Value-aware Resource Allocation for Service Guarantees in Networks

Transcription

1 Vaue-aware Resoure Aoation for Servie Guarantees in Networks Parima Parag, Srinivas Shakkottai and Jean-François Chamberand Department of Eetria and Computer Engineering Texas A&M University, Coege Station, TX 778-8, USA Emai: {parima, sshakkot, Abstrat The traditiona formuation of the tota vaue of information transfer is a muti-ommodity fow probem Here, eah data soure is seen as generating a ommodity aong a fixed route, and the objetive is to maximize the tota system throughput under some onept of fairness, subjet to apaity onstraints of the inks used This probem is we studied under the framework of network utiity maximization and has ed to severa different distributed ongestion ontro shemes However, this idea of vaue does not apture the fat that fows might assoiate vaue, not just with throughput, but with inkquaity metris suh as paket deay, jitter and so on The traditiona ongestion ontro probem is redefined to inude individua soure preferenes It is assumed that degradation in ink quaity seen by a fow adds up on the inks it traverses, and the tota utiity is maximized in suh a way that the quaity degradation seen by eah soure is bounded by a vaue that it deares Deouping soure-dissatisfation and ink-degradation through an effetive apaity variabe, a distributed and provaby optima resoure aoation agorithm is designed, to maximize system utiity subjet to these quaity onstraints The appiabiity of our ontroer in different situations is iustrated, and resuts are supported through numeria exampes I INTRODUCTION Reent years have seen an enormous growth in demand for Internet aess, with appiations ranging from persona use to ommeria and miitary operations Severa of these appiations are sensitive to a quaity of paket deivery For instane, athough arhiving data transfer an toerate ong deays, voie over Internet protoo VoIP is very sensitive to ateny Between these two extreme exampes ies a spetrum of appiations with varying servie requirements, eg eetroni ommere, video onferening and onine gaming A these appiations require the aoation of enough network resoures for satisfatory performane The design of effiient network ontro systems demands that end-user vaue be taken into onsideration when aoating resoures The Internet arhiteture is buit around the onept of a fow, whih is a transfer of data between a fixed souredestination pair How do we quantify the vaue of suh a fow? The assia formuation of the tota vaue of information transfer is a muti-ommodity fow probem, in whih eah data soure is seen as generating a ommodity aong a fixed route, and the objetive is to maximize the tota throughput under some onept of fairness, subjet to apaity onstraints Researh was funded in part by NSF grants CNS-9, CCF-776, DTRA grant HDTRA-9--, and Qatar Teeom, Doha, Qatar of the inks used [] [] If the fow from soure r has a rate x r and the system utiity assoiated with suh a fow is represented by a onave, inreasing funtion U r x r, the objetive is max U r x r st y, L where S is the set of soures, L the set of inks, the apaity of ink L Aso et R be the routing matrix with R r =if the route assoiated with soure r uses ink The oad on ink is y = R rx r Note that we refer to fows and soures interhangeaby; if there are mutipe fows between a soure and a destination, we simpy give them different names This is a onvex optimization probem that is we studied [] [] under the framework of network utiity maximization This approah to network resoure aoation often an be used to deompose the probem into severa sub probems, eah of whih are amenabe to distributed soution This soaed optimization deomposition framework has yieded a rih set of ontro shemes and protoos, whose arhitetura impiations are disussed in detai in [] For exampe, there is a strong onnetion between the so-aed prima soution to the utiity maximization probem and TCP-Reno [6], [7] haraterized in [8], [9] Simiary, one an obtain onnetions between TCP-Vegas and the dua soution of the probem [] The same approah has been taken in the design of severa new protoos suh as Saabe TCP [], [] that aows saing of rate inreases/dereases based on network harateristis, FAST-TCP [] meant for high bandwidth environments, TCP-Iinois [] that uses oss