Rate-optimal schemes for Peer-to-Peer live streaming

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1 Rate-optimal schemes for Peer-to-Peer live streaming Laurent Massoulié an Any Twigg Thomson, Corporate Research Abstract In this paper we consier the problem of sening ata in real time from information sources to sets of receivers, using peer-to-peer communications. We consier several moels of communication resources, an for each moel we ientify schemes that achieve successful iffusion of information at optimal rates. For ege-capacitate networks, we show optimality of the so-calle ranom-useful packet forwaring algorithm. As a byprouct, we obtain a novel proof of a famous theorem of Emons, characterising the broacast capacity of a capacitate graph. For noe-capacitate networks, assuming a complete communication graph, we show optimality of the so-calle most-eprive neighbour selection scheme combine with ranom useful packet selection. We then show that optimality is preserve when each peer can exchange ata with a limite number of neighbours, when neighbourhoos are ynamically aapte accoring to a particular scheme. Finally, we consier the case of multiple information sources, each creating istinct information to be isseminate to a specific set of receivers. In this context, we prove optimality of the so-calle bunle most-eprive neighbour ranom useful packet selection. I. INTRODUCTION There are nowaays many operational peer-to-peer systems supporting live streaming, i.e. real-time issemination of auio-vieo ata injecte at a source towars a set of receivers (e.g. PPLive, TVants, Sopcast, an CoolStreaming, to name a few). The etaile algorithms implemente in those systems are in general not publicly available, a notable exception being CoolStreaming (see e.g.[4]). However, some egree of reverse engineering is possible [2], an it is commonly believe that the above-mentione systems implement schemes of the following kin. Data generate at the source is partitione into winows of contiguous ata packets. Each peer then aims to ownloa all packets from a winow, before it starts ownloaing packets from the subsequent winow. To ownloa the packets from a given winow, a peer implements schemes similar to those use in BitTorrent [2] for ownloaing a file. The schemes use for etermining packet exchanges result from a combination of peer selection strategy (who to sen to) an packet selection strategy (what to sen). For instance in BitTorrent, peers serve preferentially those from whom they receive more ata in the recent The results in Sections 2-4 of this paper were announce in [7]. Early versions of their proofs appeare in Technical Report [8]. The results in Section 5 appear here for the first time.

2 2 past (this is the so-calle Tit-for-Tat mechanism, iscusse in etail in [6]). The ecision of what to sen is mae by the receiver, who selects the rarest packet, rarity being evaluate locally across neighbour peers. The aim of the present paper is to introuce mathematical moels an give proofs that simple strategies, similar to those just escribe, are capable of achieving real-time issemination of content to receivers at optimal rates. By optimal rate, we mean that issemination at larger rates is infeasible with the available communication resources. The organisation of the paper is as follows. In Section II, we introuce the moel of ege-capacitate networks. In this context, the optimal iffusion, or broacast rate, has been characterise by Emons [5] as the minimum over receivers of the minimum of the capacity of cuts between the ata source an the corresponing receiver (also known as the min-min-cut capacity). Our main result, Theorem, states that uner ranom-useful packet forwaring, the system is ergoic provie the ata injection rate is strictly below the min-min-cut capacity. The theorem of Emons [5] is retrieve as a corollary of our result. The proof of Theorem is given in Section III. In Section IV, we introuce the moel of noe-capacitate networks, an the most-eprive neighbour selection strategy. Assuming a complete communication graph (i.e., everyone can sen to anyone), we prove that the system is ergoic uner the most-eprive neighbour ranom-useful packet selection rules, when the injection rate is strictly below the maximal sustainable injection rate. In Section V, we relax the assumption of a complete communication graph. We show that rate optimality is preserve for communication graphs with small neighbourhoos that are continuously aapte over time. We then exten the results of Section IV to the case of multiple information sources, each creating istinct information to be isseminate to a specific set of receivers. In this context, we prove optimality of the so-calle bunle mosteprive neighbour ranom useful packet selection. Conclusions are rawn in Section VI. Preliminary versions of the proofs in Sections II IV appeare previously in the technical report [8], while the corresponing results have been announce in [7]. Both the results an the proofs in Section V appear here for the first time. A. System moel II. EDGE CAPACITATED NETWORKS A irecte, ege-capacitate graph G = (V, E) is given. A istinguishe source noe s wants to sen ata to all other noes, i.e. broacast information over the graph G. Data transfers consist of packet replication from some noe u to a noe v such that (u,v) E. The injection rate at the source is enote by λ, an is by efinition the rate at which the source gets new packets. The scheuling strategy we consier in this section is the following. When any noe u has some packets that one of its neighbours v has not receive yet, or is not currently receiving, an when there is no current transmission from noe u to noe v, then u picks uniformly at ranom one of the packets that it coul usefully sen to v, an sens it to v. Such a transfer procees at spee c uv, where c uv is the capacity of link (uv) E.

