Research Article Heterogeneous Resource Allocation Algorithm for Ad Hoc Networks with Utility Fairness

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1 Internatonal Journal of Dstrbuted Sensor Networks Volume 205, Artcle ID 68689, 3 pages Research Artcle Heterogeneous Resource Allocaton Algorthm for Ad Hoc Networks wth Utlty Farness Bng-Qng Han, Guo-Fu Feng, and Y-Fe Chen School of Technology, Nanjng Audt Unversty, Nanjng 285, Chna Correspondence should be addressed to Bng-Qng Han; hb@nau.edu.cn Receved 25 July 204; Accepted 3 November 204 Academc Edtor: Gacomo Olver Copyrght 205 Bng-Qng Han et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Resource allocaton s expected to be a most mportant factor especally for heterogeneous applcatons n wreless ad hoc networks. In ths paper, a novel heterogeneous resource allocaton algorthm (HRA) s presented for ad hoc networks, supportng both elastc and nelastc traffc. Frst, by combnng the frst order Lagrangan method wth pseudo utlty, the orgnal nonconvex problem s converted nto a new convex one. Then, we successfully solve the heterogeneous problem wth the dual-based decomposton approach. In addton, we ntegrate utlty farness nto the resource allocaton framework, whch can adaptvely manage the tradeoff between elastc and nelastc flows. Smulatons show and prove that HRA converges fast and can acheve the global optmum startng from many dfferent network condtons, such as elastc, nelastc, and hybrd scenaro. Wth both consderatons of flow rate and utlty farness, HRA mproves the overall network utlty and system throughput greatly.. Introducton Wth the rapd progress of moble ad hoc networks (MANETs), the nature of the network s gradually evolvng from homogeneous toward heterogeneous [, 2]. In a heterogeneous ad hoc network, the devces dffer n ther communcatonal aspects such as transmsson rate, energy consumpton, and modes of communcaton. Compared wth a homogeneous network, heterogeneous network mposes addtonal reurements. For example, t may contan specal moble nodes whch consst of both smart and omndrectonal antennas, whch can ntegrate all the physcal nformaton avalable to provde rch and versatle servces. In addton, heterogeneous ad hoc network opens up new opportuntes n multmeda systems, especally for those operatng n real tme such as vdeo and voce transmsson over wreless channels. However, due to the heterogenety n wreless networks such as ad hoc networks and wreless sensor networks (WSN), communcaton n heterogeneous networks s relatvely more complcatedthannhomogeneousnetworks.foradhoc networks, one mportant research problem s how to allocate resources to the heterogeneous nodes effectvely and farly. For sensor networks, people are usually more concerned wth system throughput and performance at the snk node. In the context of sensor networks, nodes are composed of dfferent sensor types whch can execute dverse data aggregaton tasks. Regardless of the types of wreless networks, ad hoc networks and sensor networks have much n common, such as constraned resource, lmted battery capacty, and fully dstrbuted manner. One major challenge for the wde deployment of ad hoc networks and sensor networks s to provde ualty-of-servce (QoS) support and far resource allocaton. Snce heterogeneous network may contan many dfferent applcatons assocated wth partcular nodes, flows wthn t are generally classfed as elastc and nelastc [3]. In the case of elastc traffc, the flows can adjust ther transmsson rate gradually such as fle transfer, emal, and remote termnal applcatons. Elastc flows are sutable for varable data rate and non-real-tme servces. Another class s nelastc traffc, such as vdeo and audo applcatons. Dfferent from the elastcflow,nelastcflowsverysenstvetolatencyand jtter and generally has an ntrnsc bandwdth reurement. It makes sense only when the transmsson rate exceeds a threshold.

2 2 Internatonal Journal of Dstrbuted Sensor Networks From the system perspectve, t s very mportant to emphasze the channel effcency because the avalable rado resources are very scarce and the aggregate network utlty must be maxmzed. From the users pont of vew, t s more mportant to guarantee the farness of resource allocaton, such that they wll not be n starvaton state and ther QoS demands can be met. Thus, the resource allocaton problem may encounter conflctng goals. The objectve of heterogeneous resource allocaton s no longer to solely maxmze the sum-rate of network. Instead, t s expected to fnd the optmal tradeoff strategy between resource effcency and user farness for a varety of applcatons assocated wth dfferent servces. In ths paper, we not only focus on the heterogenety of ad hoc networks, but also emphasze the tradeoff between system effcency and user farness of resource allocaton. Frst, we present our resource allocaton framework by ntegratng utlty farness nto the objectve functon. Second, we transform the orgnal nonconvex problem nto a new convex optmzaton problem, whch s euvalent to the orgnal one. Then, we extend our resource allocaton algorthm to the heterogeneous envronment, mxed wth elastc and nelastc flows. Fnally, we present the heterogeneous resource allocaton algorthm to acheve the global far allocaton of bandwdth among contendng flows. The major contrbuton of our paper s twofold: () consder the heterogenety n ad hoc networks, that s, supportng both elastc and nelastc traffc, and (2) provde feasble balance between user farness and network effcency. 