and deay signas to attain high throughput, and TRUMP [] a mutipath protoo with fast onvergene properties However, there is a growing reaization that throughput annot be onsidered as the soe vaue metri As mentioned above, in appiations suh as voie as, the data is rendered useess after a ertain deay threshod Thus, simpy ensuring that the ink apaity is not exeeded is not suffiient to provide vaue in this senario how do we ensure that the user is not dissatisfied with the quaity of servie? In many ases the quaity of data transfer over a ink dereases with oad For exampe, metris suh as the deay and the jitter experiened by pakets as they pass through a queue depend on the tota oad on the ink Suh quaity degradation might

2 aso add up over mutipe hops Indeed, the deay experiened by pakets in a fow is the sum of the deays experiened over eah hop taken One we have a ear oneption of quaity degradation as a funtion of ink oad, we ask the foowing question: an we design a simpe distributed agorithm for fair resoure aoation under whih eah users quaity is no worse than a vaue that she or he deares? We need to redefine the traditiona ongestion ontro probem to inude individua soure preferenes We denote the degradation in quaity of ink with oad y by a onvex inreasing funtion V y, and assume that degradation in ink quaity seen by a fow adds up on the inks it traverses In addition, we assume that the quaity degradation is inherent to a ink, and is identia for a fows sharing the ink Thus, there are no priorities for any partiuar fows We now maximize utiity in suh a way that the tota degradation seen by eah soure r is required to be bounded by a vaue σ r Thus, the modified objetive is max U r x r x r,y=rx subjet to R r V y σ r, where we assume that im y V y = Note that this is a onvex optimization probem where the quaity degradation on eah route is bounded Whie some of our objetives might be ahieved by existing shemes suh as DiffServ [6], they often require ompex priority management methods and per-fow information to be maintained at routers In this paper, our objetive is to design a simpe distributed ontro sheme that an ahieve this goa without maintaining perfow information or prioritizing ertain pakets at intermediate hops We overview our main ontributions beow, with detais in the setions foowing Main Resuts Cassia optimization-deomposition tehniques usuay yied a soure-rate responds to ink-prie type of ontroer [], [] [], wherein eah ink s prie inreases with the ink-oad in order to prevent the ink-apaity from being exeeded As the ink-prie inreases, soures ut down their transmission rates, where the aggressiveness of the soure ontroer is determined by its utiity funtion However, the soution to our probem has remained eusive due to strong ouping between the quaity seen at soure that requires a hard guarantee, and the ink quaity degradation that depends on the ink-oads aong its route We first present some exampes of what the quaity degradation funtions might ook ike in Setion II, and disuss exampes that we use ater in the paper We then proeed to provide a entraized soution to the probem in Setion III We deveop two agorithms to this end A prima agorithm is presented in Setion IV Our main ontribution, a dua agorithm is presented in Setion V, and stems from the reaization that it is possibe to deoupe the QoS guarantees at soures and ink quaity degradation using effetive apaity that is based on the ink-prie and user-dissatisfation One we hoose an effetive apaity for a ink, the quaity degradation depends soey on this hoie, and not on the atua ink-oad Eah soure deares its dissatisfation to the inks it uses based on the differene between the quaity it sees and what it requires The inks set a prie based on the differene between oad and effetive apaity, whih in turn depends on inkoad and tota user dissatisfation This deouping of ink-oad and effetive apaity appears to have the orret properties to aow distributed soution Finay, soures use route-prie the sum of a ink-pries on a route to determine the soure-rate We prove that the agorithm is indeed apabe of soving our resoure aoation probem using Lyapunov tehniques [7] We iustrate the optimaity of the soution reahed by our distributed dua ontro sheme in some seeted ases by direty soving the optimization