources among users running multiple applicationsat a time. In [12], a resource allocation optimization problemis presented for two groups of users. The two groups arepublic safety users group and commercial users group. In[13], the authors presented a resource allocation with usersdiscrimination algorithms to allocate the eNodeB resourcesoptimally among users and their applications. A resourceallocation optimization problem with carrier aggregation ispresented in [14] to allocate resources from the LTE Advancedcarrier and the MIMO radar carrier to each UE in a LTEAdvanced cell based on the running application of the UE.
B. Our Contributions
Our contributions in this paper are summarized as: • We present a resource allocation optimization problemwith carrier aggregation that solves for logarithmic andsigmoidal-like utility functions.
We propose a price selective centralized RA with CAalgorithm to allocate multiple carriers resources optimallyamong users. • We show that our algorithm is a robust one that convergesto the optimal rates whether the eNodeBs available re-sources are abundant or scarce. We present simulationresults for the performance of our resource allocationalgorithm.The remainder of this paper is organized as follows. SectionII presents the problem formulation. In Section III, we presentour RA with CA optimization problem. In section IV, wepresent our centralized rate allocation with CA algorithmfor the utility proportional fairness optimization problem. Insection V, we discuss simulation setup and provide quantitativeresults along with discussion. Section VI concludes the paper.II. P
ROBLEM F ORMULATION
We consider a LTE mobile system with M users and K carriers eNodeBs, one eNodeB in each cell, as illustrated inFigure 1. The users located under the coverage area of the i th eNodeB are forming a set of users M i where M i ∈{M , M , ..., M K } and M i = |M i | is the number of usersin the users set M i under the coverage area of the i th eNodeB.Each joint user j is located under the coverage area of a setof eNodeBs, as shown in Figure 1, that is given by K j where K j ∈ {K , K , ..., K M } and K j = |K j | is the number ofeNodeBs in the set K j of all in range eNodeBs for user j . ......... ... ... ...... eNodeBs Users 12 (cid:1862)(cid:1839) (cid:1861)(cid:1837) Fig. 1. System model for a LTE mobile system with M users and K carrierseNodeBs. M i represents the set of users located under the coverage area ofthe i th eNodeB and K j represents the set of all in range eNodeBs for the j th user. Each eNodeB calculates its offered price per unit bandwidth(assuming it is the primary carrier for all users under itscoverage area) and provides each user under its coverage areawith its offered price. Each joint user selects the carrier withthe least offered price to be its primary carrier and the rest ofall in range carriers to be its secondary carriers. The eNodeBwith the least offered price first allocates its resources to allusers under its coverage area based on the applications runningon their UEs. The remaining eNodeBs then start allocatingtheir resources in the order of their offered prices to all users under their coverage area based on the users applicationsand the rates that are allocated to the joint users from othereNodeBs (with lower offered prices).We express the user satisfaction with its provided serviceusing utility functions [15] [7] [16]. We assume that the j th user’ application utility function U j ( r j ) is strictly concave orsigmoidal-like function where r j is the rate allocated to the j th user. Delay tolerant applications are represented by logarithmicutility functions whereas real-time applications are representedby sigmoidal-like utility functions. These utility functions havethe following properties: • U j (0) = 0 and U j ( r j ) is an increasing function of r j . • U j ( r j ) is twice continuously differentiable in r j andbounded above.We use the normalized sigmoidal-like utility function in ourmodel, same as the one presented in [15], that is U j ( r j ) = c j (cid:16)
11 + e − a j ( r j − b j ) − d j (cid:17) (1)where c j = e ajbj e ajbj and d j = e ajbj so it satisfies U j (0) =0 and U j ( ∞ ) = 1 . The normalized sigmoidal-like functionhas an inflection point at r inf j = b j . Additionally, we use thenormalized logarithmic utility function, used in [9], that canbe expressed as U j ( r j ) = log(1 + k j r j )log(1 + k j r max ) (2)where r max gives utilization and k j is the slope of thecurve that varies based on the user application. So, it satisfies U j (0) = 0 and U j ( r max ) = 1 .III. M ULTIPLE C ARRIERS O PTIMIZATION P ROBLEM
In this section we formulate the RA problem for allocatingthe primary and secondary carriers resources optimally amongusers under their coverage areas. Each carrier first calculates itsoffered price per unit bandwidth assuming that it is the primarycarrier for all UEs under its coverage area. Then, each carrierstarts allocating its available resources optimally among allusers in its coverage area in the order of the carrier’s offeredprice, such that the carrier with a lower offered price performthe RA prior to the one with a higher offered price.