2. Related Works The resource allocaton problem has been extensvely studed n wrelne and wreless networks. As early as 998, Kelly et al. [4] proposed the network utlty maxmzaton (NUM) model, whch provdes a powerful framework for network resource allocaton. In 999, Low and Lapsley [5] realzed the basc verson of NUM framework by usng a fxed set of source nodes to predct path. However, due to the assumpton of strctly concave utltes, the NUM model merely addresses the elastc traffc whch s only sutable for non-real-tme applcatons, thus falng to capture the temporal dynamcs n heterogeneous networks. Followng the NUM model, varous resource allocaton schemes have been proposed n typcal wreless networks [6, 7], ad hoc networks [8, 9], and wreless sensor networks [0, ]. Lu et al. [7] extend the basc NUM model nto thefadngchannel.byjontlytakngntoaccountpower nterference and statstcal sgnal varatons, two types of nondetermnstc fadng channels are ntegrated nto the NUM framework. The prmary objectve of resource allocaton s to maxmze the network performance [8], whch s usually assocated wth channel capacty, end-to-end delay, system throughput, and so forth. In [0], the authors proposed a mddleware platform, whch can control runnng applcaton as well as sensng tasks n heterogeneous wreless sensor networks. Recently, several novel algorthms [2 4]werepresented to support nelastc traffc n wreless networks. In [2], a random access algorthm based on dual decomposton method was proposed n wreless networks, whch can allocatetherateandthepersstentprobabltyofnelastc flows. In [3], the authors desgn the ueue-based rate control protocol to handle the nelastc traffc n wreless sensor networks. In [4], a new method for spectrum allocaton s presented, whch can accommodate applcatons wth fxed data rate. However, these algorthms only consder the homogenous lnk capactes n ther schemes. In other words, they are based on ether the fxed channel capacty or the assumptonthatthechannelsestmatedbeforeallocaton.in fact, the channel capacty of heterogeneous networks cannot be estmated n advance, snce they are changng wth tme and depend on the condtons of other lnks. User farness s a crtcal performance metrc n ad hoc networksandshouldbenvolvedwhenaresourceallocaton scheme s desgned. A well-known farness crteron s called max-mn farness (MMF) [5, 6], whch tres to allocate bandwdth eually. That s, a resource allocaton s sad to be max-mn far, f t s mpossble to ncrease the bandwdth of any flow A wthout decreasng the bandwdth of another flow B where B s allocaton was less than or eual to A s allocaton. On the other hand, proportonal farness (PF) [4, 7] san opportunstc algorthm that allocates bandwdth to users n proporton to ther data rates. It explots the users dversty and provdes a good tradeoff between farness and network throughput. For example, PF prmarly allocates resources to users wth good channel condtons. These users can fnsh ther data transfers uckly snce they can take full advantage of network resources. Then, the remanng resources can be allocated to other users wth worse rado condtons. Very recently, a system for a famly of farness models has been proposed, ncludng a-proportonal farness [8, 9]. In fact, a-proportonal farness already contans all the prevous allocaton models such as max-mn farness and proportonal farness. For nstance, the system can acheve maxmum throughput (α =0), proportonal farness (α =), mnmum potental delay (α = 2), and max-mn farness (α = ). Thus, a-proportonal farness provdes a convenent way to acheve dfferent allocaton balance between effcency and farness by varyng the values of parameter α. A more general class of wreless allocaton s based on utlty farness [20, 2]. Utlty farness s usually defned wth a utlty functon that composes the optmzaton problem of wreless resource allocaton. Among them, the resource allocaton objectve s to fnd a feasble soluton to maxmze the utlty functon specfc to the farness concept used. However, prevous work [22 24] normally focuses on ether the rate allocaton of elastc flows or QoS demands of real-tme servces. They do not consder the two ssues as a whole. These schemes cannot be drectly appled n a heterogeneous ad hoc network whch usually contans both elastc and nelastc traffc. To the best of our knowledge, t s stll a challengng and open ssue to attan the utlty farness n ad hoc networks, especally consderng the heterogeneous nature of elastc and nelastc flows.

3 Internatonal Journal of Dstrbuted Sensor Networks 3 3. Utlty-Based Network Modelng 3.. Network Model. A wreless ad hoc network s modeled as a drected graph G =(N, L), wheren s the set of nodes and L s the set of wreless lnks. Each node n N has transmsson range d tx and nterference range d nt.forany two nodes n, n j N,fnoden s n the transmsson range of node n j,thennoden can communcate to node n j wth a transmsson lnk (n,n j ) L. We defne F to be the set of network flows, where each flow f Fcan span multple hops. These lnks are shared by a set of F flows, where flow f uses the set of L f L of lnks (f =,2,...,F). Then we can derve the L Froutng matrx (lnk-flow ncdence matrx) R ={R lf } as R lf ={, f lnk l carres flow f () 0, otherwse. For nterference we need to convert the connectvty graph G nto the correspondng contenton graph G c = (N c, L c ). G c s an undrected graph (smlar concepts are also used n [25 27]). If the source or destnaton of one lnk flow swthnthenterferencerangeofthesourceordestnatonof another one, the two flows are sad to contend wth each other. The vertex set N c contans all the lnk flows n the network. For nstance, the vertces of G c correspond to the edge set of the connectvty graph G; thats,n c = L. Anedgenset L c ndcates that two lnk flows n the