Distributed Network Resource Allocation for Multi-Tiered Multimedia Applications

Transcription

1 Dstrbuted Network Resource Allocaton for Mult-Tered Multmeda Applcatons Georgos Tychogorgos, Athanasos Gkelas and Kn K. Leung Electrcal and Electronc Engneerng Imperal College London SW AZ, UK {g.tychogorgos, a.gkelas, Abstract The contnuously growng number of multmeda applcatons n current communcaton networks hghlghts the necessty for an effcent resource allocaton mechansm to capture the unque characterstcs of mult-tered multmeda applcatons and allocate network capacty n an effcent way. Ths paper examnes the problem of sharng the network throughput under the exstence of nelastc traffc flows that follow a multtered utlty functon. Frst, the concept of mult-sgmodal utltes s ntroduced n order to descrbe user satsfacton, then, the mplcatons of the use of such utltes are dscussed for two dfferent allocaton polces; the bandwdth-proportonal and the utlty-proportonal farness allocaton polces. In the former case, the ntrnsc reasons of possble network oscllatons are analyzed n detal and a heurstc to overcome such stuatons s proposed. In the latter one, where such oscllatons are not possble, effcent ways to calculate a closed form soluton for the optmal rate allocaton are descrbed. Moreover, a novel mathematcal representaton of such a mult-sgmodal utlty s presented and closed form solutons for a number of applcaton types are calculated. Fnally, the effcency and robustness of the proposed algorthms s evaluated by smulatons for dfferent network topologes and compared aganst other work n lterature. I. INTRODUCTION The end-to-end communcaton and resource allocaton servces n current communcaton networks are provded by Transport layer protocols such as TCP, whose varous extensons have been shown to mplctly solve a resource allocaton optmzaton problem [] where all applcatons have been modelled usng concave utlty functons. Although ths was a vald assumpton n the past, the traffc generated by current applcatons has such Qualty of Servce QoS requrements that must be modelled by non-concave functons. Therefore, exstng resource allocaton schemes provde suboptmal solutons that may sgnfcantly affect both network performance and user experence. More specfcally, network traffc can be classfed nto two categores: elastc and nelastc []. Elastc applcatons nclude fle transfer FTP, emal, network management SNMP and Web access HTTP, where user satsfacton s modelled usng logarthmc and other concave utlty functons [] e.g. U x = logx. Inelastcty usually characterzes real-tme applcatons such as Vdeo Streamng, Teleconferencng, Voce over IP VoIP, Stock Tradng etc. where non-concave utltes of sgmodal shape are typcally used [][][]. The sole use of concave functons had lttle effect n the resource allocaton n prevous decades, snce elastc applcatons were responsble for almost all the traffc. In current networks though, where the majorty of the traffc s generated by nelastc real-tme applcatons [], such an assumpton may lead to sgnfcant msuse of resources whch can prove the use of TCP mpractcal []. Network Utlty Maxmzaton NUM [], contrary to the resource allocaton algorthm n TCP, can dstngush between elastc and nelastc applcatons by choosng dfferent utlty functons for each one. Snce the semnal work of Kelly et al. [], there have been several peces of work that cultvated a deep understandng n the ways that optmzaton theory can be utlzed n solvng varous convex resource allocaton formulatons n a dstrbuted way. Interested readers are referred to [] and the references theren for a complete overvew of convex network resource allocaton methods and references [][0][] and [] that ntroduce the use of sngle sgmodal utltes and examne the effects of non-convexty of the resultng optmzaton problem on the development of a dstrbuted algorthm to solve t. The sngle sgmodal utlty functons were ntroduced to model multmeda applcatons but as technology advances they may not be sutable to model many state of the art multmeda applcatons. Several vdeo streamng applcatons used nowadays offer servces at dfferent qualty levels wth each level havng dfferent bt-rate requrements and offerng dfferent Qualty of Experence QoE for the user. For example, an onlne vdeo content provder offers four dstnct levels of vdeo qualty e.g. low, medum, hgh, ultra hgh based on the vdeo resoluton and bt rate. Each qualty opton represents a dfferent level of user satsfacton. Moreover, for a specfc vdeo resoluton the allocated bt rate affects user satsfacton. For example, f low resoluton s chosen, the ncrease of bt rate above a certan level wll not result n sgnfcantly better vsual results snce the resoluton s too low for a vsble mprovement. Therefore, user satsfacton at ths qualty level s saturated and further ncrease can only be a result of the transton to a hgher resoluton profle. Such mult-tered multmeda applcatons can not be modelled satsfactorly by sngle sgmodal utltes. Pror research has shown [] that the resultng bandwdth allocatons for tradtonal NUM approaches follow the so-called bandwdth-proportonal farness. Whle ths type of farness seems to perform well when all users follow the same utlty, ths approach s responsble for some contradctory behavor n cases that users have dfferent QoS needs,.e. when they follow dfferent utltes. More specfcally, a bandwdth-