A. The Price Selection Problem and enodeB Sorting
As mentioned earlier, each carrier calculates its offeredprice assuming it is the primary carrier for all users underits coverage area. The carrier’s offered price is obtained fromthe following RA optimization problem:max r i M i Y j =1 U j ( r i,j ) subject to M i X j =1 r i,j ≤ R i ,r i,j ≥ , j = 1 , , ..., M i . (3)where r i = { r i, , r i, , ..., r i,M i } , M i is the number of UEsunder the coverage area of the i th eNodeB and R i is theaximum achievable rate of the i th eNodeB. The resourceallocation objective function is to maximize the total systemutility when allocating the eNodeB resources. Furthermore,it provides proportional fairness among utilities. Therefore,no user is allocated zero resources and a minimum QoS isprovided to each user. Real-time applications are given priorywhen allocating the eNodeB resources using this approach.Optimization problem (3) is a convex optimization problemand there exists a unique tractable global optimal solution[9]. The objective function in optimization problem (3) isequivalent to max r i P M i j =1 log U j ( r i,j ) .From optimization problem (3), we have the Lagrangian: L i ( r i,j ) =( M i X j =1 log U j ( r i,j )) − p offered i ( M i X j =1 r i,j − R i − z i ) (4)where z i ≥ is the slack variable and p offered i is the La-grange multiplier which is equivalent to the shadow pricethat corresponds to the i th carrier price per unit bandwidthfor the M i channels as in [9]. The set of all carriers inthe LTE mobile system is given by K = { , , ..., K } andtheir corresponding offered prices are given by P offered = { p offered , p offered , ..., p offered K } . The j th user set of all in rangecarriers K j (i.e. K j = { , , ..., K j } ) corresponding offeredprices are given by P j = { p j , p j , ..., p jK j } .All in range carriers K j of the j th user are arranged basedon their offered prices as follows: l j = arg min K j { p j , p j , ..., p jK j } l j = arg min K j −{ l j } { p j , p j , ..., p jK j } ... l jK j = arg min K j −{ l j ,...,l jKj − } { p j , p j , ..., p jK j } where l j is the carrier with the lowest offered price and l jK j isthe carrier with the highest offered price within the j th userset K j of all in range carriers and P j = { p j , p j , ..., p jK j } isthe set of the offered prices of all in range carriers for the j th user. The j th user sends an assignment of to the i th eNodeB that is corresponding to eNodeB l j (i.e. the eNodeBwith the least offered price among the j th user’s all in rangecarriers). On the other hand, the j th user sends an assignmentof to each of the remaining eNodeBs in its range. Once the i th eNodeB receives an assignment of from each UE in itscoverage area it starts allocating its resources to the M i UEsin M i such that the j th UE is allocated an optimal rate r j,opti from the i th eNodeB. Once the j th UE is allocated rate fromits primary carrier l j , it then sends an assignment of to the i th eNodeB that is corresponding to eNodeB l j and sends anassignment of to each of the remaining eNodeBs in its range.The process continues until the j th UE sends an assignmentof to the i th eNodeB that is corresponding to eNodeB l jK j and receives its allocated rate from that eNodeB. The j th UEthen calculates its aggregated final optimal rate r aggj . B. RA Optimization Problem
Once the carriers offered prices are calculated as discussedin III-A, each user j selects eNodeB l j to be its primary carrierand the remaining carriers in its range to be its secondarycarriers. The eNodeB with the least offered price is the first oneto start allocating its resources among all users in its coveragearea. Each of the remaining eNodeBs then starts allocating itsavailable resources after all the users in its coverage area areallocated rates from carriers in their range with lower offeredprices. Eventually, each user j is allocated rates from all of the K j carriers in its range. As discussed before, the i th carriereNodeB starts allocating its resources among all users in itscoverage area once it receives an