Flow Control for Cost-Efficient Peer-to-Peer Streaming

Transcription

1 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 21 proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM 21. Flow Control for Cost-Efficient Peer-to-Peer Streaming Dan-Cristian Tomozei Thomson Paris Research Lab Laurent Massoulié Thomson Paris Research Lab Abstract In this paper we aress the issue of network cost efficiency for live streaming peer-to-peer systems. We formalize this as an optimization problem, which features a generic cost function. The latter is appropriate to capture not only ISPspecific link weights, but also non-linear, congestion-epenent costs. Our main contribution is the introuction of the Implicit- Primal-Dual scheme for flow control in live streaming peer-topeer systems. It is fully istribute in that it relies only on local state variable exchanges. Moreover, we show that at a flui scale, combine with ranom linear network coing, it amits the cost optimal operating point as a fixe point. We also prove asymptotic bouneness of flui trajectories for particular cost functions. We finally show via experiments that these optimality properties are resilient to operational constraints such as finite generation size an finite fiel size. I. INTRODUCTION Peer-to-peer systems for live streaming, such as TVAnts an Sopcast, have recently met consierable success. As testimony for this, PPLive, which enjoys over 1 million users, has obtaine in 28 a license from the Chinese government to broacast meia content. Despite this growth an evolution to mainstream usage (as oppose to unergroun, illegal usage), peer-to-peer systems have recently been facing strong criticisms from ISPs. Inee, such systems generate massive amounts of traffic on sensitive physical links, both in the access an in the core networks. One solution to the problem, use by ISPs, consists in throttling peer-to-peer traffic at such critical parts of the network, using harware eicate to this task. Another approach, uner iscussion at the P4P working group, consists in letting the ISPs provie information to the peer-to-peer system trackers about their network costs. The hope is that the peer-to-peer system will then be able to aapt its traffic flows to reuce congestion at hot spots within the network, without sacrificing user experience. This strategy is more appealing than the previous one, in that it coul lea to a win-win outcome for network operators an en users, rather than an arms race. It is however challenging to implement. Most prominently, it requires from the peer-to-peer system esign the ability to exploit information about the unerlying network costs an regulate ata flows accoringly between large numbers of peers. The scale of such systems clearly manates lightweight, istribute solutions. Our aim in the present paper is to propose such solutions. To aress the problem, we take the following approach. We first propose a formulation in terms of an optimization problem, featuring a cost function to be minimize, given some feasibility conitions relate to cut constraints. Inspire by a primal-ual algorithmic solution to this problem, we evelop our so-calle Implicit-Primal-Dual (IPD) control scheme. The latter is fully istribute, in that neighbor peers ajust their ata rates base only on local information exchanges. It relies on the use of specific quantities of backlogge information as implicit multipliers for the cut constraints at han. We then illustrate how it might fall short of achieving a global optimum through a flui scale analysis of a particular scheuling strategy for packet transmission among peers. This motivates its combination with Ranom Linear Network Coing instea of pure store an forwar for ata exchanges. Our main result is then a proof that uner such Ranom Linear Coing, IPD amits the global system optimum as a stationary point, in a flui regime. A byprouct is a moeling an flui level escription of networks uner Ranom Linear Coing. We complement this result by a proof of asymptotic stability when the cost function is of so-calle separable type. We finally provie experimental evience that IPD, combine with Ranom Linear Coing, inee achieves close to optimal performance even in non-flui regimes. In particular we show that performance is resilient to changes in parameters essential in network coing, namely generation an fiel size. The outline of the paper is as follows: In Section III we give the problem statement an introuce Implicit-Primal-Dual. In Section IV we analyze its performance when combine with the Ranom Useful (RU) packet selection strategy. In Section V we analyze IPD in combination with Ranom Linear Coing (RLC). We prove that over coe packet networks the global optimum is an equilibrium point for IPD. In Section VI we provie numerical results. We conclue in Section VII. II. RELATED WORK In [1] the authors wish to maximize a utility function in a similar setting to the one escribe in this paper. They present a solution involving packing a polynomial number of epth-1 an -2 Steiner trees for multisource multicast. The scalability of their approach is limite as it requires all peers to maintain ata exchanges with all other peers /1/$ IEEE