3 3 Once injecte at the source, a packet p can be in a number of ifferent states. It can be replicate at all noes in the system, hence successfully broacast. Alternatively, it can be ile, that is not actively transferre, an replicate at noes u in some set S V. In this paper the symbol is use to enote strict subsets; the symbol is use to enote non strict subsets. The subsets over which packets can be replicate is not arbitrary: it must contain a spanning tree roote at s, an hence in particular it must contain s. We shall enote by S the collection of strict subsets of V that contain the source noe s. Alternatively, it can be replicate at some noes u S, for some subset S S, but also actively transferre along some eges e F, for some subset F E. We shall aopt the following escription of the system state: for all S S, X S enotes the number of ile packets, that are replicate exactly at the noes u S. In aition, an unorere list of subgraphs A = {G = (W,F ),...,G m = (W m,f m )} is maintaine, escribing the active packets : W i is the set of noes at which the i-th active packet is currently replicate; F i is the set of eges along which the i-th active packet is currently transferre. We shall assume that at any given time, at most one packet is transferre along a given ege. Thus, the total number of active packets is at most E. We shall further assume that the following constraints are met: for an active packet with escription (W,F), for each (u,v) F, then u W, v / W, an there is no other ege e F that points towars v. The physical meaning of this assumption is the following. No packet is sent towars a noe that alreay has it, an no packet is sent simultaneously from several noes to the same estination noe. One practical implementation that ensures this property consists in letting each receiver noe inform its neighbour noes of which packets it has not receive yet, an omitting from this list those packets currently being transferre. Introuce the notation: X +u v = X S. S S:u S,v/ S This counts the number of ile packets that are present at noe u an absent at noe v. Let also X+u v a enote the number of active packets that coul possibly be forware along ege (u,v), given the above constraints. That is to say, let X+u v a = u W v/ W u V,(u,v)/ F. (W,F) A The following activity conition will be enforce at all times: for any ege (u, v), either there is an active packet that is actively transferre along ege (u,v), or: X +u v = 0 an X a +u v = 0. In wors, if there is no ongoing transfer along some ege (u,v), then necessarily no packet present in the system coul be transferre along this ege. We now escribe the transitions that the system state can experience uner the propose ranom useful scheuling strategy. Primary transitions: The first type of primary transitions is ue to a fresh packet arrival at the source. After such a transition, the state variable X {s} is upate to X {s} +.

4 4 The secon type of primary transitions is ue to completion of transfer of an active packet along some ege. Let this packet be represente by (W,F), an let e = (u,v) F be the ege along which replication has just complete. Then two cases may occur. If e = (u,v) was the only ege in F, then the packet uner consieration, characterise by (W, F), is remove from the collection of active packets, an the number of ile packets replicate at S = W {v} is increase by. If instea F \ {e} is not empty, then (W,F) is replace by (W {v},f \ {e}) in the list of active packets. Seconary transitions: These happen subsequently to primary transitions, to ensure that the activity conition is met. If, after a primary transition, there is an ege (u,v) for which the activity conition is not met, this means that this ege is not actively use, while the number of packets X +u v +X+u v a which coul potentially be transferre along that ege is positive. In this case, one of these X +u v +X+u v a packets will be selecte uniformly at ranom, an start being replicate along ege (u, v). More precisely, for each S S such that u S, v / S, with probability the following state upates are mae: X S X +u v + X+u v a, X S X S, A A (S,(u,v)). For each active packet (W,F) such that u W, v / W, an for all u V, (u,v) / F, then with probability /(X +u v + X+u v a ), the active set A is upate as follows: A A \ (W,F) (W,F (u,v)). Note that all these transition probabilities sum to, as require. Moreover, these capture the uniform selection of useful packets that coul be sent along ege (u,v) that we assume throughout. A Markovian special case: A general version of the moel woul assume that the time intervals between fresh packet arrivals at the source are i.i.. ranom variables, an that packet transfer times along a given ege are also i.i.. ranom variables. Uner these assumptions, the moel we just escribe is a Markov process, provie we augment the state space to keep track of the resiual times till (i) arrival of the next fresh packet, an (ii) completion of transmission along a given ege. Of particular interest is the case where these i.i.. ranom variables are in fact eterministic. The general i.i.. case is beyon the scope of the present work. In this article, we focus on the special case where the i.i.. ranom variables involve are Exponential ranom variables, where the mean inter-packet arrival at the source equals λ, an the mean packet transfer time along ege (u,v) is c uv. In this particular case, the evolution of the state variables escribe above is Markovian, without the ajunction of resiual time variables. In the sequel we focus on this particular setup.