assignment of from eachof the M i users in M i . The rate allocated to the j th user fromits i th carrier is given by r j,opti .The RA optimization problem for the i th carrier eNodeBin K , such that the i th eNodeB received an assignment of from each of the users under its coverage area, can be writtenas: max r i M i Y j =1 U j ( r ji + c ji ) subject to M i X j =1 r ji ≤ R i ,r ji ≥ , j = 1 , , ..., M i ,c ji = K X n =1 ,n = i v jn r j,optn ,v jn = (cid:26) , the j th UE ∈ M n , , the j th UE / ∈ M n , (5)where r i = { r i , r i , ..., r M i i } , R i is the i th eNodeB availableresources, c ji is equivalent to the total rates allocated to the j th user by the carriers in its range with lower offered prices thanthe i th carrier offered price, v jn is equivalent to if the j th UE ∈ M n and is equivalent to if the j th UE / ∈ M n and r j,optn isthe optimal rate allocated to the j th user by the n th eNodeB(i.e. the n th carrier ∈ K ). Once the j th user is allocatedrate from all the carriers in its range, it then calculates itsaggregated final optimal rate r aggj = P Ki =1 v ji r j,opti .Optimization problem (5) gives priority to the real-timeapplication users and ensures that the minimum rate allocatedto each user is c ji . Optimization problem (5) is a convexoptimization problem and there exists a unique tractable globaloptimal solution [9]. The objective function in optimizationproblem (5) is equivalent to max r i P M i j =1 log U j ( r ji + c ji ) .From optimization problem (5), we have the Lagrangian: L i ( r ji ) =( M i X j =1 log U j ( r ji + c ji )) − p i ( M i X j =1 r ji − R i − z i ) (6)here z i ≥ is the slack variable and p i is the Lagrangemultiplier which is equivalent to the shadow price that corre-sponds to the i th carrier price per unit bandwidth for the M i channels as in [9]. IV. A LGORITHM
In this section, we present our price selective centralizedRA with CA algorithm. Each UE is allocated optimal ratesfrom its all in range carriers and the final optimal rateallocated to each UE is the aggregated rate. The algorithmstarts when each UE transmits its application parameters toall in range eNodeBs. Each eNodeB assigns initial values w i,j (0) to the users applications. Each eNodeB performs aninternal iterative algorithm to calculate its offered price perunit bandwidth. In each iteration, the eNodeB checks thedifference between the current value w i,j ( n ) and the previousone w i,j ( n − , if the difference is greater than a threshold δ , the shadow price p offered i ( n ) = P Mij =1 w i,j ( n ) R i is calculatedby the eNodeB. Each eNodeB uses p offered i ( n ) to calculate therate r i,j ( n ) that is the solution of the optimization problem r i,j ( n ) = arg max r i,j (cid:16) log U j ( r i,j ) − p offered i ( n ) r i,j (cid:17) . The cal-culated rate is then used to calculate a new value w i,j ( n ) where w i,j ( n ) = p offered i ( n ) r i,j ( n ) . Each eNodeB checks thefluctuation condition as in [10] and calculates a new value w i,j ( n ) . Once the difference between the current w i,j ( n ) andthe previous one is less than δ for all UEs, the i th eNodeBsends its offered price p offered i to all UEs in its coverage area.Once the j th UE receives the offered prices p offered i fromall in range carriers, it sends an assignment of to the i th eNodeB with the lowest offered price that is corresponding toeNodeB l j and an assignment of to the remaining eNodeBsin its range. The j th UE then receives its allocated rate r j,opti and shadow price p i from that eNodeB. It then updates the c ji value and sends it to the i th eNodeB that is corresponding toeNodeB l j , it also sends an assignment of to that eNodeB andan assignment of to the remaining eNodeBs in its range. Theprocess continues until the j th UE receives its allocated rate r j,opti and shadow price p K j , it then calculates its aggregatedfinal optimal rate r aggj .On the other hand, Once the i th eNodeB receives assign-ments of from all UEs in its coverage area it calculates theoptimal rate r j,opti and shadow price p i and sends them toeach UE in its coverage area. The process continues until theeNodeB with the highest offered price receives assignment of from all UEs in its coverage area, it then sends each of theseUEs its allocated optimal rate r j,opti and shadow price p i .V. S IMULATION R ESULTS