2 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 21 proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM 21. In [2] the authors consier the slightly ifferent problem of multicast over coe wire an wireless networks with no require rate at the receivers. A message passing scheme is use to implement a maximum flow algorithm. The scalability of this approach is limite ue to the fact that messages relative to each receiver nee to be passe throughout the network. Thus the number of state variables to be maintaine at each noe scales linearly with the number of receivers. In [3], the authors propose the so-calle UESSM selfish routing algorithm in a similar setting to [2]. It is shown that UESSM converges to the min-cost flow allocation for a particular type of cost function. This approach has the same scalability limitations as [2]. To our knowlege, there have not been any propose scalable istribute algorithms achieving minimum cost multicast. III. EXPLICIT AND IMPLICIT PRIMAL-DUAL Consier a peer-to-peer network escribe by a graph G =({s} V, L) with an achievable capacity region C R L + (i.e. only rate allocations c Care feasible). The source s wishes to convey information (a stream of bytes) at a rate λ to the set of receivers V. We efine c ij as the rate at which information is transferre along ege (i, j) L.Also consier an increasing strictly convex ifferentiable function Γ( c) escribing the global cost incurre from transferring information on eges at rates given by c R L +.Wewish to minimize cost while still achieving a transfer rate of λ at each of the receivers in V. More formally, enote by T s = {{s} S : S V} the set of non-trivial cuts containing the source noe an by c(s, S) for S T s the transmission rate through cut S, namely i,j:i c Sj ij. The notation i S j signifies i S, j S an (i, j) L(i.e. cut S separates noe i from noe j or, equivalently, link (i, j) crosses cut S) an will be use frequently in the remainer of the paper. We state the following optimization problem which escribes our goal: minimize Γ( c) over c (1) subject to c(s, S) λ, S T s. (2) The cut constraint (2) is clearly necessary. It is also sufficient when network coing is allowe, in view of the founing result in network coing, ientifying feasible multicast rates with the min-min-cut [4]. In the present context where all noes are receivers, the feasible broacast rate is again given by the minmin-cut conition, as follows from Emons theorem [5]. We suppose the capacity constraints are inclue in the cost function. Thus, a further conition the cost function must satisfy is Γ( c) =+ for c / C. Denote the marginal cost of a link l by p l ( c) = Γ c l ( c). We will also refer to this marginal cost as the price of link l. For example take a network with R physical links an L overlay links. The unerlay routing can be expresse as a routing matrix H {, 1} R L,forwhich { 1 if overlay link l uses physical link r, H rl = otherwise. Denote by σ r ( c) =C r H r, c the spare capacity on link r. A possible choice of cost function is network congestion, i.e. setting link prices equal to the observe packet elay. Such prices can be moele as p l ( c) = 1 σ r ( c). (3) r:h rl =1 Furthermore, ISPs wishing to avoi overloaing sensitive links can efine physical link weights w r which increase aitively the price of an overlay link: p ISP l ( c) = r:h rl =1 ( w r + 1 ). σ r ( c) The corresponing global cost function writes as: Γ ISP ( c) = R w r H R r, c log σ r( c). C r r=1 r=1 Let us now characterize the optimum rates minimizing some generic Γ. For problem (1 2) the Lagrangian is: L( c, μ) =Γ( c)+ S T s μ S ( λ c(s, S) ), (4) where the {μ S } S Ts are Lagrange multipliers associate with the cut constraints (2). Taking the partial erivative with respect to c ij for some (i, j) L, we obtain that the optimum values ( c, μ ) satisfy p ij ( c )= μ S, (5) S:i Sj where the Lagrange multipliers μ S satisfy the KKT conitions: μ S, (6) μ S(λ c (S, S)) =. (7) We are intereste in solving this problem in a fully istribute fashion. A classic solution for solving constraine optimization problems is the Primal-Dual metho. This approach consists of letting the primal an ual variables evolve simultaneously towars the sale point of (4). Srikant [6] applies this metho to the fiel of optimal resource allocation an congestion control algorithms. In the case of our problem, the Primal-Dual algorithm takes the form of the following autonomous ifferential system: ċ ij (t) + μ S (t) p ij ( c(t)), (i, j) L (8) S:i Sj μ S (t) λ i Sj Above we use a convenient notation: c ij(t) c ij (t), S T s. (9) (a) + b = { a if b>, max{a, } if b =. Proofs of asymptotic stability an optimality of the Primal- Dual algorithm (8-9) can be foun in [6](pp ). Note that

3 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 21 proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM 21. a violation of the constraint (2) for some cut S will inuce an increase in the associate quantity μ S by (9), which will in turn inuce an increase in the transmission rate through cut S by (8). This rate increase will continue until the sai cut constraint is satisfie. Despite the nice convergence an stability properties of this scheme, it is not easily implementable in practice. Since there exists an exponential number of ual variables, a irect implementation of the Primal-Dual algorithm woul require maintenance of an exponential number of state variables, which is prohibitive. In most peer-to-peer protocols ata is ivie into small pieces calle chunks. A chunk constitutes the basic ata transfer unit in such a network. For the rest of this paper we shall refer to such chunks as packets. Assume noe i has a set P i (t) of packets at time t. For all sets of noes S, enote by Z S (t) the number of packets which are foun at every noe in S an nowhere outsie of S at time t (i.e. the backlog at S). More formally, Z S (t) = i S P i(t) \ j S P j(t). Define X +i j (t) = P i (t)\p j (t). These quantities escribe the number of packets present at noe i an absent at noe j at time t. By elementary methos it can be shown that X +i j (t) = Z S (t). (1) S:i Sj We notice a strong similarity between (1) an the contribution S:i Sj μ S(t) of the ual variables to the rate aaptation rule (8) in the Primal-Dual algorithm. Furthermore, the X +i j can be compute locally to link (i, j) without specific knowlege of quantities Z S. This observation makes the backlog sizes Z S excellent caniates for approximating the ual variables μ S an eliminates the nee for transferring exponentially many control messages for computing upate rate values. Furthermore, backlog sizes evolve implicitly an their specific evolution epens on the employe packet scheuling strategy. We o not nee to keep track of them in an explicit fashion. Motivate by the previous remarks, we propose Implicit- Primal-Dual (IPD), a istribute rate control algorithm, for which we use a continuous rate aaptation metho, the Implicit-Primal rule (as oppose to the Explicit-Primal (8)): ċ ij (t) =κ [αx +i j (t) p ij ( c(t))] + c ij(t), (11) for all (i, j) L.Hereκ is a gain factor an α is a conversion factor (it can be consiere as price per useful packet). Further intuition behin this rule is as follows: the more a noe i has to offer to its neighbor j, the more it will increase the rate c ij ; at the same time, the higher the price of the link, the lower the rate c ij will become. As we previously stresse, the Implicit-Dual equations escribe the backlog size evolution an follow implicitly from the packet transfer strategy. In the next Sections we consier a classic packet forwaring network implementing the Ranom Useful (RU) scheme an a coe packet network implementing Ranom Linear Coing (RLC). We use a flui scaling of the system to euce the governing Implicit Dual rule for both cases. We fin that IPD eviates from the optimum operation point uner RU ue to reunant transmissions over congeste links. We show that RLC solves this problem an that the global optimum is a stationary point for IPD in this setting. IV. A FIRST APPROACH Consier the case in which the source generates a sequence of packets it wishes to eliver to the receivers in V at rate λ. We use the notations P i,z S,X +i j introuce in the previous Section III. Noes are assume to have the knowlege of which packets are present at their one-hop neighbors. Consier the following transmission strategy: For each link (i, j) L, with rate c ij (t) noe i picks a packet uniformly at ranom from the set P i (t) \ P j (t) of packets useful for j an sens it along the link. Such a transmission occurs if an only if X +i j (t) = P i (t) \ P j (t) >. This strategy is calle the Ranom Useful strategy. In [7] the authors prove that Ranom Useful (RU) is rate-optimal for a fixe rate allocation c. That is, when using RU if the value of the min-min-cut of the egecapacitate graph (G, c ) with source noe s is greater than λ, then rate λ is attaine at each receiver. Starting from this rate optimal strategy, we wish to characterize the behavior of the propose IPD rate control scheme (11). We moel the system as follows: We assume for simplicity that packets are generate at the source at instants of a Poisson process of rate λ. We also assume that packet transfers on links l Lare triggere at instants of Poisson processes of timevarying rates c l (t). The process pair ((c l (t)) l L, (Z S (t)) S Ts ) is jointly Markovian with a continuous an a iscrete component. Packet generation at the source increments Z {s} (at rate λ). Along any ege (i, j) Lthere are X +i j packets available for transfer from noe i to noe j. From (1) it follows that for every packet selecte for transfer along ege (i, j) (accoring to the RU strategy) there exists a unique set S T s with i S j, such that the selecte packet belongs to the backlog at S. Such a packet transfer will ecrement the corresponing Z S an increment Z S j simultaneously, an Z will occur with probability S X +i j. Moreover, since the process ictating transfers is Poisson with rate c ij, the simultaneous Z S X +i j, upates for Z S an Z S j will be performe at rate c ij provie that X +i j >. In what follows we escribe the behavior of the system at a flui time scale. Base on this scaling, we show on an example that IPD is suboptimal. We analyze the reason for which IPD eviates from the optimum point an we propose a coing-base approach. A. Flui Limits We examine a scaling of (11) for which both α an κ are small. Consier the sequence of processes (c N,Z N ) N = ((c N l (t)) l L, (ZS N (t)) S T s ) N evolving accoring to ynamics introuce above, such that the N th process follows IPD rule (11) with α = 1 N an κ = g N, for some positive constant gain g. Assume initial conitions (c N (),Z N ()) N such that lim N cn l () = c l, lim N 1 N ZN S () = z S.

4 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 21 proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM 21. s y {s} r 1 y {s,n2} l 1 l 2 y {s,n1} l 3 n 1 r 2 r 3 l 4 Fig. 1: A counterexample illustrating the suboptimality of the Ranom Useful scheme. We efine S(c, y), for c R L + an y R Ts +, the set of flui trajectories with initial conitions c l () = c l an y S () = y S := T S z T as absolutely continuous functions c l,y S : R + R +, such that: t c ij(t) = g [y +i j (t) p ij ( c(t))], (12) t y {s}(t) = λ c sj (t) y {s}(t) y +s j (t), j s (13) t y S(t) = λ c ij (t) y S(t) y S i (t), (14) y +i j (t) i,j:i Sj almost everywhere, for all links (i, j) L an all cuts S T s \{s}, where y +i j (t) :=y {j} (t) y {i,j} (t). Introuce the rescale processes ĉ N (t) = c N (Nt) an YS N(t) = T S ZN T (Nt) N for all cuts S T s. Similarly to [7] it can be shown that the processes (ĉ N,Y N ) N converge in probability for the topology of uniform