connectvty graph G contend wth each other and cannot transmt smultaneously. The contenton graph G c captures the nterference among varous lnk flows. Another mportant concept called maxmal clue susuallyassocatedwththecontentongraph. In a graph, a clue s defned as a subgraph whose vertces are adjacent to each other. A maxmal clue s referred to as a clue that s not contaned n any other clue [28]. In fact, a maxmal clue represents a collson regon where all lnk flows nterfere wth each other such that only one of them can transmt at any tme. A maxmal clue can also be nterpreted as a lmted resource contended and shared by dfferent flows. So, maxmal clues are wdely used to capture the schedulablty. For nstance, they can schedule network flows effcently. We then denote the set of maxmal clues of a contenton graph by Q. In the contenton graph, each maxmal clue Q ( =,2,...,Q)mayconsstofseverallnksnthe connectvty graph G and each lnk n graph G may belong to several maxmal clues. Accordng to the relatonshp between lnks and maxmal clues, we defne a Q L cluelnk ncdence matrx Q ={Q l } as Q l ={, f l Q 0, otherwse Utlty Functon. Utlty functon s wdely used to measure user satsfacton n the optmal flow control lterature. Theutltyfunctonofanapplcatonsauanttatvemeasure of ts QoS performance. In ths paper, we characterze utlty n terms of resource allocaton, whch emphaszes the mportant relatonshp between QoS performance and resource (2) allocaton. Moreover, the utlty has smlar characterstcs as the perceved ualty of a data flow n ad hoc networks. We denote the utlty functon of each user N by U (x ), whch s a nondecreasng functon n x.hence,we can control the level of user satsfacton by varyng x. Each user may have ether elastc or nelastc traffc. Let N E and N I denote the set of users wth elastc and nelastc traffc, respectvely. For elastc flows, we ntroduce a new concept of elastc utlty functon as U j E (x j),whchmeanstheutltyfunctonof user j wth elastc applcatons. These applcatons are rather tolerantofthroughputandtme-delays.wecanuseaconcave and nondecreasng functon to model the utlty. A typcal utlty for such elastc traffc s U j E (x j)=log(x j +).Asshown n Fgure, the utlty ncreases as the bandwdth ncreases, but the margnal mprovement s reduced. The total utlty for elastc flows can be expressed as U E = U j E (x j) (j N E ). j On the other hand, the total utlty of the whole class of nelastc flows can be defned as (3) U I = U I (x k) (k N I ), (4) k where U k I (x k) represents the nelastc utlty functon of user k. Unlke the elastc traffc, nelastc traffc depends on the QoS reurements of real-tme applcatons, whch have an ntrnsc bandwdth threshold. In a heterogeneous network, the elastc flows wll try to reduce ther rate when the network s congested, but the nelastc flows always keep sendng at ther rates wthout consderng the yeld of successful transmsson. So n the long run, the degradaton n bandwdth may cause serous packet drop and the resources are domnated by the users of nelastc flows. Thus, we can choose a sgmodal utlty functon to model these real-tme applcatons as shown n Fgure 2. Compared wth the elastc utlty functon, the nelastc functon s convex nstead of concave at lower bandwdth. There exsts another class of real-tme applcaton, whch s called hard real-tme traffc. Generally, hard real-tme applcatons have strct bandwdth reurements and do not showanyadaptvepropertes.ifthesystemdoesnotmeetthe mnmum reurements, the real-tme applcatons are not allowed to access the network. Examples nclude audo/vdeo phone, vdeo conference, and remote medcal treatment. We canusethefollowngutltyfunctontomodelthehardrealtme traffc: U (x) ={, when x x mn 0, when x<x mn, where x mn s the mnmum bandwdth reurement. Such hard real-tme applcatons can be descrbed n Fgure 2 (dash lne). Itsmportanttochooseanapproprateutltyfuncton. For nstance, f the network utlty s to be maxmzed, we (5)

4 4 Internatonal Journal of Dstrbuted Sensor Networks Utlty Utlty Elastc: concave log(x) Bandwdth (bps) Fgure : Utlty functon for elastc flows. Inelastc: hard real-tme Inelastc: sgmodal allocaton algorthm, therefore, should have the ablty to provde a good performance balance among varous applcatons. 4. Resource Allocaton wth Utlty Farness When consderng dfferent performance of varous applcatons, t may be mpossble to assgn resources smply based on the tradtonal crtera such as proportonal farness [7] and max-mn farness [5]. Therefore, we desgn a new resource allocaton framework based on utlty farness [29, 30], whch can adaptvely allocate resources to the users accordng to ther utlty reurements. Ths has nspred two new concepts assocated wth utlty farness: one s utlty max-mn farness [3] and the other s utlty proportonal farness [32, 33]. 4.. Utlty Farness Crtera. Utlty farness s defned wth a utlty functon that composes the resource allocaton problem, where the objectve s to maxmze the utlty functon specfc to the farness concept used. Defnton. A source rate allocaton x =(x,x 2,...,x n ) s utlty max-mn far (UMMF), f t s feasble, that s, x C, and f, for each node k,theutltyu k (x k ) cannot be ncreased whle stll mantanng feasblty, wthout decreasng the utlty for any other node j wth a lower utlty U j (x j ) U k (x k ). Another newly proposed crteron of utlty farness s utlty proportonal farness Bandwdth Fgure 2: Utlty functon for nelastc flows. can select the correspondng utlty functon as U n (x n ) = log(x n ). On the other hand, f the network throughput s be maxmzed, we can choose the utlty functon as U n (x n )=x n.however,systemeffcencysalwaysnconflct wth farness. A hgher effcent system (.e., hgher network throughput) always means a less far polcy. Therefore, t s necessary to fnd a tradeoff between farness and effcency Resource Allocaton Model. Consder a wreless ad hoc network whch has resource B. Let N be a set of users who wll compete for the resource. And we defne k as the cardnalty of user set N.Letx be the shared resource whch s allocated to user N. Then the resource allocaton vector s denoted by x=(x,x 2,...,x k ),where0<x <, N. As mentoned above, elastc traffc and nelastc traffc have sgnfcantly dfferent utlty functons. In a heterogeneous ad hoc network, varous applcatons wth dfferent traffc types can provde dfferent servces. Resource Defnton 2. A source rate allocaton x =(x,x 2,...,x n ) s utltyproportonallyfar(upf),ftsfeasble,thats,x C, and f, for any other feasble allocaton x, thefollowng condton s satsfed: n = U (x ) (x x x ) 0, (6) where U (x ) ndcates the utlty value to source wth the rate x, and t s an ncreasng, strctly concave functon for all N. The utlty proportonal farness (UPF) s tremendously dfferent from tradtonal proportonal farness (PF). The dfference between them s that PF tres to acheve farness n terms of rate, whle UPF strves to acheve farness n terms of utlty. Snce the utlty can reflect user satsfacton, the utlty, rather than rate, s a more meanngful metrc of QoS for far resource allocaton n heterogeneous ad hoc networks. In the followng secton, we wll study the utlty proportonal farness (UPF) n detal and propose a new resource allocaton algorthm to support utlty farness Problem Formulaton. We frst defne a clue-flow ncdence matrx A as Q R matrx, where Q = {Q l } s the clue-lnk matrx descrbed n (2) and R = {R lf } denotes the lnk-flow matrx ncorporatng the routng nformaton as descrbed n (). In fact, each element n A,thats,A f = Q l R lf, represents the number of subflows wthn a maxmal

5 Internatonal Journal of Dstrbuted Sensor Networks 5 clue. Then, the far resource allocaton problem can be formulated as the followng optmzaton problem: P: maxmze U= n = U (x ), (7) subject to Ax C, (8) where x sthesourcerateofflow and U (x ) s the assocated utlty as a user satsfacton measurement. The constrant n (8) reflects the wreless resource constrant n ad hoc networks. For each clue, A ={A f } denotes the resource usage pattern of each flow and C represents the feasble capacty regon. Sncethe objectvefunctonn(7) s strctly concave and contnuously dfferentable, there exsts an optmal soluton x to the above problem P f the feasble regon gven n (8) s compact and convex. The objectve functon n (7) s referred to as the utlty functon whch can reflect utlty farness Resource Allocaton Algorthm. To solve the prmal problem P, we turn our attenton to the dual problem and use the Lagrange dualty [5]. The frst step s to defne the Lagrangan functon L(x, λ) for the optmzaton problem as follows: L (x, λ) = n = U (x ) λ T (C Ax), (9) where λ=(λ, Q)s a vector of Lagrange multplers. The dual problem D for the prmal problem P can be defned as follows: D: mn D (λ). (0) λ 0 The objectve functon of the dual problem then becomes D (λ) = max L (x, λ) x X = max (U f (x f ) x f λ A f )+ λ C. x X f F Q Q () Let us also defne μ f = λ A f. (2) :f =0 In fact, the Lagrange multpler λ may be nterpreted as the prce per unt bandwdth consumed at maxmal clue (the shadow prce of clue ). Accordngly, μ f represents the prce of flow f whch has to pay for transmttng at rate x f. Snce the objectve functon U (x ) s ncreasng and s strctly concave over the range m x M,thedual functon D(λ) s contnuously dfferentable. Thus, we can use the gradent projecton method [7]tosolvethedualproblem D. The Lagrange multpler λ s adjusted n the opposte drecton to the gradent D(μ) as follows: + D (λ (t)) λ (t+) =[λ (t) α ], (3) λ where t s the teraton number and α > 0 s the step sze. Based on (),thegradentofd(λ) wth respect to λ s D (λ) λ Therefore, we have =C f F x f A f. (4) λ (t+) = [ λ (t) α(c x f A f )]. (5) [ f F ] The above euaton can reflect the law of supply and demand. When the bandwdth demand n the maxmal clue exceeds ts supply C, the prce for accessng the clue s ncreased (.e., to rase the clue prce). On the other hand, f the rate demand s less than the effectve capacty of clue, theclueprcewllbedecreased. Let x f (μ f ) be the demand of flow f when the flow prce s μ f ;thats, x f (μ f )=arg max x f X f (U f (x f ) x f μ f ). + (6) Here, x f (μ f ) = U f (μ f) s the nverse functon of U f (x f), whch s the dervatve functon of U f (x f ).Infact, x f (μ f ) reflects the optmal rate for flow f,wherethenetwork utlty wll be maxmzed wth a flow prce of μ f. For each elastc flow f, the source node receves shadow prce nformaton from all maxmal clues. Then t updates the transmsson rate accordng to (6). Meanwhle, the lnk updates the clue prce λ accordng to (5). Ths s an teratve procedure, where the new transmsson rate of flow f s reported to all maxmal clues every tme. It can be shown that, startng from any ntal rate, ths teratve algorthm wll converge to the optmal soluton (x,λ ) by choosng a proper step sze α. Moreover, snce the soluton s prmal-dual optmal, x s also the optmal rate for the prmal problem. In ths way, the network utlty can be maxmzed n adstrbutedmanner. 5. Heterogeneous Resource Allocaton Algorthm In ths secton, we extend our resource allocaton algorthm to the heterogeneous envronment. Partcularly, we present a dstrbuted resource allocaton algorthm to acheve utlty farness, wth consderaton of both elastc and nelastc traffc. 5.. Notatons and Defntons. In order to handle heterogeneous flows (.e., elastc and nelastc), we develop a new flow control strategy to ensure utlty farness durng the process of resource allocaton. Ths new resource allocaton mechansm, whch ntegrates utlty farness, can not only support elastc traffc,butalsobesutablefornelastctraffc. Frst, we defne a pseudo utlty functon for user : U (x )= x m U (y) dy, m x M, (7)