probem We simuate the ontroer and ondut experiments on reaisti topoogies in Setion VI We onude with pointers to future work in Setion VII II EXAMPLES OF QUALITY DEGRADATION FUNCTIONS We first begin with some ideas of ink quaity degradation with oad We make severa assumptions on the properties of ink quaity degradation funtions Mathematiay, one an ist them in the foowing manner If the tota sum-rate on any ink is y = R rx r then the quaity degradation funtion V y is non-negative and onvex inreasing in ink-oad y, the tota quaity degradation seen by fow r is R rv y ie, quaity degradation sums up over mutipe hops, and finay, in our deterministi approah to the probem, we have an impiit assumption that the servie proess at one ink does not impat the arriva proess at the sueeding ink Whie the above assumptions resut in mathematia simpiity of our optimization probem, we beieve that they provide aeptabe modes of quaity degradations in ommuniation systems with queues Beow, we justify our assumptions with some ommon exampes of quaity degradation funtions A Average deay in M/M/ queues For an M/M/ queue with arriva rate x and servie rate, x the expeted waiting time in the queue is x for a stabe queue, that is when x < In this ase, one an write quaity degradation funtion to be the expeted waiting time for any paket in the queue That is, x V x = x We note that the quaity degradation funtion is aways positive and inreases from to when x ranges in [, Further, V x = x >, V x = x >

3 Hene, the funtion is onvex inreasing B Deay-rate of a fuid buffer with on-off servie proess Consider a singe server queue with onstant-rate arriva x and a two-state On-Off servie proess where on and off times are exponentiay distributed with rates µ and λ respetivey When servie is on and the buffer is non-empty, it is servied at a onstant rate R > x suh that x < R λ λ+µ It an be shown [8] that the probabiity of buffer exeeding a threshod z is exponentiay dereasing and a possibe quaity degradation funtion in this ase oud be the inverse of this deay-rate One an write down this deay-rate expiity in terms of the above parameters as V x = im z og Pr{L > z} z = If we denote R λ λ+µ by then, one an write x λ x µ+λ V x = x λ x µ R x First, we notie that V x is aways non-negative in the interva [, Seond, we see that when x approahes, the vaue V x diminishes to zero Anaogousy, when x approahes, the vaue V x tends to infinity Additionay, V x = µ + λ V x = + µ λ x µ λµ + λ x > >, Hene, one an onude it is onvex inreasing Reent resuts [9] suggest that under appropriate onditions, even when pakets of a fow traverse from one queue to the next in a network, the deay seen in eah queue is independent of the others III CENTRALIZED RESOURCE ALLOCATION We start by deveoping ideas on how to sove our resoure aoation probem in a entraized fashion and reating mode networks that we wi use as exampes to iustrate the performane of different ontro oops throughout the paper We rea our optimization probem, repeated here for onveniene, max U r x r subjet to the onstraints y, L R r V y σ r, r S x r, r S Consider the ase, where utiity funtions assume unbounded negative vaues when x r =and quaity degradation funtions grow unbounded when sum-rate y approah Then, eary x r > and y < for optima soution Let x r be a feasibe point and there exist onstants w r suh that U rx r w s R s R r V R i x i =, r S i S σ r w r R r V i S R i x i =, r S, then x r is a goba maximum and if U r is strity onave, then x r is unique goba maximum We wi iustrate by the foowing exampes how our mode takes into onsideration a of the desired properties of the quaity degradation funtion and how they impat resoure aoation with servie guarantees A Homogeneous Tandem Network Consider n fows sharing a simpe tandem network of L inks where eah ink has a onstant apaity and identia quaity degradation funtion V assoiated with eah ink A the fows originate at the first node and their destination is the fina node as shown in Fig Fig x n x x L Tandem network of L inks being shared by n fows We assume that assoiated with eah fow i is a utiity funtion a i ogx i orresponding to its throughput x i, and a servie guarantee σ i We aso take the quaity degradation funtion to be V y = og y We hoose this quaity degradation funtion, as it satisfies the