2 proportonal far optmzaton algorthm favors users wth low demand,.e. those wth rapdly ncreasng utlty functon snce ths leads to a larger ncrease n the aggregate utlty than when allocatng to users wth hgh demand,.e. wth small value of utlty dervatve. To resolve ths contradctory behavor, authors n [] defne a new type of farness, called utlty-proportonal farness. Accordng to that, a feasble bandwdth allocaton vector x =[x,x,...,x R ]T s utlty proportonal far, f for any other feasble vector x the followng condton holds: r R x r x r U r x r 0. The utlty-proportonal farness can be acheved f the utlty functon of each user s transformed accordng to: xr U r x r = m r U r y dy, m r x r M r, where m r and M r are the mnmum and maxmum transmsson rates for user r respectvely, and the objectve functon of the NUM problem s changed to r R U r x r. The most ntutve, yet very challengng, soluton to resolve the neffcency of sngle-sgmodal utltes s the use of multsgmodal functons. Mult-sgmodal utltes, such as the one shown n black at the top plot n Fgure, are capable of capturng the step-lke behavor of user satsfacton wth respect to the varous qualty levels of modern vdeo applcatons. The development of approprate mult-sgmodal utltes that can capture the QoS/QoE characterstcs of the underlyng applcatons and the extenson of NUM to ncorporate such utltes are the man motvaton behnd our work. The organzaton of the paper n terms of the contrbuton of each secton s as follows. Secton II ntroduces the multsgmodal utlty to express user satsfacton n mult-tered multmeda applcatons and descrbes how ths affects the tradtonal NUM framework. Secton III examnes the ncorporaton of such functon whle tryng to allocate resources accordng to. Ths ncludes the mpact of such a choce on the contnuty propertes of the optmal rate allocaton functon, descrbes a detaled procedure to determne all these dscontnuty ponts and proposes an effcent heurstc algorthm n order to resolve network oscllatons, when they occur. Secton IV dscusses the changes necessary to the tradtonal NUM framework n order to be able to allocate resources accordng to the polcy. Secton V proposes a novel tangentbased mathematcal representaton of a mult-sgmodal utlty functon and analyzes how ths famly of utlty functons can lead to the approxmaton and calculaton of closed form solutons n BFP and, respectvely. Secton VI presents extended smulaton results of the proposed algorthms n varous topologes and Secton VII concludes our work. II. NUM WITH MULTI-SIGMOIDAL UTILITIES A. Propertes of a Mult-sgmodal Utlty Functon Before ntroducng mult-sgmodal utltes, t s necessary to defne a set of propertes that a functon should possess n order to be able to model user satsfacton. Such functons should: P take postve values n the range [0, ]; P be ncreasng functons of the transmsson rate, P be zero when no rate s allocated to a partcular user; P have value for rates above the maxmum rate, r max ; P be contnuous n the range 0,r max. One could argue that a potentally sxth property could be added as well. Ths descrbes the need that all qualty levels,.e. all concave parts, of the utlty to be reachable by a NUM algorthm. In other words, a mult-sgmodal utlty can ndeed model mult-tered applcatons only f all dstnct utlty levels can be optmal selectons under some condtons. Whle ths wll be explaned n more detal later, n Secton III, t s not consdered a requrement for a mult-sgmodal utlty snce the exact shape of a utlty functon s determned dependng on each user s apprecaton of the allocated btrate not affected by the operatonal characterstcs of NUM. B. Network Resource Allocaton wth Mult-sgmodal Utltes Ths subsecton extends the ntal NUM framework [] by allowng the utltes to be mult-sgmodal and dscusses the research challenges that ths mposes to the dstrbuted algorthm, whch wll be answered later n ths paper. Consder a mult-hop network where M nodes act as sources sendng streams of traffc to a set of destnaton nodes usng a set of J lnks. A sngle node can operate as source, destnaton or even as relay node that just forwards traffc to ts neghbours. When a source node sends traffc at a rate r, t enjoys utlty U r. It s assumed that all lnks n the network are wred, vector C =[C,C,,C J ] T contans the capacty of each lnk and r =[r,r,,r M ] T ncludes the transmsson rates of all sources. The optmzaton problem descrbng the Network Resource Allocaton NRA problem s: Problem Π p NRA : Fnd the optmal rate vector r max r M U r s. t. = M α,j r C j, = lnks j where routng coeffcent α,j s f user sends traffc through lnk j and 0 otherwse. We assume that the routng matrx A, contanng all routng coeffcents α,j, s known a pror and consdered fxed throughout the optmzaton process. The rates r 0, [,M], represent the transmsson rates of the respectve source nodes. Note that U r represents a transformaton of U. As mentoned later, U r =U r for bandwdth-proportonal farness but not for utlty-proportonal one. Problem Π p NRA can be solved usng Dualty Theory by constructng ts dual problem and tryng to solve the prmaldual par of problems n a dstrbuted way []. Dual problem varables are the so-called lagrange multplers and represent the prce that user has to pay to send each of the r unts of traffc through lnk j. Followng the analyss n [][], t s evdent that each user s tryng to maxmze ther Net Utlty and thus the optmal resource allocaton for user s: r =argmax { NU r =U r r },