convergence on finite intervals to the set of flui trajectories S(c, y) as N. B. Suboptimality Let us show that, in the case of Ranom Useful, the global optimum is not always a fixe point of IPD. We present a simple counterexample. Take the network in Figure 1 an consier the following cost function: Γ( c) = log(λ+2ɛ c s1 c s2 ) log(λ c 12 ) log(λ c 21 ). This means that we suppose a maximum rate of λ +2ɛ on physical link r 1 an a maximum rate of λ on physical links r 2 an r 3. Assume ɛ< λ 4. Computing the optimum yiels n 2 c s1 = c s2 = c 12 = c 21 = λ 2, μ s1 = μ s2 = 2 λ,μ s = 1 2ɛ 2 λ. We search a symmetrical equilibrium point of IPD such that c s1 = c s2 = δ an c 12 = c 21 = λ δ, where <δ<λ. Shoul an equilibrium point be asymmetrical, clearly it woul eviate from the symmetrical optimal point. Then (13 12) give the following equations (an their symmetric versions which we o not write here) from which we wish to etermine δ at the equilibrium: ċ s1 = 1 y s2, λ +2ɛ c s1 c s2 ċ 12 = 1 y s1 y s, λ c 12 ẏ s = λ c s1 y s y s2 c s2 y s y s1, y s1 y s1 y s ẏ s1 = λ c s2 c 12. y s1 y s1 y s We obtain that δ = λ 2 + 2ɛ 3. We have shown that the IPD scheme is suboptimal when Ranom Useful is use for the topology in Figure 1. The intuition behin suboptimal performance on this simple topology is as follows: The source will always sen useful packets to both n 1 an n 2. However it oes not perform any coorination on which packets it sens to one or the other. Thus packet sets at n 1 an at n 2 are not as ifferent as they coul be. This causes a lower number of transmission opportunities for links l 3 an l 4. In turn, this causes an increase in traffic on the congeste physical link r 1 an in global cost an thus a eviation from the global optimum. In the next section we solve this problem by using Ranom Linear Network Coing. While we o not characterize formally the eviation from the global optimum, we observe by numerical evaluation provie in Section VI that IPD has reasonably goo performance in the RU setting. As we previously stresse, it has a major avantage over a Primal-Dual scheme: its implementation only requires aing the local rate control rule to an alreay implemente Ranom Useful live streaming protocol. V. IPD WITH RANDOM LINEAR CODING We now consier the combination of IPD with ata transfers base on Ranom Linear Coing (RLC) [4] rather than RU packet forwaring. We first escribe the system mechanisms an a corresponing stochastic moel. We then characterize flui limits for this moel. Finally, we establish that for general convex cost functions, the optimal operating point is a stationary point of these flui limits. We further show stability of these limits in the case of separable convex cost functions. A. System Mechanisms an Moel As in the previous section, the source generates packets, which we call native packets. However it sens out linear combinations of the native packets it has generate so far, rather than native packets. Receiver noes accumulate linearly inepenent combinations of native packets. An incoming packet is kept only when it is eeme innovative, that is when it is linearly inepenent of those previously receive. At each noe, packets sent are linear combinations of the innovative packets available at this particular noe. Noes are able to ecoe a given set of K native packets as soon as they have receive K linear combinations of these packets that are linearly inepenent.

5 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 21 proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM 21. Implementations of network coing for live-streaming rely on the so-calle generation approach. There the stream is ivie into generations of K consecutive native packets. Encoing an recoing operations are performe only among packets relative to the same generation. Users are then able to play content within a generation as soon as they have receive K linearly inepenent combinations of packets within this generation. This approach has also been consiere for Vieoon-Deman applications in [8]. In the sequel we ignore the impact of finite generations, an assume instea that each packet transmitte is a ranom linear combination of all innovative packets previously receive by the sening noe. This assumption is mae for the sake of analysis, as it greatly simplifies moeling of the system. The impact of finite generations on performance will be assesse experimentally in Section VI. We also assume that the linear coefficients use in the construction of an encoe packet are taken uniformly at ranom in the finite fiel F over which linear operations are performe. Let us now introuce some notation. The i th native packet is associate with the i th unit vector in F N. Any ata packet is associate with the vector of coefficients in F N characterizing its combination in terms of native packets. For any user i, we enote by V i the subspace of F N spanne by the packets it owns. Note that the vector space V i evolves with time as new packets reach noe i. For any set S of noes, we shall also enote by V S the vector space spanne by the packets owne by all noes in S: V S = i S V i. (15) We further efine the quantity X S as the number of innovative packets that noes in S coul provie to noes in S, that can be written as X S =im(v s ) im(v S ). (16) Inee, the valiity of this expression can be reaily checke by remarking that V S V s for all sets S, an V S = V s when s S. Finally, we let X +i j enote the number of innovative packets that i coul sen to j, which can be written as X +i j := im(v i + V j ) im(v j )=X {j} X {i,j}. (17) In the present context, the Implicit-Primal-Dual aaptation rule of the