Distributed and Optimal Rate Allocation in Application-Layer Multicast

Transcription

1 Distributed and Optimal Rate Allocation in Application-Layer Multicast Jinyao Yan, Martin May, Bernard Plattner, Wolfgang Mülbauer Computer Engineering and Networks Laboratory, ETH Zuric, CH-8092, Switzerland Computer and Network Center, Communication University of Cina, Beijing , Cina Paris Researc Lab, Tecnicolor, Paris, France Abstract Application-layer Multicast (ALM) is widely used as an alternative to IP multicast for one-to-many data delivery. Besides delay or bandwidt optimized overlay tree construction, rate allocation is te most important issue: given tat relaying end osts may be limited in te available download or upload capacity, ow to optimize te aggregate utility of all receivers? Wile significant efforts ave been dedicated to efficient construction of multicast trees, tis paper focuses on te second problem. We, in tis paper, propose a fully distributed network model and distributed rate allocation approaces tat optimize te overall utility perceived by te members of a multicast application wile being TCP-friendly to co-existing flows. Contrary to previous work, tis paper presents in addition to a dual solution a primal algoritm, wic avoids oscillations during convergence. Importantly, we describe an asyncronous implementation for bot variants. To evaluate our approac, we ave implemented te proposed algoritms and run extensive simulations. Overall, our findings reveal tat te algoritms generate minimal messaging overead, and efficiently optimize te aggregate utility. I. INTRODUCTION Multicast addresses te problem of efficient delivery of data over te Internet to a large number of recipients. Applications, amongst oters, include live video streaming, Internet radio, video conferencing, and multiplayer games. Unfortunately, due to tecnical, administrative, and business related issues IP multicast is still not yet widely deployed in te public Internet. To overcome te deployment problems, a new type of multicast solution as become popular: application-layer multicast (ALM) ([1], [2] and its references) or overlay multicast, wic does not require support from underlying IP routers. In application-layer multicast, some end osts serve as relays for oter members of te multicast group. Deploying ALM is callenging for two reasons. First, algoritms to form multicast trees sould compute efficient, robust overlays tat for instance minimize te delay between peers or optimize te available bandwidt [3]. Second, even after establising a multicast tree, it may be necessary to control te sending rates of overlay nodes. After all, te available download and upload capacities of end osts (e.g., DSL customers) are generally limited. Ideally, a rate allocation strategy avoids situations were overlay nodes tat receive a ig data rate cannot provide ig data rates to many of teir cildren in te multicast tree due to lack of bandwidt. Unfortunately, in existing ALM approaces intermediate nodes determine te download rate witout considering te application quality (or utility) and information from neigbor and parent nodes. For example, in end system multicast [1], te rate for eac flow is calculated locally using te unicast TCP-friendly rate control algoritm (TRFC) [4], considering neiter te structure of te multicast tree nor te streaming quality. We, in tis paper, study te latter problem, i.e., ow to allocate rates suc tat te overall utility perceived by te members of a multicast application is optimized. One potential application is multicast video streaming [5] if overlay nodes can eiter simply forward te received video stream or truncated versions to teir cildren 1. Yet, in principle any multicast application can benefit from rate allocation algoritms. Te callenge in our work is to develop optimal, distributed, yet scalable rate allocation algoritms for multicast sessions tat optimize te overall utility of all receiving nodes. Contrary to existing work, we introduce a new notion of TCPfriendliness into our problem model, making sure tat individual multicast flows compete fairly wit any co-existing TCP traffic. Importantly, we decompose te global rate allocation problem into smaller more local problems, tus paving te way for a distributed approac to solve te optimization problem. Te algoritms we will propose ave to limit te messaging overead in te multicast tree. To tis end, we extend existing utility-pricing models [7], [8], [9] tat ave been proposed as an analytical tool for optimization based rate allocation mecanisms. Efficient solutions for te rate allocation problem are alleviated by our finding tat bottleneck links, i.e., pysical links were te rates of all its multicast flows exceed te available bandwidt, are generally only sared between nodes tat are siblings in a well-formed multicast tree. To te best of our knowledge, we are te first to devise suc a fully distributed and TCP-friendly approac to te rate allocation problem, and to even present a primal solution. Te proposed algoritms solely rely on asyncronous coordination of neigboring nodes and do not require communication of 1 Implementations suc as MPEG-4 Fine Granularity Scalable (FGS) [6] demonstrate ow to adapt te rate of video streams based on feedback from te underlying congestion control.