3 where = J j= α,j j. Equaton can be used to calculate the optmal rate of user for a gven prce vector. The optmal value of the dual varables j, j [,J], can be calculated teratvely usng a gradent method, such as the Gradent Descent [], M j t += j t s t C j α,j r. s t s the step sze of the method at tme t and affects the convergence speed and dstance from the true optmum []. Equatons and consttute a jont prmal-dual dstrbuted algorthm of NUM, whch can converge to an optmal soluton, even n the case of non-concave utltes such as sngle-sgmodal, as long as s contnuous around the optmal prce vector []-[0]. The propertes of equatons and n the case of mult-sgmodal utltes wll be dscussed n the remander of ths paper. The optmal soluton of for a partcular user s also the optmal rate for ths user for Problem Π p NRA. Ths rate s at a pont where the dervatve of the objectve functon dmnshes [], whch leads to: r =U, where U s the nverse frst dervatve functon. The two resource allocaton polces examned n ths paper dffer n the calculaton of the nverse of the frst dervatve. In U =U r and therefore the calculaton of the nverse of the frst dervatve s possble only for utltes whose dervatve s a functon. In any other case, there mght be multple optmal rates for a sngle aggregate prce. Ths s the nherent reason for the exstence of oscllatons n the rate allocaton process, as dscussed later n ths paper. On the other hand, the utlty functon n can be nverted allowng the calculaton of closed form solutons for. III. MULTI-SIGMOIDAL UTILITIES IN Ths secton provdes a detaled analyss of the effect of the use of mult-sgmodal utltes n the resource allocaton process whle preservng bandwdth-proportonal farness. For smplcty, we wll use notaton U r nstead of U r n the remander of ths secton snce U r =U r. A. Dscontnuty In the case of bandwdth-proportonal farness, r s contnuous for all prce vectors f the utlty s ether concave or convex functon of rates whle t s dscontnuous at only one pont for sngle-sgmodal utltes [][]. Equaton shows that r s n essence a functon of the aggregate prce per unt of traffc and does not depend on the ndvdual values of j, j [,J]. Therefore, we wll also refer to r as r, where s the aggregate prce for user. In addton, t turns out that the shape of a utlty functon determnes the dscontnuty ponts of the rate allocaton functon and that the dscontnuty ponts correspond to jumps from one concave regon to another or from one concave regon to zero. Moreover, there s a number of canddate dscontnuty ponts that may or may not appear as dscontnutes of r. = Utlty and Tangents Optmal Rate Allocaton y = max r 0. y = r + β A Utlty Functon and Tangents Rate r * Dscontuty of r * Aggregate Lnk Prce max Ur y = r + β y = r + β Fg. : A mult-sgmodal utlty wth dscontnuty ponts The methodology to dentfy these ponts nvolves the use of lnes that are tangent to the utlty functon U r. Intally, we draw a tangent lne y = αr + β that osculates the utlty functon at two or more ponts. Let r n, wth n =,,..., N and r <r < <r N, be the rates at whch the tangent lne y osculates the utlty functon, such that the tangent lne s graphcally always above the utlty functon. In multsgmodal utltes, a tangent lne such as y can osculate the utlty functon at most at N = K ponts, where K s the number of nflecton ponts n the utlty shape, and there can be at most KK dstnct tangents, n the case where each one of them osculates the utlty functon at exactly two ponts. Usng the example of tangent y we can prove that the canddate dscontnuty ponts are aggregate prces equal to the slopes of these tangent lnes. Theorem III.. If = α, the rates r n, n =,,...,N are all globally optmal rates for user and aggregate prce. Theorem III. shows that functon r has multple values for aggregate prce equal to the slope of the tangent y and the multplcty of the functon at that pont s equal to the number of ponts N that the slope osculates the utlty functon. Regardng the dscontnuty and monotoncty propertes of r t s possble to prove the followng theorems. The proofs of Theorems III.-III. are omtted for brevty but can be found n []. Theorem III.. If = α+δ, where δ s a very small postve constant, then the globally optmal rate r s smaller than the smallest optmal rate for = α,.e r <r. Theorem III.. If = α δ, where δ s a very small postve constant, then the globally optmal rate r s larger than the largest optmal rate for = α,.e r >rn. Theorem III.. The optmal rate functon of user, r, s a decreasng functon of. Apart from the dscontnuty of r around the ponts determned by the tangents at the utlty functon, these theorems mply that rates n the range r,rn can never be globally optmal rates and therefore r wll jump from

4 r N to r. Moreover, there wll be a maxmum value for, let max, above whch the optmal rate wll be zero. In other words, r has a postve value for 0 max and s zero for aggregate prces max. Ths maxmum nonzero aggregate prce max s called maxmum wllngness to pay for user and s a dscontnuty pont of r for all sgmodal utltes. To calculate max we can use the same procedure as for sngle-sgmodal utltes [0]. It s evdent from the above that every tangent at two or more ponts of the utlty functon represents a canddate dscontnuty pont of functon r. Each one of these ponts represents a jump from one hyperbolc tangent component to another, whle the dscontnuty pont around max represents a jump from a hyperbolc tangent component to zero rate. The latter pont wll always appear n the rate functon but the rest depend on ther relatve value compared to max. For example, f