transmission rates c ij is again efine by (11), base on the above efinition (17) of X +i j. To complete our formal moel, we make the following statistical assumptions. Native packets are create at the source s at the instants of a Poisson process with rate λ>. For each link (i, j) L, packets are transmitte from i to j at the instants of a Poisson process with time-varying rate c ij. Such packets correspon to a uniform ranom linear combination of the innovative packets receive by noe i at the time of the transition. Thus at rate λ, the vector space V s increases its imension by 1, an hence quantities X S simultaneously increase by 1 for all S T s. Also, at rate c ij,forsetss T s with i S, j/ S, quantities X S ecrease by 1 provie the ranom linear combination forme at i oes not belong to V S. With the above assumptions, the pair of processes ((c l (t)) l L, (V i (t)) i {s} V ) is jointly Markovian, with both a continuous an a iscrete component. Note that a uniform ranom selection of a vector in V i will belong to V S with probability r(s i, S) := V i V S V i = ( ) XS X 1 S i. (18) F The probability that such a transmission leas to a ecrease in X S thus reas 1 r(s i, S). It shoul be note that for two istinct sets S, T both containing i an not containing j, the occurrences of increase in X S an X T are not inepenent, their joint probability epening in an intricate fashion on the sizes of the spaces V i, V S, V T an of their intersections. These correlations will however not play any role in the behavior of the system in the flui limit, which we now characterize. B. System Dynamics in the Flui Limit We shall again consier the system in the limit where both the scale parameter α an the gain parameter κ appearing in (11) are small. The limiting ynamics will be relate to the flui trajectories that we now efine: Definition 1: The functions c ij,v S : R + R +, for (i, j) L, S T s are calle flui trajectories of the RLC streaming system if they are absolutely continuous, an amit the following erivatives almost everywhere: t c ij(t)=g [v +i j (t) p ij ( c(t))], (i, j) L, (19) λ i S,j S c ij(t)+ t v i S,j S c ij(t)a S i,s (t), S T s \{s}, S(t)= λ (2) j s c sj(t)+ j s c sj(t)a s (t), S = {s}, where g is some positive gain, v +i j (t) :=v {j} (t) v {i,j} (t), an the functions a S i,s (t),a s (t) verify a S i,s (t),a s (t), a S i,s (t)[v S (t) v S i (t)], a s v {s} (t) =. (21) In aition, the non-negative functions v S are require to verify S T v S v T, S, T T s, (22) as well as the supermoularity conition: v S + v T v S T + v S T, S, T T s. (23) For any c R L + an v R Ts +, we shall further enote by S(c, v) the set of all such flui trajectories with initial conitions c() = c, v() = v.

6 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 21 proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM 21. We then have the following: Theorem 1: Consier a sequence of processes (c N,V N ) each with the previous ynamics, where the N th process follows IPD rate aaptation rule (11) with scale parameter α =1/N an gain parameter κ = g/n. Denoting by XS N (t) the corresponing process (efine via (16)), assume that the initial conitions amit limits lim N cn l () = c l, lim N 1 N XN S () = v S for all l Lan S T s. Then the rescale processes ĉ N l (t) = c N l (Nt), l L, Y N S (t) := 1 N XN S (Nt), S T s, converge in probability, for the topology of uniform convergence on finite intervals, as N, to the set of flui trajectories S(c, v). We now provie the key steps of the proofs, the etails being omitte for brevity. Proof: Fix some S T s. One can construct the process YS N from unit rate Poisson processes P, P S,ij, i S, j S such that YS N (t) =YS N ()+ 1 N P (λnt) 1 P S,ij (Nf N N S,ij(t)), i S,j S ( ) where fs,ij N (t) := t ĉn ij (u) 1 F N(Y N S (u) Y N S i (u)) u. The uniform law of large numbers for Poisson processes enables to rewrite the previous equation as YS N (t) =YS N () + λt fs,ij(t)+ɛ N N S (t), (24) i S,j S where the error term ɛ N S (t) goes to zero with N uniformly in t on any finite interval [,T]. Using bouneness of capacities c ij suffices to establish that the sequence of processes (YS N ) is tight for the topology of uniform convergence on any finite interval [,T] (etaile arguments woul parallel closely those in [7], pp.7-1). Moreover, for any (i, j) L, it hols that ĉ N ij (t) =ĉ N ij () + t [ g v+i j(u) N p ij ( ] ĉ N (u)) u, (25) where v+i j N N (u) :=Y (u). Equation (25) together {j} {i,j} with continuity of p ij entails that the sequence of processes ĉ N is also tight. The limiting process v S of YS N verifies (22) as well as the supermoularity conition (23) by efinition (16). The latter follows from the fact that the inequality im(v S T ) im(v S V T ) hols for any S, T T s. Finally, by Equations (24 25), for any subsequence along which all processes ĉ N,Y N converge, the limiting process v S must further verify t v S (t) =v S + λt ĉ ij (u)(1 a S i,s (u))u, i S,j S where the process a S i,s must satisfy (21), as a limit of process u (1/ F ) N(Y N S (u) Y N S i (u)). Taking limits in Equation (25), expression (19) for the erivative of the limiting process of rates c ij irectly follows. C. Stationarity of the Optimal State for IPD with RLC We now establish that an optimal operating point is a fixe point of the flui trajectories (2 21). To state the result, we will use the following notation. We let (c ij, (i, j) L,μ S,S T s) enote optimal primal an ual variables of the optimization problem (1 2). We also let v S := T S μ T,S T s. (26) We then have the following: Theorem 2: Any flui trajectory (c, v) satisfying conitions (2 21) an with initial conitions c() = c an v() = v