2 any link prices to maximize utility across all receivers. In tis context, we describe two solutions, namely a typical dual algoritm, and a novel primal algoritm based on a feasible direction metod wic significantly differs from te primal algoritm in Kelly s work [8] and oter penalty algoritms suc as in [10]. Our primal algoritm always returns a feasible and stable solution and avoids oscillations during convergence. We ave implemented our algoritms in order to demonstrate teir feasibility and to study teir performance. Extensive simulation-based experiments reveal: 1) te primal and dual algoritm indeed manage to maximize te aggregate utility for all receivers. 2) wile te dual algoritm sows oscillations, te primal algoritm avoids oscillations during convergence. 3) te message overead is very small, alleviating te deployment of suc multicast systems. 4) bot algoritms are TCP-friendly to co-existing unicast and multicast traffic (see our new definition in section IV-B), wile rate allocation is proportionally fair in te multicast tree. Te rest of tis paper is structured as follows. In Section II, we briefly review related work. Ten, in Section III, we discuss te construction of overlay trees and validate our observation tat only sibling flows sare bottleneck links, wic is a prerequisite for te teoretical work in tis paper. Section IV proposes te distributed model and formulates te rate allocation problem, before we explain in Section V our proposed algoritms to solve te problem. We extend te algoritms to asyncronous environments wic better reflect te reality of large networks in Section VI. After discussing and evaluating our algoritms in Section VII, we conclude in Section VIII. II. RELATED WORK Our approac is to study te problem of rate allocation from a teoretical perspective. Yet, we see peer-to-peer Internet video broadcast (e.g., [1],[5]) as a major field of application for our results. For a review of suc tecnologies we refer to [2]. For IP multicast McCanne et al. presented rate control algoritms already more tan 10 years ago, e.g., [11]. Unfortunately, teir approac determines downstream sending rates at intermediate routers witout taking into account any information from neigbor or parent routers in te multicast tree. Even today for application-layer multicast streaming, overlay nodes generally relay traffic to cildren nodes based on standard TCP or solutions suc as TCP Friendly Rate Control (TFRC) [4]. Consequently, te overall structure of te multicast tree is ignored and aggregate utility is not optimized. In contrast, Akbari et al. [12] propose a simple optimal rate control algoritm, but tey only consider access links and teir model only allows a limited set of possible streaming rates. Te literature introduces pricing models [7], [8] as an alternative for optimizing te aggregate utility across all receivers. Kar et al. [13] ave applied suc models to IP multicast, and te autors of [14] even considered co-existing unicast TCP sessions wen optimizing te overall utility. Five years later, Cui et al. [15] applied pricing models for utility optimization in application-layer multicast systems. However, teir algoritm creates significant message overead, since tey excange information about flow rates and all pysical link prices via centralized delegate nodes. Our work avoids tis issue and, terefore, considerably minimizes message overead. Moreover, in addition to a dual solution, as proposed in a very early version of our work [16], we introduce a novel primal approac based on a feasible direction metod tat avoids oscillations during convergence. III. OVERLAY TREES Application-layer multicast requires to organize te participating nodes into a tree-like structure wit parent-cild relationsips before data can be transmitted. Wenever a parent node receives a packet, it creates a copy and forwards it to eac of its cildren. Te construction of overlay trees is callenging for two reasons. First, only overlay trees tat are optimized for properties suc as delay and bandwidt will result in good performance for te recipients. Second, joining or leaving nodes may require continuous adjustments of te dissemination tree. Before we focus on te rate allocation problem for given overlay trees, tis section briefly discusses some aspects of overlay tree construction. Given teir importance for our rate allocation solutions, one major focus of our discussion will be on bottleneck links, i.e., links in te multicast tree tat are fully saturated. A. Saring of Bottleneck Links Underlying networks can consist of eterogeneous nodes wit diverse access bandwidts. Forwarding packets from one multicast node to anoter, wic we refer to as a flow, can involve multiple intermediate pysical ops. Even if bandwidt or delay-based approaces are used to construct an overlay multicast tree, tere can remain bottleneck links. A bottleneck is a directed pysical link tat fully utilizes its available capacity. Figure 1 provides an example of an application-layer multicast tree. Te underlying pysical topology is depicted by solid lines wile arcs represent te individual flows of te multicast tree rooted at source 0. Let us furter assume tat te links l 1, l 2 and l 3 are bottleneck links, wic fully utilize teir available capacities in te indicated direction. For performance reasons 2 and for a simpler design of te algoritms temselves, it will be desirable tat te following olds: flows generally do not sare a pysical link tat is fully utilized, unless tey are sibling flows in te well-formed multicast tree. Note tat siblings are flows tat are sent by te same parent node. In our example of Figure 1 flows f 3 and f 4 sare te bottleneck link l 3, but bot flows are siblings. Figure 1 provides an example of non-sibling flows saring a bottleneck link: flows f 1 and f 5 bot pass te bottleneck link 2 limiting te number of excanged messages

3 Fig. 2. Observed trougput wen injecting iperf traffic at flow f during 50s - 100s and 150s - 200s. Te solid line represents a flow tat sees decreases in trougput wile tere is no correlation wit te second flow (dased line). Fig. 1. Example of application-layer multicast tree and bottleneck links. Solid lines correspond to pysical links wile arcs represent te individual flows of a multicast tree rooted at source 0. Te bold directed arrows l 1, l 2 and l 3 are bottlenecks according to our definition. Note tat an optimized overlay tree construction mecanism is likely to prevent f 5 from going troug bottleneck link l 1. l 1. However, f 5 is not a sibling of flow f 1. In te following, we rely on various topologies and on testbed experiments to justify te validity of te assumption tat it is predominantly sibling flows tat sare bottleneck links if tis appens at all. B. Saring of Bottleneck Links in Simulated Topologies Our goal is to study te location of bottleneck links in different pysical and overlay topologies, and to confirm our assumption. To tis end, we rely on BRITE [17] to generate ten different network topologies eac consisting of 1,000 routers 3. As for our overlay topology, we need a mecanism to construct multicast trees on top of a given pysical topology. A survey of proposed tree construction mecanisms can be found in [3]. Since applications suc as multimedia streaming are often sensitive to time and bandwidt demands, existing mecanisms frequently seek to optimize te delay wile considering bandwidt constraints, e.g., [18][19][20][21][22][23][24][25]. Hence, we rely on te same strategy and implement a tree construction mecanism tat considers bandwidt and degree constraints but mainly optimizes for delay: new peers select a peer as parent in te multicast tree if tis node is te closest in terms of end-to-end flow delay. To provide sufficient bandwidt, we furter require tat eac peer as at most four cildren. For eac of our pysical topologies, we vary te number of nodes in te multicast trees between 5 and 1,000 nodes, and apply our tree construction algoritm. Our results are clear: tere exists no single instance across our experiments were non-sibling flows sare a bottleneck link. Tis reflects our intuition tat delay- and bandwidtbased tree construction mecanism generally avoid non-sibling 3 More details of topologies are in Section VII-A flows saring bottleneck links suc as link l 1 in Figure 1. Currently, tis link is used by te non-sibling flows f 1 and f 5. However, if we optimize for delay, we expect tat node 5 will for example receive data directly from 1 and not any more from 3. C. Saring of Bottleneck Links in Testbed Experiments Wen studying potential occurrences of bottleneck links, we do not want to limit ourselves to pure simulations. Terefore, we ave implemented a proof-of-concept for an applicationlayer multicast system, wic we deploy in te PlanetLab testbed [26]. Again, te goal is to illustrate tat even for setups wit real traffic we generally cannot find bottleneck links tat are sared between non-sibling flows. More precisely, we create a PlanetLab slice wit 30 nodes located at different sites. We ave implemented te delaybased tree construction algoritm tat we introduced in te simulations. Based on tis, we build a multicast tree tat uses TCP 4 to distribute data from a source to all oter nodes of te PlanetLab slice. Finally, we run multiple experiments varying te source and number of nodes tat are part of te multicast tree. Overall, we aim at uncovering potential bottleneck links in te generated multicast trees. Our approac is as follows: For eac experiment, we artificially add load to a flow f in te multicast tree by injecting iperf TCP traffic carefully 5 between te source and destination node of te flow. Simultaneously, we examine all oter flows in te multicast tree. By doing so, we try to find out flows in te tree saring te bottleneck links tat are traversed by te flow f. If te observed trougput decreases during traffic injection for any of te oter flows (flow g), we take tis as a int tat flow g sares a bottleneck link wit flow f. Te solid line of Figure 2 provides an examples were we observe significant declines in trougput. In addition to observing canges in trougput, we run traceroutes to detect sared links. 4 TFRC is not yet practical and widely used due to several limitations [27]. 5 Iperf TCP traffic sould not introduce any additional bottleneck links.