max s smaller than all the other canddate dscontnuty aggregate prces, then none of them wll appear and there wll be only one dscontnuty pont, max. The maxmum number of dscontnuty ponts are K, as many as the nflecton ponts of the utlty. Ths can happen f there are K dstnct tangent lnes, each one touchng the utlty at two ponts that belong to two consecutve hyperbolc tangent components, wth the K th, correspondng to = max, beng graphcally represented by a tangent lne that passes from pont 0, 0 and osculates the utlty functon at ts frst hyperbolc tangent component. To provde an example of the above, the top sub-fgure of Fgure shows a utlty functon wth four dscontnuty ponts, along wth the four tangent lnes responsble for these dscontnuty ponts, whle the bottom one shows the optmal rate r wth the dscontnuty ponts clearly shown. Ths fgure llustrates the connecton between the shape of the utlty functon and the dscontnuty ponts of the prcebased rate functon r. Moreover, t llustrates that r conssts of decreasng contnuous parts and decreasng jump dscontnuty ponts. Commentng on the feasblty of all K sgmodal components to be selected as optmal choces, t s evdent that ths s possble only under the exstence of K dstnct dscontnuty ponts,.e. the utlty functon s fully reachable. In any other case, there wll be at least one sgmodal component that s unreachable durng NUM. Based on ths observaton, we can prove the followng: Theorem III.. A mult-sgmodal utlty wll have all levels reachable, and hence wll have the maxmum number of dscontnuty ponts, ff the followng condtons hold: k,k < k,j, j [,...,k ], k [,...,K] k,k < max, k [,...,K], where k,l s the slope of the tangent that osculates the utlty of user at the k th and l th sgmodal component. For an arbtrary mult-sgmodal utlty functon we can calculate all the aggregate prces for whch r s dscontnuous. To ths purpose, we assume the exstence of a tangent that osculates the utlty functon at exactly two ponts, let p and p. After calculatng all such possble tangents and ther Algorthm Calculaton of dscontnuty ponts of r : ctr = K; ctr =; Calculate max, S ; : whle true do : ndex = argmn { S ctr, :ctr } ; : tmp = mn { S ctr, :ctr } ; : f max < tmp then : break; : else : dsc ctr = tmp; ctr = ndex; : end f 0: ctr = ctr +; : end whle : dsc ctr = max; touchng ponts, a canddate dscontnuty pont s the slope of ths lne: c = U p U p. p p Usng these canddate ponts, we create the symmetrc matrx S of sze K K, where S s,s represents the slope of the tangent that osculates the s th and s th concave regon of the utlty. By conventon, we assume that the elements of the man dagonal of matrx S contan some very large postve value. Algorthm can be used to determne whch of these canddate dscontnuty ponts wll actually appear n r. Note that S ctr, :ctr denotes the frst ctr elements of the ctr th row of matrx S. The resultng vector dsc contans the dscontnuty ponts of r. Algorthm s an teratve algorthm that can be run ndependently by each user n order to determne the dscontnuty ponts of ts rate allocaton functon. Note that n case that one of the tangents osculates the utlty functon at more than two ponts, then two or more elements of matrx S wll be equal. B. Oscllatons The dscontnuty ponts calculated by Algorthm play an mportant role n the convergence of the algorthm comprsed of eq. and n the case of mult-sgmodal utltes. When the condton n [] s not satsfed, there can be oscllatons n the network. The phenomenon of oscllaton occurs when a user transmts at an excessve data rate compared to the avalable capacty n one teraton, and then, n the next teraton, the user transmts at an exceedngly low rate. An oscllaton s formed as the repetton of these two events contnues ndefntely, prevents the user from convergng to the optmal transmsson rate and leads to a wder network oscllaton. The oscllaton rates of user are n fact very close to the optmal rates for. Based on ths observaton, we propose the Oscllaton Resolvng Heurstc ORH to assure the convergence of the gradent based dstrbuted algorthm. The Oscllaton Resolvng Heurstc ORH allocates a constant non-zero rate to oscllatng users and removes them from the rest of the optmzaton process, whch contnues for the remanng users n the network. The allocated rate r osc to oscllatng users s equal to the smallest touchng pont of the tangent wth slope equal to the aggregate prce for whch the oscllaton happens. Ths approach assures that no

5 users are restrcted from accessng network resources, contrary to other approaches n lterature. Ths allocated rate satsfes the necessary condtons for optmalty and by selectng the smallest of all the optmal rates for prce we assure that there wll be more resources for the rest of the users n the network, thus leadng to hgher network utlty. The mplementaton of ORH s very smple, requres a smple oscllaton detecton mechansm wth no need for any centralzed coordnaton. Note also that, the ORH does not represent a complete soluton for solvng Problem Π p NRA and does not affect the convergence propertes of the algorthm. In fact, and are responsble for solvng Problem Π p NRA teratvely, whle the ORH s merely part of the process for resolvng an oscllaton that mght occur durng ths process. The ORH leads towards more far resource allocatons compared to mechansms such as the heurstc proposed n []. IV. MULTI-SIGMOIDAL UTILITIES IN By consderng the utlty proportonal farness transformaton of, the problem becomes