is stationary, in that: t c l() =, l L, t v (27) S() =, S T s. Before we turn to its proof, let us comment on the meaning of the result. It inicates that the IPD with RLC system, when initialize with contents at noes such that the corresponing quantities X S as efine by (16) are close to α 1 vs, an with capacities c ij close to the optimal values c ij, will not eviate from this optimal operating point, at which network cost is minimal. We believe that a stronger result hols, namely that from any initial conitions, the ynamics of IPD with RLC (at the flui level) converge to this optimal operating point. While a proof of this conjecture is currently missing, we provie a stability result in the next subsection, an experimental evience in Section VI, which support the conjecture. We now provie auxiliary results neee to establish Theorem 2. Lemma 1: Let τ be a tree roote at s, spanning all noes in V. LetA 1, A 2 be two cuts in T s that satisfy the following criticality property: there is a unique ege of τ that crosses from A i to A i, i =1, 2. Then the cut A 1 A 2 is also critical, provie its complementary set A 1 A 2 is non-empty. The proof of this Lemma is available in the companion technical report [9]. Corollary 1: Consier capacities (c l,l L) that are sufficient for streaming at rate λ, i.e. conition (2) hols. Assume further that they are minimal, in the sense that if there exist capacities c l that are also sufficient for streaming at rate λ, an that verify c l c l, l L, then necessarily c l c l. Given any two cuts A 1, A 2 T s that are c-critical, i.e. inequality (2) hols with equality, then their union A 1 A 2 is c-critical whenever A 1 A 2 is non empty. Proof: Emons s theorem [5] establishes the following result. The capacities (c l,l L) are sufficient for streaming at rate λ if an only if there exists a collection of spanning trees τ, an associate capacities λ τ >, such that these trees can be packe into link capacities c l, i.e. c l τ λ τ 1 {l τ},l L. (28)

7 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 21 proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM 21. It is then easy to see that the capacities c l are both sufficient an minimal if inequalities (28) hol with equality. Let A 1, A 2 be critical cuts in T s such that A 1 A 2 is nonempty. For any tree τ with λ τ >, criticality of A i entails that there is a single ege of τ crossing from A i to A i,fori =1, 2. Therefore, in view of Lemma 1, there is also a single ege of τ crossing from A 1 A 2 into its complement. Since this hols for all spanning trees τ with λ τ >, it easily follows that A 1 A 2 is also critical. The ingreients for the proof of Theorem 2 are now in place. Proof: (of Theorem 2). Consier first the erivative of c ij. In view of (19) an of the choice of initial conitions (26) an c() = c, it reas t c ij() = g [ v+i j p ij ( c ) ]. However, v+i j coincies with S:i S,j / S μ S. Thus the optimality of ( c, μ ) implies that the above time erivative equals zero. Consier now the erivative of v S. Let us first consier the case where the associate multiplier μ S is strictly positive. Then, by the complementary slackness conition (7), necessarily c (S, S) =λ. In aition, for all i S, vs v S i μ S >. Thus, the corresponing term a S i,s in Equation (2) is necessarily zero, in view of (21). This establishes that the time erivative of v S is inee zero. We then consier the case where the multiplier μ S equals zero. We shall procee by inuction on the size S of S T s to show that the time erivative of v S must be zero. To start the inuction, consier first the smallest cut in T s, that is S = {s}. Assume that μ {s} =(the other case being alreay treate). The erivative of v {s} is necessarily non-negative, for otherwise, with initial conition v {s} () =, process v {s} woul take on negative values, which contraicts the properties of the flui limits. It cannot be strictly positive, for otherwise, for small enough t>, one woul have v {s} (t) >, an hence a s (t) =in (21), in which case this erivative woul rea, for small enough t, t v {s}(t) =λ c sj (t) =λ c sj + o(t) o(t), j s j s a contraiction with the assumption that this erivative is positive at t =. Consier now S T s \{s}. The inuction hypothesis is that for all T S, T S, t v T () =. Distinguish two cases. First assume that i S, v S v S i >. (29) Thus necessarily, the terms a S i,s must be zero, for all i S. In aition, (29) entails that for all i S, there exists a subset T i S such that i T i an μ T i >. In turn, this implies that each such cut T i must be c -critical, in the sense of Corollary 1. Thus, applying inuctively the result of the corollary, it follows that i S T i is also c -critical. However, since i T i, the set i S T i coincies with S, which is then c -critical. Combine with the fact that a S i,s =,this implies the esire result that t v S() =. The last case to consier is when for some i S, Conition (29) is violate, that is vs i = v S. By the inuction hypothesis, for all such i, v S i (t) =vs i + o(t). Thus the erivative of v S at zero cannot be negative, for otherwise it woul violate inequality (22): v S v S i. Assume then that it is positive. It then follows by the inuction hypothesis that, for small enough t, a S i,s (t) must be zero for all i S. Thus the erivative of v S (t) for small t reas λ c ij (t) =λ c (S, S)+o(t) o(t), i S,j / S a contraiction with the assumption that t v S() is positive. D. Stability of IPD-RLC for Separable Costs So far we establishe that at the flui scale, IPD-RLC oes not eviate from the optimal operating point once it has reache it. This begs the question of whether it is attracte to this