4 (a) Part of application-layer multicast tree (b) Lineal flows (f 2 and f 4 ) case (a) First possible optimized tree for lineal flows (b) Second possible optimized tree for lineal flows (c) Collateral flows (f 1 and f 3 ) case Fig. 3. Two cases of non-sibling flows saring a bottleneck link. Bold arrows are bottleneck links in te direction (c) First possible optimized tree for collateral flows (d) Second optimized tree for collateral flows Fig. 4. Possible metods to avoid cases of non-sibling flows saring bottlenecks wile te overall delay is reduced and degree constraints is inviolated. Bold arrows are bottleneck links in te direction Note tat false positives can occur were we decide tat a flow f sares a bottleneck wit flow g altoug tis is not true in reality. However, across all PlanetLab experiments we cannot find a single case were non-sibling flows sare a common bottleneck. Again, tis is in accordance wit our intuition tat delay- and bandwidt-based tree construction approaces avoid suc situations. D. Cases of non-sibling flows saring bottleneck links We run more tests on random overlay trees wic determine parent nodes for eac joining peer randomly bot in simulations and PlanetLab experiments. We are aware tat suc a tree construction mecanisms is unlikely to be used in practice. Yet, we want to confirm or refute our expectation tat suc an approac can cause non-sibling flows to sare a common bottleneck link. Fig.3(b) and 3(c) illustrate two general cases for all nonsibling flows sare bottleneck links cases experienced during our experiments. One is lineal flows saring bottleneck links case sown Fig.3(b) and te oter one is collateral flows saring bottleneck links case sown 3(c). Moreover, we find intuitive and pratical metods to optimize te overlay tree for reducing te overall delay and to eliminate cases of nonsibling flows saring a common bottleneck link. However, te details are out of te scope of tis paper. We illustrate intuitive metods in Fig.4(a) and Fig.4(b) for cases of linear flows saring bottleneck links, and in Fig.4(c) and Fig.4(d) for cases of collateral flows saring bottleneck links. E. Summary We ave found in tis section tat bottlenecks, i.e., links tat fully utilize teir available capacity, are generally only sared between sibling flows if at all. Te following section will explain ow to leverage tis finding to design optimal, distributed, yet scalable rate allocation algoritm. Even if tis assumption does not old, our algoritms can outperform previously proposed solutions for rate allocation, see Section VII. Last but not least, we point out tat tere exist teoretical approaces for eliminating situations were non-sibling flows sare a common bottleneck link [28]. IV. DISTRIBUTED AND OPTIMAL RATE ALLOCATION Now, we propose a fully distributed network model and establis a formulation of te rate allocation problem, based on wic we will develop our algoritms in Section V and VI. Importantly, we outline ow to decompose te global problem into smaller pieces, tus paving te way for a distributed approac to solve te optimization problem. In contrast to existing work, we extend utility-pricing models wit a new notion of TCP-friendliness (Section IV-B), before we formulate te optimization problem for maximizing te aggregate utilities of all receivers (Section IV-C). A. Distributed Model In te following we borrow some notations from te utility pricing model described in [7], [15], [8]. Consider an overlay network of n + 1 nodes, denoted as H = { 0, 1,..., i,..., n }. Node 0 is te source of te multicast cannel, wile all oter nodes are receivers of te multicast cannel. We assume tat te overlay tree as already been computed. Non-leaf nodes are forwarding data to teir cildren and are able to scale-down te flows wit tecniques suc as fine granular scalable coding/transcoding tecniques [6]. Te multicast cannel consists of n end-to-end unicast flows, denoted as F = {f 1,...,f i,...,f n }. Flowf i (or f i, for simplicity s sake) is te flow tat terminates at i. Eac flow f i F as a rate x fi (or x i, for simplicity s sake). We collect all tex i into a rate vectorx = (x i,f i F). We denote U f (x i ) as te utility of flow f i, wen f i transmits at rate x i.