desired onvexity property It aso aptures the effet that for ahieving apaity over erroneous ommuniation inks, one needs to use arbitrary ong odes eading to unbounded variane in avaiabe servie Thirdy and very importanty, this hoie of quaity degradation funtion gives us anaytia expression for x i s, that offers vauabe insight into trade-off between throughput and servie guarantees Under these assumptions, we have the foowing optimization probem n max a i og x i 6 subjet to the onstraints n L og x i σ i, i =,,, n n x i =, =,,, L x i, i =,,, n The og funtion ensures that a x i s are aways positive and aso that the sum-rate onstraints are never ative Let 7

4 w i be the Lagrange mutipiers assoiated with the quaity degradation onstraint, then the Lagrangian is given by n n n k= Lx, w = a i og x i + w j L og k + σ j x x j= 8 Fig Three fows sharing a two-ink network Differentiating the Lagrangian with respet to x i s and equating them to zero, we obtain n x j = L n k= w From above equations, we an derive the KKT onditions k x i 9 a j= i a i x w i + w i i x i Defining row vetor of ones of size n, and n n matrix D =, i =, x with rea entries suh that D jj = + L P n k= w k a j and D ij = a w i + w for i j; we an equivaenty write the above equation in the x i x i =, x foowing ompat form, w x i og x i + x +σ i =, i =, D = i Let i = arg min{σ i : i =,,, n}, then w i =,i i and n j= x j = exp σ w i /L og x i + x +σ = by KKT onditions Then, i n x Let us onsider the i = a i / a i exp σ i /L ase when σ < min{σ,σ } In this ase, xi+x w = w =and og i +σ = For the This exampe verifies that our modeing intuition is right for simpe ase of a i =, i =,, we have optima rates resoure aoation with servie guarantees Here are some observations from this simpe exampe: x = x =x = exp σ / For any finite servie requirement, sum-rate is aways ess than the apaity of eah ink Throughputs derease with number of hops due to servie requirements When quaity degradation is inherent to a ink, fow with most stringent servie requirements imits the throughput This exampe verifies our modeing intuition for resoure aoation with servie guarantees, with same onusions as in previous exampe We wi use above two simpe exampes for numeria studies of our dua agorithm in Setion V IV PRIMAL ALGORITHM for every other fow B Three-Fows Two-Hop Network Consider the network in Fig in whih three soures transmit over two inks Let ink i have apaity i Assuming og utiity and quaity degradation funtions as in previous exampe, the resoure aoation probem beomes n max a i og x i subjet to the onstraints og x i + x σ i, i =, i og x i + x σ i Let w i be the Lagrange mutipiers orresponding to quaity degradation onstraint of fow i, then the Lagrangian is written Lx, w = a i og x i + w i og x i + x +σ i + w og x i + x i +σ i x We now deveop an agorithm that oud potentiay be used to obtain an approximate soution of our optimization probem The approah that we use is aed the Prima method, as it foows from the Prima formuation of the probem The main idea is to reax the onstraints by inorporating them as a ost into the objetive Essentiay, the idea is that there is a prie to vioating the quaity onstraints, and we maximize utiity minus prie Thus we onsider the funtion Jx = Ur x r B r R rv y, 6 where B r is a barrier funtion assumed to be onvex inreasing, from zero to unbounded vaues when argument inreases from zero to σ r To minimize this funtion, we an use a gradient desent approah, ie, ẋ r = k r x r U rx r q r, q r = B s R s V y R s R r V y 7 Sine the probem is onvex, it is straightforward to show using Lyapunov tehniques [], [9], [7] that the above agorithm onverges, and eads to one maximizer of 6 To this end, note that Jx as defined in 6 is a strity onave funtion We denote its unique maximizer by ˆx Then, Jˆx Jx is non-negative and equas zero ony at x =ˆx This makes