optimal operating point, from arbitrary initial configurations. Lacking an answer at this stage, we consier instea the weaker property of convergence to a boune set containing the optimal state. We establish that this inee hols, for specific separable cost functions. For this result we consier a ifferent scaling than the one we presente in Subsection V-B. We suppose that the X(t) vary slowly an that the rates c ij (t) are compute on a more rapi time scale. This woul correspon to the flui trajectories obtaine in the limit of small scale parameters α, but with fixe, non-vanishing gain parameters κ in (11). We thus consier flui trajectories as in Definition 1, with the exception that Equation (19) is replace by the conition [v +i j (t) p ij ( c(t))] + c ij(t) =, (i, j) L. (3) Furthermore, we assume that the cost function is separable, i.e. it can be explicitly written as Γ( c) = l L Γ l (c l ), where the iniviual cost functions Γ l are assume convex an ifferentiable in the omain [, c l ), with c l >, where they are finite. With such separable costs, the above conition (3) simplifies to c ij (t) = ( Γ ij) 1 (v+i j (t)). (31) We finally assume that streaming is feasible at rate λ + ɛ for some positive ɛ, which in the present context is equivalent to the following property: There exist capacities ĉ ij, (i, j) Lfor which streaming at rate λ + ɛ is feasible, an a positive value v min > such that l L, v v min (Γ l) 1 (v) ĉ l. (32) We are now in a position to establish the following

8 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 21 proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM 21. s r 1 l 1 l 2 n 1 n 2 r 2 l 3 l 4 n 3 n 4 l 5 Fig. 2: A network with share physical links. Theorem 3: Consier separable cost functions as above. Then any flui trajectories with fast rate aaptation as escribe by (31) are asymptotically boune, i.e. there exists a constant A > such that the quantities v S (t) verify lim sup t v S (t) A. Before proving the Theorem, we nee the following Lemma 2: Let r (, 1) an v R Ts + such that the supermoularity property (23) is verifie. Denoting v +i j = v {j} v {i,j}, then for all S T s,alli S, j S, the following inequality hols: v +i j <rv S (1 r) S 1 v S i > (1 r) S v S. (33) Proof: Note that by efinition of v +i j an by writing (23) for cuts S an {i, j}, one has v S v S i + v +i j. Using the assumption v +i j < rv S,itfollowsthatv S < v S i + rv S, an the conclusion easily follows. Proof: (of Theorem 3). Fix some r (, 1). Consier the following Lyapunov function: L(v) = sup(1 r) S v S. S T s If we enote by S the cut in T s achieving the maximum in the efinition of L at a certain point in time, then for all i S, j/ S,wehavev +i j rv S. Inee, if this i not hol, by Lemma 2 the cut S = S i woul satisfy (1 r) S v S > (1 r) S v S, a contraiction of S s maximality. Moreover, for all i S, necessarily v S (1 r) 1 v S i >v S i, since r<1. Thus the quantities a S i,s in the erivative (2) are zero. Finally, whenever rv S is larger than v min, necessarily v +i j v min, an thus by the stability assumption (32), it follows that t v S λ i S,j / S ĉ ij ɛ. We have thus establishe that L(v) strictly ecreases whenever rv S v min. A fortiori, it ecreases whenever L(v) r 1 v min. It therefore follows that asymptotically, L(v) must be below r 1 v min. Thus for each S T s, one has lim sup t (1 r) S v S (t) r 1 v min, an the result of the Theorem follows by taking the constant A equal to r 1 (1 r) V v min. VI. NUMERICAL EVALUATIONS In this section we present numerical evaluations for IPD an we escribe the influence of various implementation parameters on the algorithm s performance. For our simulations we consier the network epicte in Figure 2. For this network share physical links r 1,r 2 an r 3 have a maximum physical rate of 11. The source s generates content at rate λ = 5. We wish to minimize congestion. We use r 3 a cost function which sets overlay link prices to the sum of quantities inversely proportional to the spare capacities on the physical links use. This price function (3) has been iscusse in Section III. The optimal rate allocation in this case is c s1 = c s2 = c 24 = c 43 =5 an c 13 =. Noe 3 can receive all of its content from noe 4. Any other strategy woul have a higher global cost, ue to the strict convexity of the cost function. Intuitively, instea of imposing a high loa on physical link r 2, it is globally better to make use of the free resource r 3. We test IPD uner the following scenarios: a) Iealize Coing: We simulate explicitly the evolution of quantities X S (t) escribe in Section V. These simulations are a best case scenario for the achievable performance of Ranom Linear Coing. The simulation is feasible, since there are only =16variables associate with the backlogs. It is an iealize case in the sense that we use an infinite coing fiel size. Thus the only limitation is the content bottleneck (i.e. for every link (i, j) such that X +i j > an every S T s such that i S j, we consier r(s, S i) =1 {XS=X S i} instea of (18) from Section V). Furthermore, noes nee to receive all the ata from the estination to be able to ecoe the content (an start playing it). b) Generation Coing: We use a iscrete packet-level simulator for which we implement Ranom Linear Coing using generations of a fixe size K. In this setting noes nee to receive K linearly inepenent combinations within a generation to be able to ecoe it an start playing. For all links (i, j), noes i select a useful generation for j at ranom, compute a ranom linear combination within this generation an then forwar it to noe j. This is a realistic setting. We consier several generation sizes, as