convex and always satsfes the condton n []. More mportantly ths allows us to calculate a closed form soluton for. Ths stems from the fact that the frst dervatve of the utlty functon can be easly calculated as U r = U r, whch s nvertble as long as the utlty s contnuous and monotonc, whch are both true for any concave utlty and the mult-sgmodal utlty consdered n ths paper. In ths case, the optmal rate s gven by: r = U. As explaned n the next secton, based on, we can calculate a closed form soluton for utltes that satsfy these two propertes. Ths s a sgnfcant advantage of the utlty proportonal farness approach whch leads to the development of algorthms that calculate the optmal soluton even for non-concave utltes and converge sgnfcantly faster than the tradtonal teratve approach. V. A NOVEL MULTI-SIGMOIDAL FUNCTION A. A Hyperbolc Tangent Based Utlty Functon Based on the desred propertes of a mult-sgmodal utlty, presented n Secton II, we propose the use of the followng famly of mult-sgmodal functons: { U r = K } r ck tanh + K, K k= where r s the transmsson rate, c k s the k th nflecton pont, wth c >c > >c K, and s a postve desgn parameter that determnes the steepness of the k th component of the mult-sgmodal functon. K s the number of sngle sgmodal components comprsng the mult-sgmodal functon, each one of them havng a sngle nflecton pont. For example, the mult-sgmodal functon n black n the top plot of Fgure conssts of four hyperbolc tangent components. Hyperbolc tangent functons have been extensvely used n neural networks research area [] but ther convenent propertes make them also applcable wthn the context of mult-tered multmeda applcatons for the followng reasons: They possess the fve propertes descrbed n Secton II. They can be combned to create mult-sgmodal shapes of arbtrary number of rate levels. They can be calbrated usng the nflecton vector c and the steepness vector b to acheve the desred shape. Ther frst dervatve can be easly nverted to calculate the optmal rate allocaton for a specfc prce vector. The hyperbolc tangent functon, tanh x, s a symmetrc, contnuous property P, dfferentable and ncreasng property P functon, whch s centered around ts nflecton pont at r =0and has two horzontal asymptotes, the lnes y = and y =. Each tangent component can be scaled and shfted approprately so that the resultng utlty has values wthn the range [0, ]. The resultng mult-sgmodal functon has horzontal asymptotes the lnes y =0and y =property P. Note that nflecton ponts c k can be used as desgn parameters to create the step-lke behavour of the utlty around the rate values of each applcaton qualty level. Parameters, k =,...,K, can be used to calbrate the steepness of the respectve tangent components. In general, larger values for lead to smoother shapes. In partcular, they can be used to brng U 0 and U r max as close to the bounds 0 and respectvely as necessary, where r max s the maxmum rate above whch the utlty s equal to. Specf- cally, for r =0equaton becomes tanh c k bk, for k =,,...,K. Gven that y = s an asymptote, the equaton wll never be satsfed n the equalty but we can select varables, k {,,...,K}, so that the maxmum error ɛ k of the k th tangent component s bounded. More specfcally, t s possble to calculate an upper bound for each n order to meet property P accordng to c k c k tanh +ɛ k tanh ɛ k and snce tanh s negatve around r =, c k tanh ɛ k. 0 By selectng the component bounds approprately, t s possble to bound the total error ɛ = K k= ɛ k below a maxmum threshold. In addton, t can be shown that the effect of parameter b,.e. the sgmodal component that s closer to the pont r =0, s domnant over the rest and therefore the calculated bound for b s expected to be much tghter for the same error. Workng n the same way, t s possble to calculate the upper bounds for parameters to assure that property P s also satsfed and, as seen later, to mnmze the approxmaton error of the optmal rate. B. Approxmatng the Optmal Rate n The famly of mult-sgmodal utltes descrbed n s a non-concave functon wth multple concave and convex regons. Its frst dervatve s gven by { V r = K } r sech ck, K k=

6 whch s not a - functon snce the same value V corresponds to more than one rates and therefore s not nvertble. Fgure shows the utlty dervatve for the mult-sgmodal example n black of Fgure, whch llustrates that a sngle value of utlty dervatve corresponds to at most K dstnct rates. It s possble however to approxmate these rates effcently. The approxmaton methodology reles on the fact that V n s a summaton of a number of ndependent hyperbolc secant components. Moreover, they are symmetrc, they can be nversed separately, and by takng nto account that the rate that maxmzes Problem Π NU can only be n a concave regon or at zero rate, t s possble to calculate a sngle rate for each component by r c,k=b k sech K b k + c k, where sech s the nverse hyperbolc secant, b k, k =,,...,K, form steepness vector b and nflecton ponts c k, k =,,...,K, form nflecton vector c of user.an addtonal canddate soluton s at r c,k + = 0, whch must be also taken nto account. Consequently, the optmal rate of user for vector wll be the one that yelds the maxmum net utlty,.e. r =argmax {NU r c,k k =,,...,K +}. The use of equaton to approxmate the optmal rate for any prce vector transforms the dstrbuted algorthm descrbed n Secton II to use nstead of. The resultng algorthm comprsed of and s an extenson of the standard gradent