5 W x Jˆx Jx, a natura andidate Lyapunov funtion and we use it in the foowing proposition, whih has a simiar proof to that of [9] Proposition Consider a network in whih a soures foow the prima ontro agorithm 7 Let Jx be as defined in 6 and funtions U r,k r,v and B s be suh that W x grows unbounded as x, and ˆx i > for a i Then, the ontroer in 7 is gobay asymptotiay stabe and the equiibrium vaue maximizes 6 Proof: Differentiating W x with time t, we get Ẇ = J x r ẋ r = k r x r U rx r q r <, x ˆx, and Ẇ =for x = ˆx Here, seond ine of the equation foows from equations 6 and 7 Thus, a the onditions of the Lyapunov theorem are satisfied and we have proved that the system state onverges to ˆx However, prima ontroer suffers from the foowing handiaps The approah is not optima sine the reaxation woud yied an aeptabe soution ony if the barrier vaues at the optima soution of our origina objetive were sma Further, the above formuation woud not aow for optima points on the boundary of onstraint set We now approah the probem from a dua perspetive to see if we an obtain any better insight V DUAL ALGORITHM We start by writing down Dua version of our resoure aoation probem defined in in the hope it yieds insight on how to obtain a distributed method of ahieving optima resoure aoation The Dua probem orresponding to the probem defined in is given by Dw = max R s V y σ s x s U s x s w s Let x r be the optima maximizer, then U rx r= w s R s R r V y This gives us a system of equations that needs to be soved to find the optima x r for eah w However, this is in an impiit form that requires the knowedge of oad on every ink that a fow traverses Therefore, this approah is not ompetey distributed Nevertheess, this formuation gives us the hint that instead of ink oad and ink-degradation being dependent on eah other direty with oad y = Rx and degradation V y, we oud break up their ouping We do this by introduing a new variabe ỹ that we refer to as effetive apaity The reaxed version of the resoure aoation probem is now max U r x r 8 subjet to the onstraints R r x r = y ỹ, L R r V ỹ σ r, r S x r, r S 9 Assuming that our onave utiity and onvex quaity degradation funtions ensure that respetivey x r s and ỹ s are aways positive, we an express the Dua probem in terms of positive Lagrange mutipiers p s and w r s min Dp, w p,w where Dp, w is the maximum of the Lagrangian Lx, ỹ, p, w with respet to x, ỹ Dp, w = max x s,ỹ U s x s p w s R s V ỹ σ s R s x s ỹ Let x, ỹ be the maximizers for the above probem for any p, w, then U rx r= R r p, p = V ỹ R r w r Now, we find the partia derivatives of Dp, w with respet to variabes p and w D =ỹ R s x p s, L D = σ r R r V ỹ r S w r Then the update equations for soving the Dua minimization of the reaxed probem are ṗ = h p R s x s ỹ + p, + ẇ r = k r w r R r V ỹ σ r L w r r S, where h,k r are positive funtions and the notation z + ρ is used to denote the funtion { z + z ρ ρ = max{z,} ρ =

6 A Distributed Agorithm We an now easiy see that the above agorithm is distributed in nature At any time during evoution of our agorithm, we an treat Lagrange mutipiers p and w r as ink-prie and route-dissatisfation, respetivey A fow r needs to pay ink-prie p for ongesting ink if it uses the ink with the route-prie being the sum of a suh p s, and w r is its end-to-end dissatisfation under the urrent system state The effetive apaity of ink is ỹ and is deouped from the atua oad on this ink y = R rx r We denote the sum of ink-pries by q r = R rp for any fow r, and sum of route-dissatisfation on a ink by ν = R rw r to yied the tota dissatisfation on that ink Note that suh a tota impies that there is no need to maintain per-fow information at the ink We denote the effetive quaity degradation seen by any fow r by σ r = R rv ỹ Notie again that due to deouping through ỹ, the pereived quaity degradation is a funtion of effetive apaity, and not the atua ink-oad The agorithm is iustrated in Fig Athough the diagram is reminisent of traditiona soure-rate responds to inkprie [] [] orresponding to the ongestion ontro probem defined in, the system is atuay very different The system may be desribed as foows: Eah fow r, as it traverses its route, it arues the prie q r that it needs to pay for using eah of the inks Using this route-prie, eah soure omputes a feasibe rate x r = U r q r Furthermore, eah soure deares its dissatisfation w r to the inks it uses based on the differene between the quaity degradation σ r that it sees and σ r what it is wiing to toerate The dissatisfation is updated using ẇ r = k r w r σ r σ r + w r Eah ink detets the tota dissatisfation ν of fows sharing it and omputes effetive apaity ỹ = V p /ν and updates the ink prie by ṗ = h p y ỹ + p Further, the ink ensures that