Algorithm (1) and Algorithm (2) were applied in C++ todifferent sigmoidal-like and logarithmic utility functions. Thesimulation results showed convergence to the global optimalrates. In this section, we present the simulation results fortwo carriers and UEs shown in Figure 2. Three UEs { UE1,UE2,UE3 } (first group) are under the coverage area ofonly Carrier eNodeB, another three UEs { UE4,UE5,UE6 } (second group) are joint users under the coverage area Algorithm 1
The j th UE AlgorithmLet c ji = 0 ∀ i ∈ { , , ..., K } Send the UE application utility parameters k j , a j and b j toall in range eNodeBsReceive offered prices that are equivalent to P j = { p j , p j , ..., p jK j } from all in range carriers eNodeBs loopfor m ← to K j do l jm = arg min K j −{ l j ,...,l jm − } { p j , p j , ..., p jK j } is carrier l jm for the j th UE end forend looploopfor m ← to K j − do Send Flag assignment of to the i th eNodeB andan assignment of to the remaining carriers in K j { eNodeB i = eNodeB l jm } Send c ji to the i th eNodeB { eNodeB i = eNodeB l jm } Receive the optimal rate r j,opti from the i th eNodeB { eNodeB i = eNodeB l jm } Receive shadow price p i from the i th eNodeB { eNodeB i = eNodeB l jm } Send the optimal rate r j,opti to the i th eNodeB { the i th eNodeB corresponds to eNodeB l jm +1 } Calculate new c ji = P Kn =1 ,n = i v jn r j,optn for the i th eNodeB that corresponds to eNodeB l jm +1 end forend loop Send c ji to the i th eNodeB { the i th carrier corresponds tocarrier l jK j } Receive the optimal rate r j,opti from the i th eNodeB { the i th carrier corresponds to carrier l jK j } Receive shadow price p i from the i th eNodeB { the i th carrier corresponds to carrier l jK j } Calculate the aggregated final optimal rate r aggj = c ji + r j,opti { the i th carrier corresponds to carrier l jK j } of both carrier and carrier eNodeBs and three UEs { UE7,UE8,UE9 } (third group) are under the coverage areaof only carrier eNodeB. UE1 and UE7 are running thesame real-time application that is represented by a normalizedsigmoidal-like utility function, that is expressed by equation(1), with a = 5 , b = 10 which is an approximation to astep function at rate r = 10 . UE2 and UE8 are runningthe same real-time application that is represented by anothersigmoidal-like utility function with a = 3 and b = 20 . UE3and UE9 are running the same delay-tolerant application that isrepresented by a logarithmic function with k = 15 . The jointusers UE4 and UE5 are running delay tolerant applicationsthat are represented by logarithmic functions with k = 3 and k = 0 . , respectively. The joint user UE6 is runningreal-time application that is represented by sigmoidal-likeutility function with a = 1 and b = 30 . Additionally, Weuse r max = 100 for all logarithmic functions, l = 5 and lgorithm 2 The i th eNodeB AlgorithmLet w i,j (0) = 0 ∀ j ∈ M i Receive application utility parameters k j , a j and b j fromall UEs under the coverage area of the i th eNodeB loopwhile | w i,j ( n ) − w i,j ( n − | > δ for any j = { , ...., M i } where the j th UE under the coverage area of the i th eNodeB do Calculate p offered i ( n ) = P Mij =1 w i,j ( n ) R i for j ← to M i do Solve r i,j ( n ) = arg max r i,j (cid:16) log U j ( r i,j ) − p offered i ( n ) r i,j ( n ) (cid:17) Calculate new w i,j ( n ) = p offered i ( n ) r i,j ( n ) if | w i,j ( n ) − w i,j ( n − | > ∆ w then w i,j ( n ) = w i,j ( n −
1) + sign ( w i,j ( n ) − w i,j ( n − w ( n ) { ∆ w ( n ) = l e − nl } end ifend forend while Send the i th eNodeB’ shadow price p offered i = p offered i ( n ) = P Mij =1 w i,j ( n ) R i to all UEs in the eNodeB coverage area end loopif The i th eNodeB received Flag assignment of from eachUE (the j th UE where j ∈ M i ) in its coverage area thenloop Let w ji (0) = 0 ∀ j, j = { , ...., M i } while | w ji ( n ) − w ji ( n − | > δ for any j = { , ...., M i } do Calculate p i ( n ) = P Mij =1 w ji ( n ) R i for j ← to M i do Receive c ji value from the j th UESolve r ji ( n ) = arg max r ji (cid:16) log U j ( r ji + c ji ) − p i ( n ) r ji ( n ) (cid:17) Calculate new w ji ( n ) = p i ( n ) r ji ( n ) if | w ji ( n ) − w ji ( n − | > ∆ w then w ji ( n ) = w ji ( n −