algorthm [] and any oscllatons that are lkely to appear due to dscontnutes can be resolved usng the heurstc presented n Secton III-B. The procedure descrbed above has transformed, whch nvolves the soluton of a non-convex optmzaton problem, nto a smple selecton out of K + choces of the rate that maxmzes the net utlty usng. However, snce ths s an approxmaton method, t s necessary to determne the approxmaton error and propose methods to mnmze t. It s easy to verfy from Fgure that the approxmaton error depends on the degree of overlap of the hyperbolc secant components and, moreovoer, t has ts maxmum values at the ntersecton ponts x k of two consecutve components. The effects of ths overlappng can be restrcted effcently. The nflecton ponts of the utlty s sgmodal components are determned by the technology used at the source node and they are assumed that can not be changed. However, there s often more freedom n selectng the steepness parameters of a mult-sgmodal utlty. In such cases, the steepness parameters, k =,,...,K, can be used as desgn parameters to assure that the approxmaton error s small. In ths way, t s possble to calculate some addtonal bounds for the values of the parameters of the utlty functon so that the The mathematcal dervaton of s presented n detal n []. We assume that two hyperbolc secant components c and c are not overlappng f f c x c=f c x c 0 at ther ntersecton pont x c Utlty Dervatve and Hyperbolc Secant Components Approx. { Error 0 0 x 0 Rate Mb/s Utlty Dervatve V r st Hyperbolc Secant Component nd Hyperbolc Secant Component rd Hyperbolc Secant Component th Hyperbolc Secant Component Fg. : Example of a mult-sgmodal utlty dervatve and ts Hyperbolc Secant Components hyperbolc secant components of the utlty dervatve are nonoverlappng. In general, the smaller the values of are, the more concentrated the respectve hyperbolc secant component s around the nflecton pont. Clearly, the choce of for component k affects the range of choces at the neghborng ones and therefore t s not possble to determne analytcally a sngle steepness vector b to assure low approxmaton error. However, t s possble to formulate optmzaton problems that calculate the optmal steepness vector b accordng to varous crtera, such as the maxmum tolerable approxmaton error. Such an optmzaton problem s formulated n []. C. Calculatng Optmal Rate n The exstence of varous types of user applcatons wth dverse QoS requrements complcates the process of calculatng a generc closed form soluton for the optmal rate. It s however possble to derve applcaton-specfc analytcal solutons for n the case of Utlty Proportonal farness. Based on the analyss above, t s possble to derve the optmal rate allocaton for browsng, fle transfer and vdeo streamng applcatons usng the utlty functons suggested n [], [] and []. These optmal rate allocaton functons are demonstrated n Table I. r mn and r max represent the mnmum and maxmum transmsson rates of a user, and parameters α and β are calbraton parameters of the snglesgmodal utlty. The calculaton of analytcal solutons for concave and sngle-sgmodal utltes s relatvely easy and wll be omtted for brevty. However, the calculaton for multsgmodal utltes such as those descrbed n s more complcated and, therefore, wll be descrbed n detal n the remander of ths secton. Eq. conssts of K hyperbolc tangent components that have been scaled and shfted so that the resultng utlty has values n the range [0, ]. Therefore, each of the scaled components s restrcted n a dfferent non overlappng regon. For example, values n the range 0., 0. correspond to the thrd hyperbolc tangent component of the utlty n the top plot of Fgure. Ths mples that a value of utlty belongs to only one of the hyperbolc tangent components, whle the rest

7 TABLE I: The Optmal Resource Allocaton Functon for Wdely Used Types of Applcatons Applcaton Type User Utlty Functon Optmal Rate Allocaton Functon HTTP U r = log r r mn log r max r mn r =rmn r max r mn FTP U r = logr + logr max + r =rmax + Sngle-tered Vdeo Applcaton U r = r +exp αr β = α β log { α Mult-tered Vdeo Applcaton U r = K } K k= tanh xr ck + K r b k = b j atanh K j + + c j of the components have value ether or. To calculate the nverse of, we wrte: { y = K } r c k tanh + K K k= K r c k Ky K = tanh k= r c j Ky K = μ + tanh b j ϕ. Index j represents the ndex of the hyperbolc tangent component that corresponds to the requested pont. Term μ represents the components before j that have value,.e. μ = j, and term ϕ represents the components after j that have value,.e. ϕ = K j. Based on these, becomes: r c k Ky j+=tanh, and by solvng wth respect to r, we fnd that: r y =b j arctanh Ky j + + c j. Moreover, by combnng and we calculate the optmal rate allocaton of user wth respect to the aggregate network prce for as = b j arctanh K j + +c j. r Eq. s a closed form of the optmal rate allocaton for a specfc aggregate prce when the utlty functon has multsgmodal shape,.e. when t models mult-tered multmeda applcatons. In order to evaluate, t s necessary to determne the hyperbolc tangent component that the specfc aggregate prce corresponds to,.e. determne the value of j. Accordng to the frst order necessary condton for optmalty [], at the optmal soluton U r =, whch leads to U r =, whch mples that the regons of utlty values can be easly mapped to regons of aggregate prce values. Specfcally, for a mult-sgmodal utlty wth K nflecton ponts of the form descrbed n, the