the quaity degradation suffered by sharing fows is V ỹ by adding to or dropping part of the tota fow as needed In summary, aong with the two traditiona eements of sourerate x r and ink-prie p, we have two additiona ontro variabes: soure-dissatisfation w r and effetive apaity ỹ with ink-degradation V ỹ that provide two further dimensions of ontro that are required for distributed soution Now, we show that under reasonabe assumptions over V, the sakness ondition of sum-rate being ess than or equa to effetive apaity is aways satisfied with equaity at equiibrium x r = U r q r x, w y, ν R Soures ẇ r = k r σ r σ r + w r R T ỹ = V Links p ν ṗ = h y ỹ + p q, σ p, V ỹ Fig A bok diagram of vaue-aware resoure aoation that aows deouping of user-dissatisfation on the soure side, and quaity on the ink side Proposition Let us assume that V is strity onvex and inreasing Then, at the equiibrium y =ỹ for a L Proof: Our proof is by ontradition Let us assume that there is a L, suh that y < ỹ Then for this, we have p = Note that V is a non-negative inreasing funtion That is, either V = or V z > for a z [, ] For the former ase, y ỹ ==V, ie, y =ỹ For the atter ase, p annot be zero sine V is not in the feasibe range of y and hene this wi fore y =ỹ at the equiibrium We have in effet, shown that the equiibrium onditions of our ontro oop satisfy the KKT onditions of our origina optimization probem defined in The onditions are easy to verify, and we may state this resut as a oroary of Proposition Coroary The stationary point of is a maximizer of the onvex optimization probem desribed by B Goba Stabiity of Distributed Agorithm It is quite easy to show that the above agorithm is gobay asymptotiay stabe To show this, we hoose our Lyapunov funtion to be Qp, w =Dp, w Dˆp, ŵ, where ˆp, ŵ are the unique minimizers of Dp, w It is ear that Qp, w for a vaues of p, w Aso, it is easiy seen that Dp, w grows radiay unbounded in p, w for our hoie of V Therefore, to show that the above agorithm is stabe it suffies to show Ḋp, w, with equaity iff p =ˆp and w =ŵ Note that at ˆp, ŵ, one woud have ṗ = ẇ r = Proposition Let Qp, w be as defined in and funtions U r,v,k r and h be suh that Qp, w grows unbounded with p and w Then the ontroer in

7 is gobay asymptotiay stabe and the equiibrium vaue maximizes Proof: Differentiating D with respet to time, we get Ḋp, w = D ṗ + D ẇ r p w r Soure Rates x x = h p y ỹ y ỹ + p k r w r σ r σ r σ r σ r + w r Fig 6 8 Distributed resoure aoation for Homogeneous Tandem network <, p, w ˆp, ŵ, and Ḋˆp, ŵ = Here, the seond ine of equation foows from equations and Thus, a the onditions of the Lyapunov theorem [7] are satisfied and we have proved that the Lagrange mutipiers onverge to ˆp, ŵ Hene, the system onverges to the minimizer of From the onvexity of our origina probem, and Coroary, this in turn impies that the stabe point is the maximizer of Now, we study our primary exampes of homogeneous tandem network and three-fows two-inks network to ompare our dua agorithm s performane with the optima soution C Homogeneous Tandem Network Consider the same setting as in III-A Then, for an optima x,y we woud have x i = a i Lp n y = w i p Here, we have assumed that p = p and y = y by symmetry We woud aso have additiona equations from KKT onditions w i L og n p x i y = y + σ i = D Three-Fows Two-Hop Network Consider the same setting as in III-B Then, for an optima x,y we woud have x i =, i =, x = p i p + p y i = w i + w p i, i =, Here, we have assumed that p = p and y = y by symmetry We woud aso have additiona equations from KKT onditions n p x i y = w i og y i + σ i w og y i + σ =, i =, = Finiteness of x i from throughput and quaity degradation onstraint ensures that p and the soution degenerates to the soution of origina resoure aoation probem sine yi = x i + x, i =, For numeria study of onvergene of our proposed agorithm for this topoogy, we used the foowing parameters in our Simuink mode, σ = σ =,σ =, = = and a i =, i =,, For this ase, σ >σ + σ, and hene we woud have w = Therefore, x = x =x = exp σ = 7 We have potted the onvergene of soure rates with iteration time for a fows in Fig Finiteness of x i from throughput and quaity degradation onstraint ensures that p and the soution degenerates to the soution of origina resoure aoation We an aso verify that by simuating dua agorithm on Simuink and using our proposed distributed agorithm to find the optima operating point as predited This exerise aso shows that proposed agorithm indeed onverges We used the foowing parameters for our Simuink mode, n =,L =,σ =,σ =, = = = and a = a = For this ase, we have x = x = exp = 8 We have potted the onvergene of soure rates with iteration time for both the fows in Fig Fig Soure rates x x x 6 8 Distributed resoure aoation for three-fow two-hop network