5 Let I f = [m f,m f ] denote te rate range of flows. F is te set of flows sent from. If a ost H is te destination of a flow f and te source of anoter flow f F, ten f is te cild flow of f, denoted as f f. We denote as te cild of and p as te parent node of, i.e., p. For te remainder of tis paper, we rely on te following two assumptions: Assumption 1: U f is strictly increasing and concave, and twice continuously differentiable on I f. Assumption 2: Te curvatures of U f are bounded away from zero on I f : U f 1/κ > 0. Formally, we define a bottleneck link as follows, Definition 1: A link l is a bottleneck link for te time being if, and only if, c l = f F(l) x f. i.e., te link is fully utilized were te rate summation of all flows in te cannel tat go troug te link l reaces te available bandwidt. c l is available bandwidt 6 for te multicast cannel at link l. Please note links and flows in our model are unidirectional. For eac bottleneck link l, F(l) = {f F l(f) = l} is te set of flows in te cannel tat pass troug it in te direction. Most flow f as a single bottleneck link at a particular point of time denoted as l(f). We find in our experiments tat flows rarely experience multiple bottleneck links at te same time. Te location of te bottleneck links of f can be inferred by topology tools [29][30]. Bottleneck locations may cange over time. However we assume tat te bottleneck link locations are more stable tan dynamics of flow traffic. Now, suppose tat te overlay network consists of L bottleneck links, denoted as Γ = {1,2,...,L}. Note tat a bottleneck link is a directed link and can be any link in te flow pat and not necessary an access downlink or uplink. We store te c l of all bottleneck links in vector C = (c l,l Γ). Sibling flows are flows from te same node, i.e., all f F, towards different cildren nodes. In accordance wit our discussion of Section III bottleneck links are generally only sared between sibling flows, tat means if l(f i ) = l(f j ), ten ( i ) p = ( j ) p. To capture wic flows are affected by wic bottleneck links, we define a bottleneck constraint matrix A wit dimensions Γ F. Te value of its entries A lf is 1 if flow f traverses te directed bottleneck linkl, i.e.,l = l(f),f F(l), oterwise te value is 0. Summing all flow rates of flows using a particular bottleneck link l cannot exceed te available capacity c l of te link: A x C (1) We envision a distributed approac and algoritm to solve te rate allocation problem. Fortunately, we can generally decompose te set of inequalities in Equation (1) into smaller independent subsets tat are local to individual end osts. Tis is possible since bottleneck links are generally only sared by sibling flows. To tis end, for a ost wit cildren flows f we define a matrix A wit dimensions Γ F, were Γ is 6 Te available bandwidt of bottleneck links can be measured in an endto-end manner by tools suc as patcar and patrate. Fig. 5. Example of TCP-friendly available bandwidt for ALM. (a) Measure te unicast TCP-friendly trougput T f1 for flow f 1 and T f2 for flow f 1 going troug bottleneck l. TCP-friendly available bandwidt for ALM is C = A T = T f1 + T f2. (b) Re-allocate te rate for flows f 1 and f 2 wile x 1 +x 2 T f1 +T f2 olds, tus te co-existing TCP traffic acieves no less trougput. te set of bottleneck links for all f flows. Again, te value A l f is 1 if flow f traverses te bottleneck link l Γ, oterwise te value is 0. Accordingly, we can decompose te problem as follows: A x F C, H were x F = (xf,f F ),C = (c l,l Γ ) H. (2) Wile Equation (1) and Equation (2), respectively, ensure tat te sum of all flow rates for a bottleneck link does not exceed te available capacity, we now add a second class of constraints. Evidently, te sending rates of cildren nodes cannot be iger tan te rate at wic tey receive traffic from teir parents. To formalize tis, we define te data constraint or flow preservation matrix B wit dimensions F F : its entries B f1f 2 are 1 if f 1 is a cild flow of f 2 (i.e., f 1 = f 2 F 2), are 1 if f 1 is f 2 and as a parent flow, and are 0 oterwise. Using tis notation, te constraint can be described as follows: B x 0 (3) Table I summarizes te introduced variables and notations. Moreover, an example in Appendix A illustrates te model we ave defined in tis section for te rate allocation problem. B. TCP-friendliness For Co-existing Traffic Te model described so far does not consider te fact tat tere may exist unicast TCP traffic and oter multicast sessions using te same pysical links. Now, we describe ow to extend our current problem formulation to include a new notion of TCP friendliness for application-layer multicast, wic we define as follows: Definition 2: A rate control algoritm is TCP-friendly, if and only if any co-existing TCP traffic does not obtain less

6 trougput tan it would if all flows of te applicationlayer multicast cannel were using unicast TCP as congestion control algoritm [16]. Let Tf be te TCP-friendly available bandwidt for te unicast flow f at te bottleneck link l(f ) determined by a TFRC algoritm. We simply collect all Tf in vector T = (Tf,f F ). Hence, we obtain C, te vector of TCPfriendly available bandwidts for te multicast cannel for F, as C = A T. Ten, te constraints of Equation (2) look as follows: A x F A T, H were x F = (xf,f F ), H (4) Figure 5 illustrates our approac to add TCP-friendliness. Assume two sibling flows f 1 and f 2 in te multicast tree saring bottleneck link l. In addition, tere is a TCP cross flow tat sares te link l. Wit our TCP-friendliness for ALM model, we determine te unicast TCP-friendly trougput T f1 for flow f 1 and T f2 for flow f 2 by TFRC algoritm wic allow te TCP cross flow getting its fair bandwidt. TCPfriendly available bandwidt for tis multicast tree is te sum of T f1 and T f2. Our distributed rate allocation approaces may re-allocate te rate between flow f 1 and flow f 2, wile te coexisting TCP traffic acieves no less trougput tan its fair bandwidt. C. Optimization Problem Finally, we formulate te rate allocation problem to maximize te aggregate utility, i.e., te overall utilities of all flows in te application-layer multicast tree: max U f (x f ) (5) m f x f M f f F fulfilling te following distributed/local constraints: { A x F A T B x 0 were x F = (xf,f F ), H. V. ALGORITHMS We first present a typical dual and ten our novel primal approac for te optimization problem in Section IV-C. A. Dual Algoritm Te literature suggests pricing models [7], [15], [13], [8] to maximize te aggregate utility across all receivers. Our goal is to develop a dual algoritm tat is distributed, i.e., tat leverages te fact tat we ave decomposed te global problem in te preceding section and ave formulated local constraints. For details on our proposed dual algoritm, we refer te reader to Appendix B. First we obtain te maximizer by applying te Kun-Tucker teorem: x f (µ α,µ β ) = [U 1 f (µ α l(f) +µβ f m f (6) f f µ β f )] Mf Initialization sending data wit te data constraints applied unicast TCP-friendly rate for eac flow. Link Price Update (by bottleneck link l = l(f i ) Γ i) at t = 1,2...: update price of te bottleneck link: µ α l(f i )(t+1) = [µα l(f i )(t)+γ( f i F(l)x f i (t) c l(f i ))]+ relay Price Update (by flow f i F ) at t = 1,2...: i update relay price of f i wit updated flow rate x f i : µ β f i (t+1) = [µ β f i(t)+γ(x f (t) x fi (t))] + i flow rate Adaptation (by flow f i F ), At t = 1,2... i 1 receive relay prices µ β f j (t) from all cildren flow {f j f i f j} 2 calculate: λ α f i(t) = µ α l(f i )(t) λ β f i(t) = µ β l(f i )(t) f j f i µβ f j (t) 3 adjust rate: x f (t+1) = [U 1 i f i (λ α f i (t)+λ β f i (t))] M f m f communicates wit te rate x f (t+1) for flow f i i 4 send µ β f i(t+1) to ( i ) p TABLE II DUAL ALGORITHM OF END HOST i µ α = (µ α l,l Γ) and µβ = (µ β f,f F) are vectors of Lagrangian multipliers. µ α l can be understood as te link price of bottleneck linklandµ β f can be understood as te relay price tat f must pay its parent flow. We solve te dual problem using te gradient projection metod. Te dual algoritm for eac end ost is presented in Table II. We assume te network is syncronous, i.e., updates occur in rounds at times t = 1,2... Eac end ost is capable of communicating wit neigbors, of measuring A, T, and of computing and adapting te sending rate for eac flow f (i.e., sender-based flow). Te bottleneck link price is locally calculated by end ost wit A, T and te sending rates of sibling flows saring te bottleneck. End ost first calculates te relay price wit te updated flow rate x f. Ten, end ost calculates te data price of flow f wit te updated relay benefit from its cildren nodes. End ost adapts te sending rate for eac flow f in te final step. Terefore, te algoritm solely depends on te coordination of end osts, and avoids canging te existing infrastructure. We allocate te unicast rate for eac flow and ten apply data constraints for te initial rate in te algoritm, i.e., x f (0) = Tf. B. Primal Algoritm - A Feasible Direction Algoritm Besides te typical dual approac, we propose a novel primal approac in tis paper. Primal algoritms work on te original problem directly by searcing te feasible region for an optimal solution. Tanks to our fully distributed model, directly solving te optimization problem (5) only requires coordination among tose sibling flows saring bottleneck links. Te proposed novel primal algoritm is a feasible direction metod [31]. Te primal algoritm continues to move in

7 Notation H = { 0, 1,..., n} p H H f F = {f 1,f 2,...,f n} f i i x i l Γ Γ = 1,2,...,L c l C, l Γ f i f i F = (f ) Γ = {l l(f ),f F } l(f) Γ F(l) A = (A lf ) L F B = (B f f) F F A = (A lf ) Γ F T f T TABLE I SUMMARY OF NOTATIONS IN THE MODEL Definition End Host p is te parent node of, is a cild of Set of cildren of Unicast flow in ALM cannel Flow f i terminated at i Flow rate of f i F Bottleneck link l Set of bottleneck links Available bandwidt for te cannel of bottleneck link l f i is a cild flow of f i Set of flows sent from in te cannel Set of bottlenecks of flow f Te bottleneck link tat f goes troug Set of siblings flows tat go troug bottleneck link l Bottleneck constraint matrix Data constraint matrix Bottleneck constraint matrix of F TCP-friendly available bandwidt for unicast for f at l(f ) Collection of T f for f F C = A T I f = [m f,m f ] Vector of TCP-friendly available bandwidt for ALM cannel for F Feasible range of U f (x f ) U f (x f ) Utility function of streams at rate x f x F = (x f,f F ) Flow rate set of F finite steps in te determined direction, obtaining a new and better rate allocation. Te process is repeated until te optimal utility is reaced. We point out tat eac rate allocation computed in tis process is feasible. Moreover, te aggregate utility continuously improves. In general, te convergent rate allocation is te global maximum of te convex optimization problem. However, it is a constrained local maximum of a general (non-convex) problem (see Capter 11.1 in [31]). Terefore, te primal algoritm is more general and it does not require te convex property of te application utility. Definition 3: Te data sadow price of a flow is te cange in te aggregate utility of te flow itself and its subtree by relaxing te data constraint by one unit (a small move). We name a flow a data constrained flow f if it is actively constrained by its parent flow, i.e., x f = x f p; oterwise it is a data unconstrained flow, i.e., x f < x f p and actively constrained by its bottleneck link. For a data constrained leaf flow f te data sadow price is: p f = U f / x f = U f(x f ) (7) For a data constrained intermediate flow f, te data sadow price is: p f = U f(x f )+ f F p f (8) Wen a flow f is data unconstrained flow, its data sadow price is zero (p f = 0). For example, te data sadow price of eac dased line flow in Figure 6 is zero. We call a node a data constrained node (Figure 6(b)) if its incoming flow is a data constrained flow, oterwise it is a data unconstrained node (Figure 6(a)). In our primal algoritm, data sadow price of a flow is computed by its receiver using its data constraint information and data sadow price of cildren flows. Te data constraint information from te parent node is a 1-bit information and indicates te flow and weter nodes are data constrained or not. We assume te network is syncronous suc tat updates occur in rounds at times t = 1,2... Again, eac end ost is capable of communicating wit neigbors, of measuring A, T, and of computing and adapting te sending rate for eac flow f (i.e., sender-based flow). We coose te data constraints applied TCP/TFRC rate as te initial rate in te algoritm, i.e., x f (0) = Tf Te closer te initial rate is to te optimal. rate, te faster te algoritm converges to te optimal rate. We present te primal algoritm of an intermediate node in Table III. Te algoritm purely depends on te coordination of neigboring nodes. Eac node receives te data sadow prices from its cildren nodes. Te algoritm reallocates te bandwidt of te bottleneck link wit step sizeγ from cildren flows wit lower sadow prices to cildren flows wit iger sadow prices suc tat te available bandwidt constraints are not violated and flows wit iger data sadow price get more bandwidt; and tus obtain a better rate allocation after eac step wit an improved aggregate utility. Ten, eac node sends an update of its data sadow price to its parent node, and sends an update of te flow rate and te data constraint information for eac cildren flow to its cildren. In te following we discuss te termination of te primal algoritm. Teorem 1: For any application-layer multicast session, te rate allocation by te primal algoritm in Table III wit sufficient small step size γ converges to te optimal allocation at most in l Γ (c(l)/γ) steps.