1) + sign ( w ji ( n ) − w ji ( n − w ( n ) { ∆ w ( n ) = l e − nl } end ifend forend while Send rate r j,opti = w ji ( n ) R i to all UEs in the eNodeBcoverage areaSend the shadow price p i = p i ( n ) to all UEs in itscoverage area end loopend if Carrier 1 UE2UE1UE3 UE6UE4UE5 Carrier 2 UE9UE7 UE8
Fig. 2. System model with two carriers eNodeBs and three groups of users.UE1,UE2 and UE3 under the coverage area of only carrier . UE4, UE5 andUE6 under the coverage area of both carriers. UE7, UE8 and UE9 under thecoverage area of only carrier . r j U j ( r j ) Sig1 a = 5, b = 10Sig2 a = 3, b = 20Sig3 a = 1, b = 30Log1 k = 15Log2 k = 3Log3 k = 0.5
Fig. 3. The users utility functions U j ( r j ) . Sig1 represents UE1 and UE7applications, Sig2 represents UE2 and UE8 applications, Log1 represents UE3and UE9 applications, Log2 represents UE4 application, Log3 represents UE5application and Sig3 represents UE6 application, r j is the rate allocated tothe j th user from all in range eNodeBs. l = 10 in the fluctuation decay function of the algorithm and δ = 10 − . The utility functions corresponding to the nine UEsapplications are shown in Figure 3. A. The i th carrier offered Price p offered i for ≤ R ≤ and R = 100 In the following simulations, carrier eNodeB availableresources R takes values between and with stepof , and carrier eNodeB available resources is fixed R = 100 . In Figure 4, we consider each carrier to be theprimary carrier for all UEs under its coverage area and showthat carrier offered price p offered is higher than carrier offered price p offered when R ≤ R where R = 100 . Onthe other hand, Figure 4 shows that p offered > p offered when R < R ≤ . This shows how the carrier’s offered pricedepends on its available resources, the shadow price increaseswhen the carrier’s available resources decreases for a fixednumber of users. As mentioned before, the joint users selectthe carrier with the lowest offered price to be their primarycarrier. Therefore, in this case the joint users select carrier to be their primary carrier and carrier to be their secondarycarrier when R ≤ whereas they select carrier to be theirprimary carrier and carrier to be their secondary carrier when < R ≤ . B. Aggregated rates r aggj for ≤ R ≤ and R = 100 In the following simulations, carrier available resources R takes values between and with step of and carrier −3 −2 −1 R p o ff e r e d i p offered p offered Fig. 4. Carrier offered price p offered for different values of R and fixednumber of users and carrier offered price p offered for R = 100 assumingthat each carrier is the primary carrier for all UEs under its coverage area.
50 100 150 200020406080100120 R r a gg j UE1UE2UE3UE4UE5UE6UE7UE8UE9
Fig. 5. The aggregated final optimal allocated rate r aggj for each user fromits all in range carriers versus carrier available resources ≤ R ≤ with carrier available resources fixed at R = 100 . eNodeB available resources is fixed R = 100 . In Figure 5,we show the aggregated final optimal rates for the nine userswith different available resources R of carrier . The finaloptimal rates r aggj for the first group of UEs are allocated tothem by only carrier as they are under the coverage area ofonly that carrier and do not have secondary carriers. Similarly,the final optimal rates r aggj for the third group of UEs areallocated to them by carrier as they are under the coveragearea of only that carrier and do not have secondary carriers.On the other hand, the second group of UEs are joint users andare allocated rates from both carriers. The joint users selecttheir primary carrier l j to be the carrier with the lowest shadowprice l j = arg min { , } { p offered , p offered } and the other carrier witha higher offered price to be their secondary carrier l j . Theaggregated final optimal rate allocated to each joint user isthe aggregated rate of its primary carrier allocated rate andits secondary carrier allocated rate. Figure 5 shows that usersrunning real-time applications are given priority over usersrunning delay tolerant applications and are allocated higherrates in the case of low carrier’s available resources.VI. S UMMARY AND C ONCLUSIONS