8 r VI SIMULATIONS We utiize two reaisti topoogies presented in [] to ondut numeria experiments Our objetive is to study the performane of our vaue-aware ontroer in different networking senarios We simuated our distributed resoure aoation agorithm in Matab using disrete-time evoution of ink-pries and end-to-end dissatisfation Soures send pakets at the rate generated by the ontroer and inks average it out by a forgetting fator α Links base their deision on this average rate A Aess-Core Topoogy Our first network is an aess-ore topoogy It represents a paradigm simiar to ommeriay avaiabe Internet aess, wherein users have a reativey sma aess bandwidth from homes and businesses, onneted together by resoure rih ore-network User bandwidth is onstrained, either direty at the fina hop into the home, or at a neighborhood head-end Appiations suh as PP fie transfers ow quaity onstraint, as we as voie and video as higher quaity onstraints resut in end-to-end traffi on suh a topoogy Fig 6 Aess-Core Topoogy We onsider the situation when nodes and wish to ommuniate to node and simiary nodes and to node 6 over an aess-ore network as shown in Fig 6 The abes on the inks denote their respetive apaity We refer to a fow by its origin node The QoS onstraints on quaity of degradation for fows - are,,,, respetivey We pot the onvergene of soure rates in Fig 7a We aso pot oad y and effetive-apaity ỹ for the diagona ore ink used by fows & in Fig 7b We aso potted the quaity seen by the fow and it s onstraint σ =in Fig 7 Note, we have different rate of onvergene of different parameters We assumed that ore inks have a apaity of muh higher order than that of aess inks Thereby, every ink on the ore is taken to be of apaity, where aess node onnets to the ore with apaity Simiar apaity for nodes - are,, respetivey We hose nodes, 6 to have identia aess ink apaity of B Abiene Topoogy Our seond network represents the major nodes of the Abiene network topoogy [] The network onsists of high bandwidth inks, and onnets severa universities and researh abs Traffi onsists of arge sae data transfers ow quaity 6 9 Fig Abiene Topoogy onstraints and distributed omputation where fows have strit deay onstraints We onsider fows over the Abiene network as shown in Fig 8 with abes denoting the apaity of the orresponding ink Note, they are of same order We a the fow on bottom to be fow and one on the top, fow These two fows have QoS onstraint on dissatisfation 8 and respetivey Fow has the zigzag path and has the most stringent QoS onstraint of We pot the onvergene of fow rates in Fig 9a We aso pot oad y and effetive-apaity ỹ for the ink of apaity shared by fows and, in Fig 9b We have aso potted the quaity seen by the zigzag fow and it s aeptabe onstraint σ =in Fig 9 The onusions that we draw from our simuations are i Our vaue-aware resoure aoation agorithm onverges to a stabe soution, ii user quaity onstraints are satisfied at equiibrium, ie, the agorithm performs as designed and iii the effetive apaity is identia to the atua ink oad at equiibrium showing that our reaxation produes a tight soution VII CONCLUSIONS In this paper we onsidered the design of a distributed resoure aoation agorithm that woud aow eah individua fow to speify its measure of vaue We assumed that every fow passing through a ink suffers a ertain quaitydegradation due to the oad on the ink, and that suh degradation adds up over the mutipe inks that the fow traverses The objetive is to ensure that the system throughput is maximized in a fair manner, subjet to eah fow s quaity of servie satisfying a hard onstraint Our aim was to ensure that the agorithm shoud be simpe, use oa information, and the reays need not maintain per-fow information We first showed that attempting to sove this probem by the usua optimization