6 6 Internatonal Journal of Dstrbuted Sensor Networks where M < and m 0are the maxmum and mnmum transmsson rates needed by user,respectvely. By substtutng the pseudo utlty U (x ) nto the orgnal optmzaton problem P, we can derve a new optmzatonproblem P2 as follows: P2: maxmze U= n = U (x ), (8) subject to Ax C. (9) Snce the orgnal utlty functon U (x ) s contnuous, s strctly ncreasng, and s nonnegatve, the pseudo utlty U (x ) must be ncreasng and be strctly concave over the range m x M. After the transformaton, the new problem P2 staysasaconvexoptmzatonproblemands capable of handlng both elastc and nelastc flows Heterogeneous Allocaton Problem Formulaton. Snce the flows n heterogeneous ad hoc networks are composed of elastc and nelastc traffc, we can dvde the utlty functon nto two parts. One s the utlty functon U e (x e ) for elastc flow e and the other s the utlty functon U (x ) for nelastc flow. Then, the heterogeneous resource allocaton problem (HP)nadhocnetworkscanbeformulatedasfollows: HP: max U= U e (x e )+ U (x ), (20) e subject to A e x e C e, A x C, 0 x e M e, m x M. (2) Here, C e and C denote the effectve capacty of clue for elastc flow e and nelastc flow, respectvely. Inelastc traffc can be further classfed nto two subclasses: hard realtme traffc and soft QoS traffc. For nelastc real-tme traffc wth dscontnuous utlty, m s the threshold rate. When the source rate exceeds the threshold, that s, x m,thenetwork mantans a constant performance U (x )=U 0.Otherwse, the utlty drops to zero when t s below the threshold. For nelastc soft traffc wth sgmodal utlty, m s the mnmum transmsson rate wth certan QoS guarantee. The utlty functon s convex at low transmsson rates and becomes concave when x m. The formulaton of (20) s an extenson of the resource allocaton problem descrbed n (7). The man advantage ofthenewformulatonsthattsupportsbothelastcand nelastc traffc. Snce the constrants are lnear and the objectve functon s concave over the range m x M,wecansolve the heterogeneous resource allocaton problem by convex programmng. The Lagrangan of the regularzed prmal problem HP can be wrtten as L(x e, x, λ, μ) = U e (x e ) λ T (C e Ax e ) e + U (x ) μ T (C Ax ) = e + (U e (x e ) x e (U (x ) x λ A e )+ λ C e μ A )+ μ C, (22) where λ and μ are the Lagrangan multplers wth the nterpretaton of shadow prce for elastc and nelastc flows, respectvely. Then we can decompose the maxmzaton problem nto the followng two subproblems: where D (λ) = max L x (x e,λ) e D(λ,μ)=D (λ) +D 2 (μ), (23) = max ( (U x e (x e ) x e λ A e )+ λ C e ), e e (24) D 2 (μ) = max L x 2 (x,μ) = max x ( (U (x ) x μ A )+ μ C ). (25) The frst subproblem D (λ) can be regarded as the bandwdthallocatonforelastcflowsandthesecondsubproblem D 2 (μ) may be nterpreted as the rate control for nelastc flows. Thus, by solvng the two subproblems, we can obtan the optmal resource allocaton n heterogeneous ad hoc networks wth consderaton of both elastc and nelastc traffc. Accordng to (24), the subgradent of D(λ, μ) at pont λ e s gven by D (λ, μ) =C e λ x e A e. (26) e e The subgradent for D(λ, μ) at μ can be obtaned n a smlarway.thus,wecanobtantheprceadjustmentscheme wth step szes α and β at teraton tme t: + D (λ, μ) λ e (t+) =[λ e (t) α ] λ e =[λ e (t) α(c e e + x e A e )], (27)

7 Internatonal Journal of Dstrbuted Sensor Networks 7 + D(λ, μ) μ (t+) =[μ (t) β ] μ =[μ (t) β(c x A )]. + (28) 5.3. Elastc Rate Allocaton. The subproblem D (λ) represents an optmal resource allocaton problem for elastc flows. The gradent of L (x e,λ)wth respect to x e s as follows: L (x e,λ) =U e x (x e) λ. (29) e e E(f) Thus, we can derve the followng algorthm of source rate adjustment for elastc flow e: x e (t+) =[x e (t) +γ L + (x e (t),λ(t)) ], (30) x e where γ>0s a constant step sze and [z] + = max{z, 0} Inelastc Rate Allocaton. The objectve of the subproblem D 2 (μ) s to obtan the optmal rate allocaton for nelastc flows, such that ther QoS demands can be met. We defne the objectve functon of subproblem D 2 (μ) as L 2 (x,μ)= (U (x ) x μ A )+ μ C. (3) Note that the objectve functon L 2 (x,μ) s dfferental wth respect to varable x over the range m x M. Accordng to (3),wehave L 2 (x,μ) x =U (x ) μ. (32) I(f) Thus, the nelastc rate allocaton algorthm can be derved as follows: x (t+) =[x (t) +ω L M 2 (x (t),μ(t)) ], (33) x m clue reserves necessary bandwdth x f A f to support real-tme servce. Note that hard real-tme flows do not change ther rates durng the teraton, so the bandwdth reservaton process wll not be affected by elastc flows. On the other hand, f the traffc has sgmodal utlty, the source node calculates the effectve capacty C for nelastc flow. Then t updates the new transmsson rate and broadcasts ths message to all the neghborng nodes. Algorthm HRA Elastc Rate Allocaton () Receve rates x e (t) from all elastc flows e, wheree E(f). (2) Update shadow prce for elastc flows as λ e (t+) =[λ e (t) α(c e e + x e A e )]. (34) (3) Send λ e (t + ) to all flows e such that e E(f). (4) Adjust the elastc flow rate at tmes t =, 2,...: x e (t+) = [x e (t) +γ L + (x e (t),λ(t)) ]. (35) x e (5) Send x e (t + ) to all lnks on ts path. Inelastc Rate Allocaton () For nelastc flows wth hard real-tme traffc, each clue frst reserves necessary bandwdth x f A f andthenallocatestheremanngbandwdth. (2) Loop