hyperbolc component j s wthn regon [ j K, j ] K, wth j =,,...,K, of the utlty values and corresponds to prces n the regon K j, K j,wth K 0. In other words, dependng on the value of the aggregate prce, we can determne the component that the optmal rate belongs to and specfy j. For example, Table II shows the utlty value regons and ther respectve aggregate prce regons for a mult-sgmodal utlty gven by for K =. Note, that aggregate prces wthn [0, correspond to U =and therefore to component j = K. By splttng the summaton of hyperbolc tangent components and calculatng the nverse of only one of them, we create some dscontnutes on the boundares of the aggregate prce regons. These dscontnutes are caused by the fact that arctanhx ± when x ± respectvely. Specfcally for the dscontnutes appear on the ntermedate boundares snce, by defnton of the utlty functon, r 0 = rmax and r =0. For example, n the case of a mult-sgmodal utlty wth K =, the dscontnutes exst for =, =and =. In order to handle these dscontnutes and assure contnuty of the rate allocaton functon, one could assgn an approxmaton of the optmal rate for these boundary cases based on neghborng rate values. In other words, the optmal rate r for the boundary aggregate prces can be calculated by a transformaton of the form: r = f r,r +, where = ɛ, + = +ɛ and ɛ s a very small postve constant. A potental approach could be a weghted average of the rates for prces and + accordng to: r = w r +w r +, w + w where w and w are weghtng parameters wth w k > 0, k {, }. The relatve values of the parameters w and w can be used to select a rate value that s closer to one or the other dscontnuty end. For example, w >w mples that r wll be closer to r than to r +. For the numercal results later, we wll use w = w = and ɛ =0 to calculate the optmal rate for boundary aggregate prces. Ths weghted averagng of neghborng ponts for the estmaton of the optmal rate s a way to make a contnuous functon of the aggregate prce. Ths contnuty for all aggregate prces also mples that when usng utlty proportonal farness all rates wthn the range [ r mn,r max] Component Utlty Value Regogon Aggregate Prce Re- [ ] 0, [, K, ] [ K, K, ] [ K, K, ] [ 0, TABLE II: Tangent Components and the Respectve Utlty and Aggregate Prce Value Regons for a Utlty wth K =

8 Fg. : Example of a network topology wth a sngle bottleneck lnk are feasble contrary to the bandwdth proportonal farness case, where only a small part of the total rate range s feasble see bottom sub-plot n Fgure. Ths shows that the rate allocaton functon has the robustness and elastcty to adjust to any changes n the lnk prces and take advantage of the full range of the avalable rate regon n order to maxmze user satsfacton n the network. VI. SIMULATION RESULTS The optmzaton framework presented n the prevous sectons was smulated n MATLAB to study ts performance. Several examples where network oscllatons occurred were examned to evaluate the effcency of ORH to stablze the network n the case of and llustrate the ablty of to provde stablty and lead to far allocaton of resources when heterogeneous applcatons compete. The smulaton results are organzed n two sectons; a sngle bottleneck network case and a multple bottleneck network one. The smulaton setup ncluded a varety of types of applcatons, ncludng FTP, HTTP and multmeda applcatons. Ths dctated the use of dfferent utlty functons, concave or mult-sgmodal, accordng to the type of applcaton. All multmeda applcatons were modelled usng multsgmodal utltes accordng to for dfferent nflecton and steepness vectors. Furthermore, the calculaton of the steepness parameter vector b for each mult-sgmodal utlty was done by solvng the optmzaton problem n [] for a maxmum approxmaton error σ =0 usng the Global Optmzaton Toolbox n MATLAB, and all utltes where desgned so that ther maxmum transmtted rate r max s 0Mb/s and U r max =for all source nodes. A. Sngle bottleneck lnk Fgure shows an example topology of a network that has a sngle bottleneck lnk. The traffc flows are desgnated by a dfferent lne style. The capactes of lnks and where selected to be larger than r max, whle the capacty of lnk was chosen so that t s nadequate for all sources to transmt at ther maxmum rate r max, thus creatng a bottleneck. Source nodes and have mult-sgmodal utltes of four hyperbolc tangent components whle sources and represent HTTP and FTP traffc respectvely and are modelled usng logarthmc utlty functons []. Several dfferent values for the capacty of the bottleneck lnk were used n order to examne cases of network oscllaton or stablty. In essence, by ncreasng the bottleneck lnk capacty, one can decrease 0 Rate Mb/s Rate Mb/s 0 Source Iteratons Source Iteratons Rate Mb/s Rate Mb/s 0 Source Fg. : Convergence of Rates Iteratons Source Iteratons the optmal lnk prce due to the avalablty of more resources and the weakenng of the competton among users. Fgure shows the convergence of the rates of the source nodes for bottleneck capacty C =Mb/s. All lnks apart from lnk have zero prce and =0.0. Ths happens as lnk saturates and ts lnk prce ncreases. In, user starts oscllatng and the ORH heurstc s nvoked. The algorthm allocates some rate to User and removes hm from the rest of the optmzaton process n order to resolve oscllatons and assure convergence of the dstrbuted algorthm. As shown n blue, the oscllaton of user also causes other users to oscllate but at a smaller extend. On the other hand, when s