8 initialization sending data wit te data constraints applied unicast TCP-friendly rate for eac flow. update te data sadow price from cildren get sadow price for cildren flows f i : p f j (t),f j F i compute te median sadow price of cildren flows for eac bottleneck links: pj med (t) update information from te parent node get te flow rate x i (t) and te data constraint information re-allocate te rate among te cildren flows for f j F i for f j saring te same bottleneck j := 1 to n do if p fj (t) > pj med (t) x j (t+1) = x j (t)+γ else if p fj < pj med x j (t+1) = x j (t) γ end if end if update Data Sadow Price to parent node for data constrained node i : p fi (t+1) = U f (x i) for f j saring te same bottleneck j := 1 to n do if x j (t) x i (t) x j (t+1) = x i (t);p fi (t+1) = p fi (t+1)+p fj (t) else x j (t+1) = x j (t) end if for data unconstrained node i : p fi =0 send p fi (t+1) up to parent node p i Update Streaming and information to cildren for f j F i stream media to cild j wit updated rate x j (t+1) update te data constraint information and x j (t+1) to j TABLE III PRIMAL ALGORITHM OF END HOST i (a) data unconstrained node (b) data constrained node Fig. 6. Nodes and flows in te primal algoritm (dased line means data unconstrained flow, data constrained flow oterwise). Proof: For te subtree rooted at end ost denoted as Tree(), given te receiving rate f, te primal algoritm improves te aggregated utility by re-allocating te rate of sared bottleneck. In particular, te primal algoritm relocates te rate for eac sared bottleneck link from cildren flows wit low data sadow prices (f,p f < p (t)) to flows med wit ig data sadow prices (f,p f > p (t)). Tus, med i T ree() U f (x fi (t)) < i T ree() U f (x fi (t+1)) Eac allocation generated in te primal algoritm process is feasible and te value of te aggregate utility of te subtree U(xfi (t)) improves constantly. Given te receiving rate f and te concave utility, tere is a limit for te aggregate utility of te subtree. Te algoritm will finally converge to te maximum point. For a convex optimization problem, te convergent rate allocation is te global maximum (te optimality) of te subtree(see Capter 11.1 in [31]). Since eac subtree iteratively converges to its local optimality for given receiving rate, te entire multicast tree rooted at 0 will be eventually optimal. Eac bottleneck link will move at most c(l)/γ steps to reallocate te rate among te sibling flows, tus te algoritm will converge at most in l Γ (c(l)/γ) steps. In contrast to dual approaces [15], [32], a feasible direction algoritm sows steady convergence towards te optimal state. VI. TOWARDS ALGORITHMS IMPLEMENTATION IN ASYNCHRONOUS NETWORK Bot te dual algoritm and te primal algoritm assume a syncronized network environment. Tis means tat relay price and flow rate updates in te dual algoritm, as well as te flow rate and data sadow price updates in te primal algoritm, occur in rounds. In tis section, we describe ow to deal wit an asyncronous environment, were different peers and flows update teir data sadow prices, sending rates, or prices at arbitrary times. A. Asyncronous Model For our work we use te asyncronous model as introduced in [7], [15]. Let T = {0,1,2,...} be te set of time instances at wic eiter a flow rate, data sadow price, or a relay price is updated. We define: 1) Tf T te set of time instances at wic a flow f (its sender) updates its rate x f. 2) Tβ f T te set of time instances at wic a flow f (its sender) updates its relay price µ β f. 3) Td f T te set of time instances at wic a flow f (its receiver) updates its data sadow price. We furter define T is a time window size suc tat te time between any consecutive updates is bounded by T for eac kind of updates.

9 B. Asyncronous Dual Algoritm Link price update: End ost locates te bottleneck links l(f ) Γ, measuresa andc, and adapts ratesx f (t). Te bottleneck link price is calculated as in te syncronous model. End ost locally computes te link price µ α l(f )(t) wit rates x f (t) for sibling flows saring te bottleneck l, namely f F (l) and C using Equation (29) in Appendix B. Relay price update: End ost keeps track of all recent rate updates of flows x f (t ), (t T) t t and t T f and computes its estimated rate ˆx f (t) by using a weigted average of tese values: ˆx f (t) = t =t T b f (t,t)x f (t ), t =t T b f (t,t) = 1. (9) Ten, end ost using tis estimated rate for flow f computes te relay price for f, namely µβ f (t + 1), as in Equation 30. Flow rate update: End ost collects all recent relay price updates from cildren nodes µ β f (t), were (t T) t t and t T β f. To update te flow rate for flow f, we first compute te prices µ α l(f )(t) of all bottleneck links it traverses, and te estimated prices ˆµ β f (t) of all its cildren flows, were f f. ˆµ β f (t) = t =t T b β f f (t,t)µ β f (t ), t =t T b β f f (t,t) = 1. (10) First end ost computes te prices µ α l(f )(t) of bottleneck links tat flow f traverses, and wit µα l(f )(t) end node calculates λ α f (t). Wit estimated ˆµ β f (t) and µ β f end ost (t), calculates λ β f and furter computes cildren flow rates (t), according to Equations (23). Algoritm: To upgrade te syncronous dual algoritm to an asyncronous one, we estimate te flow rate ˆx f (t) in Equation (9) to compute te relay price during step Relay Price Update, and estimate te relay price ˆµ β f (t) in Equation (10) to compute te sending rate x f (t + 1). Now, one can easily extend te syncronous dual algoritm in Table II to an asyncronous one. We omit presenting te details of te asyncronous dual algoritm. C. Asyncronous Primal Algoritm Data sadow price update from cildren: End ost collects all recent data sadow price updates p f (t ) of its cildren flows f, were (t T) t t and t T d f, and computes its estimated data sadow price ˆp f (t) by using a weigted average of tese values: ˆp f (t) = t =t T a f (t,t)p f (t ), t =t T a f (t,t) = 1. (11) Flow rate update from parent node End ost collects all recent flow rate updates of flow x f (t ), (t T) t t and t T f, computes its estimated rate ˆx f (t) by using a weigted average of tese values: ˆx f (t) = t =t T b f (t,t)x f (t ), t =t T b f (t,t) = 1. (12) Algoritm: To upgrade te syncronous primal algoritm to an asyncronous one, we only need to estimate data sadow prices ˆp f (t) in Equation (11) to redistribute te rates among cildren and estimate te parent flow rate ˆx f (t) in Equation (12) to compute its sadow price. We again omit presenting te details of te asyncronous primal algoritm. D. Weigted Average Policies Te weigted average policy in te model is