untl the bandwdth allocaton of all the hard real-tme traffc s completed. (3) For nelastc flows wth sgmodal utlty, the source nodes of clue calculate the effectve capacty C for nelastc flow and then broadcast ths message to all the neghborng clues. (4) Update the shadow prce of nelastc flows accordng to (28). (5) Adjust the nelastc flow rate at tmes t =, 2,...: where ω s a postve scale step sze and [z] j projecton onto the range [m,m ]. denote the x (t+) =[x (t) +ω L M 2 (x (t),μ(t)) ]. (36) x m 5.5. Algorthm for Heterogeneous Resource Allocaton. To sum uptheaboveanalyss,wepresenttheheterogeneousresource allocaton algorthm (HRA) as follows. For each elastc flow e, λ e may be nterpreted as the shadow prce of flow e.frst,thesourcenoderecevesshadow prce nformaton from all elastc flows. Then, t calculates the shadow prce accordng to (27) and updates the source rate for elastc flow e accordng to (30). Fnally, t reports the new transmsson rate to all lnks on ts path. For nelastc flows, source rates are allocated accordng to dfferent types of traffc. If the traffc s hard real-tme, each (6) Send new x (t + ) to all nodes along the lnk. The HRA algorthm wll proceed wth a two-stage teraton. Frst, the nelastc rate allocaton process s executed snce the QoS reurements for real-tme applcatons should be met n advance. Then, the elastc flow allocaton s executedbasedontheremanngbandwdthfornon-real-tme servce. Furthermore, f we choose approprate utlty functons U I and U E for elastc and nelastc flows, respectvely, dfferent utlty farness models such as utlty proportonal farnessandutltymax-mnfarness canbeacheved.

8 8 Internatonal Journal of Dstrbuted Sensor Networks f2 300 f f3 Fgure 3: 6-node topology wth elastc flows. Flow rate (Kbps) Performance Evaluaton In ths secton, we present the smulaton results for HRA algorthm n heterogeneous ad hoc networks consstng of both elastc and nelastc flows. 6.. Smulaton Setup. In order to evaluate the performance of HRA algorthm presented n Secton 5,wemplementthe algorthm n NS-2 (verson 2.32). In addton, we set up dfferent types of traffc generators over dfferent scenaros. Thenetworkscenarosarecomposedofdfferenttraffc,such as Case : elastc traffc, Case 2: nelastc traffc, and Case 3: mxed network traffc, respectvely. Conseuently, users have dfferent utlty functons n dfferent network scenaros. A 200 m 200 m ad hoc network consstng of 60 nodes has been bult to test the resource allocaton scheme. Each node hasabandwdthcapactyofmbpsandthetransmsson range of the node s 30 m. Moble nodes can travel n dfferent drectons wth eual probablty and usng three dfferent speed models: 0 m/s, 3 2 m/s, and 9 20 m/s Case : Elastc Network Scenaro. In Fgure 3, wesetup the elastc network scenaro composed of sx users, where each user s utlty functon s set to U(x n )=log(x n )(x n >0) for mposng utlty proportonal farness. In ths case, we can study the optmal utlty farness among the users. As shown n Fgure 3, the network conssts of sx nodes and three undrectonal lnks through 3 4, , and , wth the lnk capacty eual to Mbps. In the smulaton, we choose the step sze γ = 0. and the smulaton tme s set to 200 seconds. Fgure 4 shows the calculated rates of the elastc flows change wth tme. We observe that the date rates of all flows change sharply at the begnnng of the teraton. In a sense, flowandflow3arerelatvelystable,whleflow2smuch more ntense than the other flows. Because flow 2 goes across the four nodes (.e., nodes 2, 3, 4, and 5) and they are all n the mddle of the network, the wreless channel competton wll be more ntense, whch causes bgger fluctuatons of flow 2. After a perod of fluctuaton, the algorthm converges to the optmal soluton uckly, whch shows the effectveness of our algorthm. In addton, we can fnd somethng nterestng from Fgure 4. Wth the passage of tme, the data rate of flow drops rapdly, whle flow 2 and flow 3 ncrease ther data rates gradually. To ensure the utlty farness among Tme Flow Flow 2 Flow 3 Fgure 4: Smulaton results for elastc flows. A f B C D E F f2 G f3 Fgure 5: 7-node topology wth nelastc flows. the elastc flows, the three flows need to adjust ther data rate accordng to the network utlty. Fnally, they all converge totheoptmalvaluewthsatsfactoryspeed,whchfurther verfy the convergence of the HRA algorthm Case 2: Inelastc Network Scenaro. In Case 2, for comparson purpose, we set the nelastc network scenaro whch s composed of seven nodes and four flows. As shown n Fgure 5, the network conssts of four undrectonal data flows A C D, G E D, F E, and B C D E F wth capactes fxed, respectvely, at 2 MB/s, MB/s, 3 MB/s, and MB/s. There are four sources S A, S B, S F,and S G wth a sgmodal utlty functon, whch represents an nelastcdataflowtypcallyassocatedwththereal-tme applcaton. In the smulaton, we run the HRA algorthm wth γ = 0.2 and the results are gven n Fgure 6. It can be seen that the network utlty vares over tme. At the begnnng, the ntal utlty of each user s set to a lower value, respectvely. Later, each source user gradually ncreases ther utlty to mprove the network throughput. However, the source rate of each f4