appled all users are shown to converge smoothly wthout any oscllatons. The top plot n Fgure shows the convergence of the aggregate utlty n the network,.e. the summaton of the ndvdual utltes llustratng the effect of the oscllaton of User n the objectve functon of the optmzaton problem. Ths nstablty s resolved successfully by ORH. The bottom plot n Fgure shows that when the oscllatng user s removed, the remanng users compete for the rest of the network resources whch leads to hgher ndvdual utltes for these users. On the other hand, there are no oscllatons when s used and the resultng rate allocaton leads to exactly the same degree of satsfacton for all sources. Therefore, the utlty of all users s depcted usng a sngle lne n red. B. Multple bottleneck lnks Fgure llustrates a topology wth three bottleneck lnks where eght flows are competng for network resources. The dfferent traffc flows are dstngushed by a dfferent lne style and colour combnaton. Lnks, and are the bottlenecks whle the rest are suffcently large to accommodate traffc even at the maxmum rate r max. Nodes, and measure user satsfacton usng concave utltes, whle the remanng fve flows model mult-tered multmeda applcatons. Fgure shows the convergence of the aggregate objectve functon when and are appled. As the blue lne shows,

9 Objectve Functon User Utltes Convergence of the Objectve Functon Iteratons Convergence of Utltes Iteratons U, U, U, U n U, U n U n U n Fg. : Convergence of Objectve Functon and Indvdual User Utltes the ORH can successfully resolve user oscllatons that occur durng the optmzaton process whle the manages to prevent any oscllatons and allow the dstrbuted optmzaton algorthm to converge smoothly to the optmal rates even for ths more complex network scenaro. The convergence of rates was omtted n ths case as t resembles the behavour n the prevous smulaton example, however, t s worth notng that, n general, gves prorty to users wth hgher rate requrements whle allocates more rate to users that are satsfed easer n an attempt to acheve hgher aggregate utlty n the network. Ths behavor can be verfed n all the aforementoned fgures. VII. CONCLUDING REMARKS Ths paper studed the resource allocaton problem motvated by the fast growng number of multmeda applcatons n current communcaton networks. We ntroduced the concept of mult-sgmodal utltes and proposed effcent methods to overcome any challenges that ther use mposes for two dfferent allocaton polces, and. We proposed a novel mathematcal representaton of such utlty functons and a dstrbuted algorthm to optmze the allocaton of bandwdth by explotng the specal structure of the utlty functon. Fnally, the performance and robustness of the proposed framework were evaluated through extensve smulatons for varous network topologes. Objectve Functon... Convergence of the Objectve Functon Iteratons Fg. : Convergence of rate allocaton wth oscllaton REFERENCES [] S. Low, A dualty model of tcp and queue management algorthms, IEEE/ACM Transactons on Networkng, vol., no., pp., Aug. 00. [] W. Stallngs, Data and Computer Communcatons, th ed. Pearson Custom Publshng, 00. [] S. Shenker, Fundamental desgn ssues for the future nternet, IEEE Journal on Selected Areas n Communcatons, vol., no., pp., September. [] J.-W. Lee, R. R. Mazumdar, and N. B. Shroff, Non-convex optmzaton and rate control for mult-class servces n the nternet, IEEE J. on Selected Areas n Commun., vol., no., pp. 0, August 00. [] C. Lu, L. Sh, and B. Lu, Utlty-based bandwdth allocaton for trpleplay servces, n Unversal Multservce Networks, 00. ECUMN 0. Fourth European Conference on, feb. 00, pp.. [] Csco, Vsual networkng ndex: Global moble data traffc forecast update, 000, ns/ns/ns/ns0/ns/whte paper c-0.pdf, Tech. Rep., February 0. [] G. Tychogorgos, A. Gkelas, and K. Leung, A non-convex dstrbuted optmzaton framework and ts applcaton to wreless ad-hoc networks, Wreless Communcatons, IEEE Transactons on, vol., no., pp., September 0. [] F. P. Kelly, A. Maulloo, and D. Tan, Rate control n communcaton networks: Shadow prces, proportonal farness and stablty, Journal of the Operatonal Research Socety, pp.,. [] M. Chang, S. Low, A. Calderbank, and J. Doyle, Layerng as optmzaton decomposton: A mathematcal theory of network archtectures, Proceedngs of the IEEE, vol., no., pp., jan. 00. [0] J.-W. Lee, R. Mazumdar, and N. Shroff, Nonconvexty ssues for nternet rate control wth multclass servces: stablty and optmalty, n INFOCOM 00, vol., March 00, pp.. [] P. Hande, S. Zhang, and M. Chang, Dstrbuted rate allocaton for nelastc flows, Networkng, IEEE/ACM Transactons on, vol., no., pp. 0, December 00. [] G. Tychogogos, A. Gkelas, and K. K. Leung, A new dstrbuted optmzaton framework for hybrd ad-hoc networks, n IEEE GlobeCom 0 Workshop on Heterogeneous, Mult-Hop, Wreless and Moble Networks, Houston, USA, December 0. [] W.-H. Wang, M. Palanswam, and S. H. Low, Applcaton-orented flow control: Fundamentals, algorthms and farness, IEEE/ACM Transactons on Networkng, vol., no., pp., December 00. [] D. P. Bertsekas, Nonlnear Programmng. Athena Scentfc,. [] S. Boyd and L. Vandenberghe, Convex Optmzaton. Cambrdge Unversty Press, 00. [] Optmzng resource allocaton for mult-tered multmeda applcatons, Onlne, Tech. Rep., July 0. [] J. Drakapoulos, Mult-sgmodal neural networks and backpropagaton, n Artfcal Neural Networks, Fourth Internatonal Conference on, June, pp Fg. : Example of a network topology wth multple bottleneck lnks 0