very general. In particular, we implement two popular policies for bot te primal algoritm and te dual algoritm: latest update only: only te most recently received data in (t T) t t is used for te estimation. update average: all data in time window (t T) t t are averaged for te estimation. Trougout te remainder of tis paper, we will rely on te update average policy. VII. EVALUATION After presenting our distributed rate allocation algoritms, we now evaluate te performance in terms of teir ability to optimize te overall utility (Section VII-B), in terms of teir convergence beavior (Section VII-C), and finally in terms of te number of messages tat are excanged (Section VII-D). To tis end, we present an implementation of our asyncronous algoritms (Section VII-A) and compare it against Cui s algoritm [15] for wic we received an implementation from te autors. A. Evaluation Setup Te main focus of tis paper is to study distributed and optimal rate allocation from a teoretical perspective, not to present an actual implementation of a real multicast system. Yet, we ave implemented and tested our asyncronous algoritms in a simulation environment. For tis purpose, we rely on BRITE [17] to generate network topologies consisting of 1, 000 routers. Tis topology size appears sufficient to study and compare te performance of our algoritms against existing approaces. Te available bandwidt of all links is uniformly distributed between 10 Mbps and 1, 000 Mbps and te average delay is 0.6 ms. Overall, we create 10 different network topologies. As described in Section III-B, we design and implement a mecanism to construct multicast trees on top of a given pysical topology for our experiments. Our designed tree construction mecanism is delay-based and takes into account bandwidt requirements and degree constraints. For every pysical topology, we vary te number of nodes in te

10 Fig. 8. Average Utility for Assumption Violated Trees Fig. 7. Average utilities. multicast trees between 5 and 1, 000 nodes and repeat te experiment. Te available bandwidt for te access links of multicast nodes are limited to 100 Mbps and are uniformly distributed between 1 Mbps and 100 Mbps. Tere was no single instance across our experiments were non-sibling flows sare a bottleneck link. Our delay- and bandwidt-based tree construction avoids suc situations. As utility function we coose U f (x f ) = ln(x f ), wic is concave and strictly increasing for x f. Intuitively, tis reflects te fact a user can benefit from a rate increase from 1 to 2 Mbps muc more tan from a rate increase from 10 to 11 Mbps. As a matter of fact, our distributed model can easily be adapted to support oter utility functions, e.g., te utility function of TCP [9] or TFRC). Generally, we assume tat a tecnology suc as MPEG-4 Fine-Grained Scalable Video Streams [6] is used to adapt sending rates. To study te performance of te suggested algoritms, we coose an average update interval of 10 ms. Te average measurement interval for detecting bottleneck links is set to 1, 000 ms, as we assume a slowly time-varying asyncronous network were network dynamics are slower tan te dynamics of flow traffic. Overall, tis results in low overead for measuring te bottlenecks. Finally, we set te step size γ to Mbps and time window T for te weigted average updates to 50ms for bot te primal algoritm and te dual algoritm in te experiments. B. Utility We start by studying te ability of various algoritms to maximize te aggregate utility perceived by all multicast receivers. In tis context, we compare our distributed primal and dual solutions against two oter approaces. Te standard unicast algoritm allocates te rates independently as unicast flows using te TCP/TFRC algoritm, and ten applies te data constraint as described by Equation (3). Moreover, we apply Cui s algoritm in its corrected version, see [33]. In all cases, we obtain a set of rates and can determine for every experiment te aggregate utility across all receivers. Figure 7 summarizes te acieved utilities, namely te average across all our experiments. We find tat bot te primal and te dual algoritm result in te optimal utility wic is te same utility as Cui s algoritm. Yet, our approac by design is TCP-friendly to cross traffic and incurs less overead as we will see in te next subsection. Te standard unicast approac is far from acieving te same utilities as te oter tree approaces, empasizing te benefits of a distributed rate allocation approac. Please note rate allocation is proportionally fair for flows in te multicast tree. Moreover, we find tat te minimum rates allocated by our algoritms no less tan tose obtained by a standard unicast approac in our experiments. We assume tat bottleneck links are in te vast majority of cases only sared between sibling flows, see Section III. Neverteless, to study ow well our algoritms maximize utility if tis assumption is violated, we do te following: instead of minimizing delays, we construct random multicast trees for our pysical topologies. By doing so, we obtain a few cases were non-sibling flows sare a common bottleneck, and ten apply our algoritms. Even for suc adverse overlay topologies, our algoritms as sown in Figure 8 acieve aggregate utilities tat are only marginally worse tan tose obtained in an optimized tree after eliminating non-sibling flows saring a bottleneck link [28]. Tis small degradation is due to te fact tat non-sibling flows are regarded as coexisting traffic tat is not part of te multicast stream. Yet, even in suc adverse scenarios our algoritms significantly outperform tose of standard unicast algoritms. C. Convergence Time Convergence beavior of unicast congestion control algoritms ave been extensively studied in te literature[7], [8], [34]. Now, we analyze for various step sizes te time tat our algoritms take to converge to a stable allocation of rates. Our primary interest is not to study te absolute convergence times, but rater to compare convergence beavior of te primal and dual algoritm. Since Cui s algoritm is a dual algoritm wit te same convergence properties as our proposed dual algoritm, we refrain from presenting results for Cui s algoritm. Figure 9 sows tat te distributed allocation algoritms converge to an optimal rate wen new nodes join te multicast tree at time t = 10s and t = 40s. Moreover, we observe: 1) Te dual algoritm introduces oscillations during te convergence process. Tis leads to ig link prices (i.e., ig packet loss rates). Te primal algoritm results in good rate allocations, avoiding oscillations during te convergence process.