9 Internatonal Journal of Dstrbuted Sensor Networks Utlty Flow FE Flow ACD Tme Flow GED Flow BCDEF Fgure 6: Smulaton results for nelastc flows Fgure 7: 60-node heterogeneous network wth mxed flows. user should be also controlled well due to the lnk capacty lmtaton. Moreover, the utltes allocated to four sources S A, S B, S F,andS G are dfferent. At frst, the utltes of all sources are almost the same. Then,thesourcenodeS F rapdly mproves ts data rate, thereby ncreasng the utlty and surpassng the other nodes uckly. In the whole smulaton, the data rates of flow GED and flow BCDEF are very close, overlappng constantly. Ths s determned by the network topology and user utlty. As we can see, flow BCDEF travels across multple nodes and flow GED s n the mddle of the network. In ths way, each node can adjust ts data transfer rate accordng to the networktopologyandlocatonautomatcally,whchensuresthe overall optmzaton of network utlty and the user farness. As expected, the oscllaton s observed untl the flows all converge to the optmal soluton. Therefore, ths confrms that the HRA algorthm gven n ths paper can provde an effcent resource allocaton scheme wth both consderatons of network utlty and farness for heterogeneous ad hoc networks Case 3: Mxed Network wth Elastc and Inelastc Flows. In Case 3, we evaluate the dynamc performance of our heterogeneous resource allocaton algorthm (HRA) n largescale networks and compare our soluton wth two other algorthms. As shown n Fgure 7, we desgned a hybrd wreless ad hoc network scenaro, ncludng 60 nodes randomly generated n the area of sze 000 m 000 m. The network s composedofthreeparts,wthansolatednodenthemddle. Nodes of dfferent colors represent dfferent applcatons, whch belong to the heterogeneous network and can provde versatle servces. The transmsson range and nterference range of all nodes are 250 m and 550 m, respectvely. To test the performance of HRA algorthm under elastc and nelastc condtons, we ran dfferent types of traffc generators over the wreless ad hoc scenaro. There are varous data sources n the network, whch are composed of two elastc flows and two nelastc flows. The utlty functons of all sources are gven by: U (x ) = log(x ), U 2 (x 2 ) = log(x 2 +), U 3 (x 3 ) = /( + e 2(x 3 4) ),andu 4 (x 4 ) = /( + e 2(x 4 7) ). Each utlty functon s multpled by a nonnegatve weghtng factor w, to adjust the utlty rato of elastc and nelastc flows. The frst two functons, whch are logarthmc utlty functons, represent elastc applcatons, whereas the last two sgmodal utlty functons approxmate real-tme applcatons. All the sources have ther maxmum data transfer rate at 0 Mbps. Fgure 8 shows how the average transfer rates for nelastc and elastc flows converge when HRA algorthm s used, where the rates are allocated to two elastc and two nelastc flows. From ths fgure, we can see that the QoS reurements of all flows are satsfed. The average servce rate of the nelastc flow on each lnk l s larger than the mnmum threshold, whch euals 350 Kbps. Moreover, for each lnk of the elastc flow, ts average servce rate s larger than the mnmum transmsson rate reuest. Therefore, HRA algorthm not only can allocate elastc flow effectvely, but alsocanbeusednthenelastcflow,whchguaranteesthe transmsson of real-tme applcatons Comparson between HRA and Other Algorthms. As utlty s our man performance metrc, we wll compare our soluton HRA wth two other nonheterogeneous allocaton schemes. One s a smple resource allocaton algorthm called SRA, n whch the heterogeneous characterstcs are not exploted for rate allocaton. SRA always handles elastc and nelastc flows n the same way. Therefore, t cannot dstngush elastc and nelastc flows well. The other s smlar to the Utlty-Based Adaptve Resource Allocaton algorthm proposed by Rodrgues and Casadevall [34]. We make some modfcatons and call t UBARA. It manages the tradeoff between system resource effcency and user farness

10 0 Internatonal Journal of Dstrbuted Sensor Networks Average flow rate (Kbps) Inelastc flow Inelastc flow 2 Iteratons Elastc flow Elastc flow 2 Fgure 8: Average elastc and nelastc flow rates. Elastc utlty Speed (m/s) HRA UBARA SRA Fgure 9: Elastc utlty comparson n moble scene. and can be appled to wreless network scenaros. Snce the orgnal algorthm cannot deal wth hybrd network scenaro, we take the followng changes to handle both elastc and nelastc traffc. For nelastc flows, each lnk l transmtsat the same rate level n all network condtons. For elastc flows, t allocates flow rate based on the user utlty. In ths part, we wll verfy the mpact of node moblty on the performance of HRA. The moblty model s as follows: a node randomly selects a destnaton wthn the network lmts andthenmovestowardstataspeedof4,8,2,6,and20m/s. The smulaton lasts 50 seconds and the delay lmt s set to 5. The network topology and other parameters are the same as Case 3. Fgure 9 shows how the utlty of elastc flow changes wth the node moblty. We can derve the followng observatons. Frst, when the moblty speed s low, the utlty of three algorthms s very close. Among them, HRA performs best. Second, as the node moblty speed ncreases, UBARA and SRA decrease very uckly. At last, when the node speed ncreases up to 20 m/s, HRA can stll keep a hgh system utlty, whle SRA and UBARA fall to less than 3. In Fgure 0, we can fnd somethng nterestng. For nelastc flows, HRA exhbts averagely twce hgher utlty than UBARA and SRA. When the node speed ncreases more than 2 m/s, the utlty of SRA and UBARA declnes less than, whch means they cannot meet the rate demand of nelastc flows. On the contrary, HRA stll keeps a hgher utlty more than 2. That means HRA can completely meet the demand of real-tme applcatons, whch are composed of nelastc flows. In all experments, HRA performs best among the three algorthms. Specfcally n nelastc network scenaros, HRA performs much better than SRA and UBARA. These results are due to the followng reasons. One s that HRA tres to allocate resources to the urgent nelastc flow wth hgher prorty, snce nelastc flow s delay-senstve and elastc flow s delay-tolerant. When the channel s n good Inelastc utlty Speed (m/s) HRA UBARA SRA Fgure 0: Inelastc utlty comparson n moble scene. condton, HRA tres to allocate a hgh rate to elastc and nelastc flows accordng to the user utlty. If the channel s n poor condton, HRA decreases the servce rate of elastc flows to guarantee the transmsson demand of nelastc flows. In ths way, the servce rate of nelastc flow can exceed the threshold. Thus, the aggregate utlty of nelastc flows s always mantaned at a hgh level (.e., greater than two), whch fully meet the demand of real-tme applcatons. However,SRAandUBARAalwaystrytoallocateeual bandwdth to elastc and nelastc flows, no matter how poor the network state s. Ths greatly affects the transmsson of nelastc flows, leadng to the overall nelastc utlty less than