deomposition tehniques in Prima formuation yied approximate soutions, and in Dua formuation entraized soutions However, the observation that deouping the ink-oad from the quaity degradation using a seondary variabe that we a effetive-apaity, aows us to design suh a ontroer Under our sheme, the soure hooses its rate based on a route prie, and it deares a dissatisfation based on the quaity of servie that it sees Links hoose an effetive-apaity based on dissatisfation and ink-prie, and modify the prie as if the effetive-apaity were the atua apaity of the ink The ontro sheme ony requires that inks be aware of aggregate quantities of the fows using 7 6 8

9 8 x x x x y ỹ σ σ Soure rates 6 Link Load Dissatisfation x a Soure-Rate b Load & effetive-apaity QoS onstraint for fow Fig 7 Convergene performane on Aess-ore topoogy x x x y ỹ σ σ Soure rates Link Load Dissatisfation x a Soure-Rate Fig 9 x b Load & effetive-apaity Convergene performane on Abiene topoogy x QoS onstraint for fow them, and the soures perform omputations soey based on the parameters obtained from the inks they traverse, hene satisfying our requirements We studied iustrative exampes of quaity-degradation funtions, and ertain anonia networks that heped us gain insight into the working of the system We showed in eah ase that our distributed ontroer performs in a near-optima manner Finay, we performed simuations on more reaisti topoogies and iustrated the good performane of our agorithm In the future we wi study protoo deveopment based on our distributed ontro ideas, whih we woud then test on a rea network REFERENCES [] F P Key, Modes for a sef-managed Internet, Phiosophia Transations of the Roya Soiety, A8,, pp 8 [] F P Key, Mathematia modeing of the Internet, Mathematis Unimited - and Beyond Editors B Engquist and W Shmid, Springer- Verag, Berin,, pp 68 7 [] R Srikant, The Mathematis of Internet Congestion Contro, Birkhauser, [] S Shakkottai and R Srikant, Network Optimization and Contro, Foundations and Trends in Networking, Now Pubishers, vo, no, 8 [] M Chiang and S H Low and A R Caderbank and J C Doye, Layering as optimization deomposition: A mathematia theory of network arhitetures, Proeedings of IEEE, January, 7 [6] R Stevens, TCP/IP Iustrated, Voume, Addison-Wesey, 99 [7] L L Peterson and BS Davie, Computer Networks: A Systems Approah, Morgan-Kaufman, 999 [8] F P Key, Charging and rate ontro for easti traffi, European Transations on Teeommuniations, 997, vo 8, pp 7 [9] F P Key and A Mauoo and D Tan, Rate ontro in ommuniation networks: shadow pries, proportiona fairness and stabiity, Journa of the Operationa Researh Soiety, 998, vo 9, pp 7 [] S H Low and D E Lapsey, Optimization fow ontro, I: Basi agorithm and onvergene, IEEE/ACM Transations on Networking, Deember, 999, pp [] G Vinniombe, On the stabiity of networks operating TCP-ike ongestion ontro, Proeedings of the IFAC Word Congress, Bareona, Spain, [] T Key, Saabe TCP: Improving Performane in Highspeed Wide Area Networks, ACM SIGCOMM Computer Communiation Review, vo, no, Apri,, pp 8 9 [] D X Wei and C Jin and S H Low and S Hegde, FAST TCP: motivation, arhiteture, agorithms, performane, IEEE/ACM Transations on Networking, Deember, 6 [] S Liu and T Başar and R Srikant, TCP-Iinois: a oss and deay-based ongestion ontro agorithm for high-speed networks, vauetoos 6: Proeedings of the st internationa onferene on Performane evauation methodoogies and toos, Pisa, Itay, Otober, 6 [] J He and M Suhara and J Rexford and M Chiang, Rethinking Internet Traffi Management: From Mutipe Deompositions to A Pratia Protoo, in Pro CoNEXT, New York, NY, Deember, 7 [6] S Bake and D Bak and M Carson and E Davies and Z Wang and W Weiss, An Arhiteture for Differentiated Servies,RFC 7 June, 997 [7] H Khai, Noninear Systems, nd edition, Prentie Ha, Upper Sadde River, NJ, 996 [8] Lingjia Liu and Parima Parag and Jia Tang and Wei-Yu Chen and Jean-François Chamberand, Resoure Aoation and Quaity of Servie Evauation for Wireess Communiation Systems Using Fuid Modes, IEEE Transations on Information Theory, May, 7, vo, no, pp [9] W Kang and F P Key and N H Lee and R J Wiiams, State spae oapse and diffusion approximation for a network operating under a fair bandwidth sharing poiy, to appear in Annas of Appied Probabiity [] Internet, avaiabe at 9