11 (a) (b) (c) Fig. 9. Convergence beavior for various step sizes. Nodes join at t = 10s and t = 40s) on average for Cui s algoritm wile tis number is only approximately 20 million for our algoritms. Generally, te fewer bottleneck links are traversed by te flows in te underlying pysical topology, te more our algoritms reduce te number of excanged messages compared to Cui s algoritm. Actually, we find tat most price messages in Cui s algoritm include link prices wit a value of zero. Tese are cases, were our algoritms avoids sending a message. Fig. 10. Number of excanged messages. 2) Te primal algoritm converges more slowly tan te dual algoritm in our experiment setting. However, te convergence rate of te primal algoritm is restricted, see Teorem 1. Note tat te primal algoritm in Figure (9a) also converges to te optimal rate, but only after a certain time, wic is not sown in tis plot. 3) Large step sizes result in faster convergence tan small step sizes. However, large step sizes may lead to oscillation and instability, in particular for te dual algoritm as sown in Figure (9c). D. Messaging Overead Section VII-B reveals tat our dual and primal solution indeed maximize te aggregate utility of all receivers. Now, we sow tat tis can be acieved wit low message overead compared to Cui s algoritm. To tis end, we rely on te same experiments as in te preceding sections and count te number of messages tat are excanged witin a time period of 600s. Figure 10 presents te mean number of excanged messages for various multicast tree sizes and compares our primal and dual solutions against Cui s algoritm. As expected te number of messages is te same for te primal and dual variants of our algoritms. Overall, we find tat our algoritms produce significantly fewer messages tan Cui s algoritm, but as sown in Section VII-B, acieves similar results for te aggregate utility. For multicast trees wit 500 nodes, tere are more tan 180 million messages VIII. CONCLUSION In tis paper, we formalize and solve te problem of optimal rate allocation for a given application-layer multicast tree as a convex optimization problem wit constraints. Moreover, our observation tat bottlenecks are generally sared between sibling nodes in a multicast tree, allows to formulate localized constraints for distributed algoritms. Our proposed model is TCP-friendly to co-existing traffic in te network. We furter propose a novel primal approac and a typical dual approac, and design te according algoritms to solve te optimization problem. Our experiments reveal tat bot proposed algoritms are optimal in terms of global utility and are fully distributed wit a very small messaging overead. We point out tat our algoritms can be used for multi-tree multicast systems, by applying our algoritms for eac tree individually. A possible topic for future work is to derive a matematical description of convergence rates for te two proposed algoritms in asyncronous environments. Finally, we plan to integrate te proposed algoritms into large application-layer multicast systems to test teir performance. IX. ACKNOWLEDGMENTS We tank Dr. Y. Cui for providing us is implementation and Mr. H Huang for is discussion and elp wit implementing te algoritms. REFERENCES [1] Y. Cu, S G. Rao, and H. Zang, A Case for End System Multicast (keynote address), in SIGMETRICS 00: Proceedings of te 2000 ACM SIGMETRICS international conference on Measurement and modeling of computer systems, New York, NY, USA, 2000, pp. 1 12, ACM. [2] J. Liu, S. G. Rao, B. Li, and H. Zang, Opportunities and Callenges of Peer-to-Peer Internet Video Broadcast, in In (invited) Proceedings of te IEEE, Special Issue on Recent Advances in Distributed Multimedia Communications, 2007.

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Tsitsiklis, Parallel and Distributed Computation, Prentice-Hall, APPENDIX A ILLUSTRATION OF THE PROPOSED NETWORK MODEL We illustrate our distributed network model wit a detailed example in Figure 11. For simplicity, no TCP cross traffic is introduced in tis example. Te full matrix of link capacity constraints in te example is given in Inequalities (13). Please note tat (i) links in te model are directed links; and (ii) te link capacity constrains te flows tat pass troug it in eac direction independently x 1 x 2 x 3 x 4 x (13) In te example, tere are five end-to-end unicast flows (F = 5). Te network is composed of four directed bottleneck links (L = 4). Te available bandwidt for te cannel is sown in Fig. 11(c). Hence, te available bandwidt constraint at te bottleneck link, i.e., te inequality (1), becomes: x 1 x 2 x 3 x 4 x (14)

13 (a) Application-layer multicast tree (b) Pysical network topology (c) Available bandwidt constraints Fig. 11. Example of te proposed network model. Te unit used for bandwidt is Mbps. Te bold links in (c) are bottleneck links in te arrow direction, no TCP cross traffic introduced in tis example. Let te utility function be U f (x f ) = ln(x f ). Te optimal rates (Mbps) allocated by proposed algoritms are x 1 = 2.0,x 2 = 4.0,x 3 = 4.0,x 4 = 2 and x 5 = 2. Ten te total utility is f F U f(x f ) = If we allocate te rates independently as unicast flows using TFRC and apply data constraints, we get a different set of rates: x 1 = 3,x 2 = 3,x 3 = 3,x 4 = 2 and x 5 = 2. Tus, te total utility is f F U f(x f ) = 4.682, wic is worse tan te optimal result According to our observation, only x 1 and x 2, x 4 and x 5 may sare bottlenecks. As sown in Equation 2, Inequality (14) can be decomposed into: ( ) ( ) x ( )( ) x 2 ( )( ) ( 1 x3 )( ) 1 8 ) 1 0 x4 x x 1 +x 2 6 x 3 8 x 4 2 x 5 2 ( )( 2 2 (15) Now, we find tat Inequality (15) from our model is muc simpler tan te full link capacity constraint (Inequality (13)). Te decomposed Inequality (15) turns our proposed algoritms into fully distributed algoritms wit te lowest message complexity. In tis example, Inequality (3) becomes: x 1 x 2 x 3 x 4 x 5 APPENDIX B DUAL APPROACH Te dual problem is formalized as follows: 0 (16) min D(µ α,µ β ) = min max µ α,µ β 0 µ α,µ β 0 x L(x,µα,µ β ) (17) Te Lagrangian form of te primal optimization problem is: L(x,µ α,µ β ) = f F(U f (x f ) µ α (A x c) µ β (B x) µ α = (µ α l,l L) and µβ = (µ β f,f F) are vectors of Lagrangian multipliers. = (U f (x f ) f F l L L(x,µ α,µ β ) µ α l ( lf x f c l ) f FA µ β f ( f fx f ) f F f FB = (U f (x f ) x f µ α l A lf f F f F l L x f f F f Fµ β f B f f + µ α l c l l F (18) Vectors λ α = (λ α f,f F) and λβ = (λ β f,f F) are defined as: λ α f = µ α l = µ α l(f) (19) l=l(f) λ β f = µβ f Furter, we get, L(x,µ α,µ β ) = f F f f µ β f (20) (U f (x f ) (λ α f +λ β f )x f)+ l L µ α l c l (21) µ α, µ α l can be understood as te link price of bottleneck link l. Consequently, λ α, λ α f is te bottleneck link price tat f as to pay for its single bottleneck, namely µ α l(f).for µβ, µ β f is te relay price tat f must pay its parent flow f p for relaying data to f. If f as no parent flow, ten µ β f = 0. Meanwile, for f p, µ β f can be understood as its relay benefit from f. For λ β, we interpret λ β f as data price of f, wic is te relay price µ β f subtracts te relay benefit from all its cildren flows f f µβ f. Now, we ave D(µ α,µ β ) = max x L(x,µα,µ β ) = max f (x f ) (λ x f F(U α f +λ β f )xf)+ µ α l c l (22) l L