11 Internatonal Journal of Dstrbuted Sensor Networks. Therefore, the demand of real-tme applcatons cannot be met by SRA and UBARA. The other reason s that when nodes move at a hgh speed,thenetworktopologyschangngdramatcally.snce SRA and UBARA do not dstngush between elastc and nelastc flows, they may not fnd the global optmum. In all experments, UBARA performs slghtly better than SRA, because UBARA s a utlty-based resource allocaton scheme, consderng the user farness. However, UBARA cannot satsfy the QoS reurement of real-tme applcatons, as t cannot dstngush the two types of traffc. On the other hand, HRA can fnd the optmal tradeoff strategy between resource effcency and user farness for elastc and nelastc flows, although the network states are changng very fast. Wth both consderatons of flow rate and utlty farness, HRA performs much better than SRA and UBARA especally n hghly dynamc and moble networks. In contrast, SRA and UBARA nether explot the heterogeneous characterstc nor optmze the servce rates of nelastc flows, whch lead to the low accumulated system utlty n hybrd network scenaro mxed wth elastc and nelastc flows. To further show the performance of HRA, we evaluate the throughput of HRA and compare t wth the other two algorthms. The smulaton results are shown n Fgure. Whenthenumberofusersslessthan8,SRAandUBARA are very close and ther throughput s hgher than 2 Mbps. However,whenthenumberofusersslargerthan0,the throughputofsraandubaradeclnesucklybecauseof the absence of farness consderaton. When the user number reaches 8, the throughput of SRA s below Mbps. In all experments, HRA acheves the hghest aggregate throughput among the three algorthms. The explanaton for the above results s straghtforward: HRA can guarantee the utlty farness among elastc and nelastc flows, whle the other two algorthms cannot. Fnally, we evaluate the farness ndex of HRA, whch s a partcularzaton of the well-known Jan farness ndex proposed by Jan et al. n [35]. The dfference s that the farness ndex here consders both elastc and nelastc flows, and we call t mxed farness ndex (MFI). From Fgure 2,we observe that HRA has a better farness ndex than both SRA and UBARA. When the user number s eual to, the three algorthms all acheve the same farness ndex close to. They do not need to allocate rate among dfferent users f the user s only one; therefore, the farness ndex must be. On the other hand, f users ncrease, they need to allocate extra bandwdth to other users, so the farness ndex wll be reduced. When the number of users exceeds 9, the gap between HRA and the other two algorthms becomes larger. The above smulaton results ndcate that, compared wth SRA and UBARA, HRA can allocate resources more farly and more effcently n a dstrbuted manner, whle mantanng a hgher system throughput for real-tme flows. In fact, HRA can manage the tradeoff between elastc and nelastc flows, whch guarantees the utlty farness among varous applcatons. More mportantly, the smulaton confrms that the flow control algorthm proposed n ths paper can provde an effcent utlty far resource allocaton n heterogeneous Average throughput (Mbps) Farness ndex of flows Number of users HRA UBARA SRA Fgure : Average throughput of network users Number of users HRA UBARA SRA Fgure 2: Farness ndex of network users. networks, not only sutable for common elastc data traffc but also applcable to nelastc real-tme applcatons. 7. Conclusons In ths paper, we have developed a heterogeneous resource allocaton algorthm (HRA) for wreless ad hoc networks, supportng both elastc and nelastc traffc. The proposed algorthm can allocate resource for heterogeneous ad hoc networks, whch may contan dverse node types and execute varous tasks. Besdes rate allocaton, utlty farness s also ntegrated nto the framework to provde a good performance

12 2 Internatonal Journal of Dstrbuted Sensor Networks balance among varous applcatons. We have shown that HRA algorthm converges fast and can acheve the global optmum startng from many dfferent network condtons, such as elastc network scenaro, nelastc network scenaro, and hybrd network scenaro. As hghlghted, HRA can manage the tradeoff between elastc and nelastc traffc, whch guarantees the utlty farness among competng flows. Compared wth the other two algorthms, HRA can allocate resource more farly and more effcently, whle mantanng a hgher network throughput for the system. Therefore, our algorthm s well suted for heterogeneous ad hoc networks. Moreover, even though the development of ths research s n the ad hoc network envronment, our algorthm can be generally extended and appled to wreless sensor networks duetothegoodscalabltyoftheframework. Conflct of Interests The authors declare that there s no conflct of nterests regardng the publcaton of ths paper. 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