well as two finite fiels: F 2 an F 2 8 for evaluating the performance of our scheme. c) Ranom Useful: Finally, we implement Ranom Useful in our packet-level simulator. In our simulations receivers become active at t = 3 time units. By this time, the source generates about 15 packets, a elay from which the receivers will want to recover. We simulate several scenarios with three generation sizes an two finite fiels use. Define the Relative Mean Error (RME) of an execution as the relative error of the mean global cost observe γ with respect to the optimal value γ, namely γ γ γ. In Figure 3 we consier the Iealize Coing case for two values of α (we use κ =.6). We plot the receiver elays with respect to the source (X +s j ) an the rates at which overlay links are being use for the two cases. The curve converging to the lowest value correspons to noe n 3 for the elay plots an to link l 3 =(1, 3) for the rate plots. The cost function we are consiering is not separable (as it was assume in the proof of Theorem 3). Nevertheless, we notice that our system stabilizes after roughly 5, time units. This observation reinforces our conjecture that IPD is stable for more general cost functions. Furthermore, we observe that the instance using a lower α is closer to the optimum in two ways: First, link l 3 is less utilize than in the case of the instance

9 This full text paper was peer reviewe at the irection of IEEE Communications Society subject matter experts for publication in the IEEE INFOCOM 21 proceeings This paper was presente as part of the main Technical Program at IEEE INFOCOM Delay 2 Delay 15 noe 3 15 noe Reception rates 8 Reception rates link 3 link link 3 link (a) α =1/1, relative mean error = % (b) α =1/6, relative mean error =.5295% Fig. 3: Influence of α elay of noe n 3 an rate usage of links l 3 =(1, 3) an l 5 =(4, 3) are inicate. using a higher α. Secon, the relative error is almost 1 times smaller in this case. However, the price we pay for approaching the optimum is a ifference in elays at receiver noes of almost an orer of magnitue between the two instances. Since in a real implementation low elays are esirable, for the next part of this Section we choose the higher value of α =.1. Let us now observe performance of the Generation Coing scheme versus the Ranom Useful scheme. As previously state, we use α =.1 an κ =.6 in (11). In Table I we Scheme Fiel size Generation size RME Iealize Coing 5.95% Generation Coing % % % % % % Ranom Useful 12.94% TABLE I: Relative Mean Error give the Relative Mean Error of an execution instance for each of the cases, for a time span of 2, time units. We notice that in this setting, when ruling out the Iealize Coing case, the best performance is obtaine by performing coing over F 2 8. The generation size parameter has the anticipate impact on performance: the higher the generation size, the closer we are to the optimum. It is surprising that IPD in a Ranom Useful setting, while proven suboptimal in Section IV, gives similar performance to IPD in a Generation Coing setting for F 2 8 an outperforms it when the fiel size is equal to 2. Inee, while for F 2 8 virtually all transfers are linearly inepenent combinations, for F 2 the number of transfere linearly epenent combinations becomes non-negligible. VII. CONCLUSION In this paper we propose a fully istribute cost-efficient rate control scheme for live streaming peer-to-peer systems. We formulate our goal as an optimization problem having a convex cost function. Using the backlog sizes as approximations for the ual variables we introuce Implicit-Primal- Dual, our rate control scheme. We manage to show the avantage of using Network Coing over a classic Ranom Useful approach. Namely, while the two have the same feasible capacity region in the consiere setting, we prove that for Ranom Linear Coing IPD has a fixe point at the global optimum while this oes not hol in the Ranom Useful setting. Our numerical evaluations showe that IPD still performs reasonably well for Ranom Useful packet networks. REFERENCES [1] Minghua Chen, Miroslav Ponec, Suipta Sengupta, Jin Li, an Philip A. Chou. Utility maximization in peer-to-peer systems. In SIGMETRICS 8: Proceeings of the 28 ACM SIGMETRICS international conference on Measurement an moeling of computer systems, pages , New York, NY, USA, 28. ACM. [2] Desmon S. Lun, Niranjan Ratnakar, Muriel Méar, Ralf Koetter, Davi R. Karger, Tracey Ho, Eba Ahme, an Fang Zhao. Minimumcost multicast over coe packet networks. IEEE/ACM Trans. Netw., 14(SI): , 26. [3] S. Bhara, S. Shakkottai, an P. Gupta. Min-cost selfish multicast with network coing. Information Theory, IEEE Transactions on, 52(11): , 26. [4] Raymon W. Yeung, Shuo-Yen Robert Li, Ning Cai, an Zhen Zhang. Network coing theory: single sources. Commun. Inf. Theory, 2(4): , 25. [5] Jack Emons. Ege-isjoint branchings. R. Rustin (E.), Combinatorial Algorithms, Algorithmics press, pages 21 31, [6] Rayaurgam Srikant. The Mathematics of Internet Congestion Control (Systems an Control: Founations an Applications). SpringerVerlag, 24. [7] Laurent Massoulié an Anrew Twigg. Rate-optimal schemes for peerto-peer live streaming. Perform. Eval., 65(11-12):84 822, 28. [8] Sihartha Annapurey, Saikat Guha, Christos Gkantsiis, Dinan Gunawarena, an Pablo Roriguez. Exploring VoD in P2P Swarming Systems. In Proceeings of the 26th Annual IEEE Conference on Computer Communications Mini Symposiums (INFOCOM 7 Mini Symposium), Anchorage, Alaska, May 27. [9] Dan-Cristian Tomozei an Laurent Massoulié. Flow control for cost efficient peer-to-peer streaming. Technical report, Thomson, 29. Available as antom/pub/nc-rate.pf.