In this paper, we introduced a novel RA with CA optimiza-tion problem in cellular networks. We considered mobile userswith elastic or inelastic traffic and used utility functions torepresent the applications running on the UEs. We presenteda novel price selective centralized algorithm for allocatingresources from different carriers optimally among users. Ourprice selective algorithm guarantees the minimum possible price for the aggregated final optimal rates. The algorithmuses proportional fairness approach to provide a minimumQoS to all users while giving priority to real-time applicationusers. Our centralized algorithm is performed mostly in theeNodeBs. Therefore, it requires less transmission overhead andless computations in the UEs. We analyzed the convergenceof the algorithm with different carriers available resources andshowed through simulations that our algorithm converges tooptimal values. R
EFERENCES[1] H. Ekstrom, “Qos control in the 3gpp evolved packet system,”
Commu-nications Magazine, IEEE , vol. 47, no. 2, pp. 76–83, 2009.[2] Y. Wang, K. Pedersen, T. Sorensen, and P. Mogensen, “Utility Max-imization in LTE-Advanced Systems with Carrier Aggregation,” in
Vehicular Technology Conference (VTC Spring), 2011 IEEE 73rd , pp. 1–5, May 2011.[3] G. Yuan, X. Zhang, W. Wang, and Y. Yang, “Carrier aggregationfor LTE-advanced mobile communication systems,” in
CommunicationsMagazine, IEEE , vol. 48, pp. 88–93, 2010.[4] S. Parkvall, A. Furuskar, and E. Dahlman, “Evolution of LTE towardIMT-advanced,”
Communications Magazine, IEEE , vol. 49, pp. 84–91,February 2011.[5] H. Shajaiah, A. Abdel-Hadi, and C. Clancy, “Utility proportional fairnessresource allocation with carrier aggregation in 4g-lte,” in
MilitaryCommunications Conference, MILCOM 2013 - 2013 IEEE , pp. 412–417, Nov 2013.[6] F. Kelly, A. Maulloo, and D. Tan, “Rate control in communicationnetworks: shadow prices, proportional fairness and stability,” in
Journalof the Operational Research Society , vol. 49, 1998.[7] S. Shenker, “Fundamental design issues for the future internet,”
SelectedAreas in Communications, IEEE Journal on , vol. 13, no. 7, pp. 1176–1188, 1995.[8] R. Kurrle and C. Clancy, “Resource Allocation for Smart Phonesin 4G-LTE Advanced Carrier Aggregation,” Master’s thesis, VirginiaPolytechnic Institute and State University, 2012.[9] A. Abdel-Hadi and C. Clancy, “A Utility Proportional Fairness Approachfor Resource Allocation in 4G-LTE,” in
ICNC Workshop CNC , 2014.[10] A. Abdel-Hadi and C. Clancy, “A Robust Optimal Rate AllocationAlgorithm and Pricing Policy for Hybrid Traffic in 4G-LTE,” in
PIMRC ,2013.[11] A. Abdel-Hadi, C. Clancy, and J. Mitola, “A Resource Allocation Algo-rithm for Multi-Application Users in 4G-LTE,” in
MobiCom Workshop ,2013.[12] H. Shajaiah, A. Abdel-Hadi, and C. Clancy, “Spectrum sharing betweenpublic safety and commercial users in 4g-lte,” in
Computing, Network-ing and Communications (ICNC), 2014 International Conference on ,pp. 674–679, Feb 2014.[13] H. Shajaiah, A. Abdel-Hadi, and C. Clancy, “Multi-Application Re-source Allocation with Users Discrimination in Cellular Networks,” in
Accepted in PIMRC, 2014 , 2014.[14] H. Shajaiah, A. Khawar, A. Abdel-Hadi, and T. Clancy, “Resourceallocation with carrier aggregation in lte advanced cellular systemsharing spectrum with s-band radar,” in
Dynamic Spectrum AccessNetworks (DYSPAN), 2014 IEEE International Symposium on , pp. 34–37, April 2014.[15] J.-W. Lee, R. R. Mazumdar, and N. B. Shroff, “Downlink powerallocation for multi-class wireless systems,”
IEEE/ACM Trans. Netw. ,vol. 13, pp. 854–867, Aug. 2005.[16] G. Tychogiorgos, A. Gkelias, and K. K. Leung, “Utility-proportionalfairness in wireless networks,” in