5 5 B. Ege capacities: main result We shall enote by λ (G) the min-min-cut of graph G with source noe s. That is, λ (G) = min min c vw. () u V S V :s S,u/ S v S w/ S The main result in the present context is the following Theorem : The Markov process ((X S ) S S,A) corresponing to ranom useful packet forwaring is ergoic uner the conition λ < λ (G). (2) The proof of this result will be given in Section III. It relies on the so-calle flui limits approach, introuce an popularise by [0] an [4]. Informally, the approach consists in first establishing that trajectories of the original Markov process, after joint rescaling of both time an space, evolve accoring to some simpler, flui ynamics, an then to prove that trajectories of the flui ynamics converge to zero in finite time. We efine the spanning tree packing number, π(g) of graph G with istinguishe source noe s as the solution of the following optimization problem: Maximize λ T (3) T T over λ T 0, T T (4) subject to λ T c ij, (ij) E(G), (5) T T :(i,j) T where T enotes the collection of spanning trees of G, roote at s. A irect consequence of Theorem is the following Corollary : (Emons, 972 [5]) For any oriente graph G with ege capacities c ij, (ij) E(G), an source noe s, the spanning tree packing number π(g) is equal to the min-min-cut number, λ (G). Proof: To any i V (G) an any collection of non-negative numbers λ T satisfying inequalities (5), one can associate a flow from s to i with total capacity T T λ T. Thus, necessarily π(g) is at most the maximum flow between s an i, for any i V (G). The celebrate max-flow min-cut theorem states that this maximum flow coincies with the minimum cut capacity between s an i. This establishes that π(g) is at most λ (G). We now establish the converse inequality using Theorem. Let ɛ > 0 be arbitrary, an consier the injection rate λ = λ (G) ɛ. Theorem guarantees that the Markov process keeping track of the number of packets in any possible state is ergoic. As a consequence, there exists an equilibrium istribution for the time it takes a packet to be successfully broacast, an a steay state istribution for the spanning tree along which a packet is effectively broacast. Let q be the iscrete probability istribution over the collection T characterising the tree along which a packet is broacast in equilibrium. Let also F T enote the cumulative istribution function of the time to broacast a packet in equilibrium, conitionally on the fact that it is broacast along tree T, for all T T.

6 6 Let δ > 0 an M > 0 be some fixe positive numbers. Let τ be some time inex, that we shall let increase to infinity. Let N T (τ) enote the number of packets that have been injecte at the source uring [0,τ], an which have been successfully broacast along tree T by time ( + δ)τ. Clearly, provie M δτ, N T (τ) is larger than the number N T (τ) of packets injecte at the source uring [0, τ], that have been successfully broacast with a broacast time no larger than M. By the ergoic theorem, the following hols: lim τ τ N T (τ) = λ q(t)f T(M), In turn, this implies that: lim inf τ τ N T(τ) λq(t)f T (M). almost surely. Let N ij (( + δ)τ) enote the number of packet transmissions along ege (ij) uring [0,( + δ)τ]. Again by the ergoic theorem, an the fact that the time for a packet transmission along ege (ij) is exponentially istribute with mean /c ij, it hols that lim sup τ ( + δ)τ N ij(( + δ)τ) c ij, almost surely. On the other han, for any ege (ij) E, the process ynamics are such that necessarily: τ 0, N T (τ) N ij (( + δ)τ), almost surely. T:(ij) T Diviing this last inequality by τ, an letting τ ten to infinity, the two previous inequalities entail that λq(t)f T (M) ( + δ)c ij, (ij) E(G). T:(ij) T Let successively M ten to infinity an δ ten to zero to obtain: λq(t) c ij, (ij) E(G). T:(ij) T Thus, the numbers λ T := λq(t) satisfy the constraints (5). Since they sum to λ, this ensures that λ π(g). Since this is true for any positive ɛ = λ (G) λ, the esire inequality λ (G) π(g) follows. III. PROOF OF THEOREM The proof consists of three main parts. We first characterize the flui trajectories that are vali limits of the process trajectories after joint rescaling of both time an space. We then establish that, uner the stability conition λ < λ (G), these must converge to zero in finite time. We finally euce the ergoicity result by applying a suitable version of Foster s criterion.

7 7 A. Flui ynamics: characterization an convergence Let us introuce the following efinition. Definition : The real-value non-negative functions t y S (t), S S, are calle flui trajectories of the above Markov process if they satisfy the following conitions. For all S S, all u S, all v / S, there exist non-negative functions t φ S,(uv) (t) such that y {s} (t) = y s (0) + λt v V \{s} φ {s},(sv)(t) v S\{u} φ S\{v},(uv)(t) S {s} : y S (t) = y S (0) + u S (6) u S v/ S φ S,(uv)(t), an that are non-ecreasing, Lipschitz continuous with Lipschitz constants c uv. In aition, for all (u,v) E, it hols that: φ S,(uv) is c uv -Lipschitz. S S:u S,v/ S Moreover at almost every point t, the function φ S,(uv) is ifferentiable, an the following hols: y +u v (t) > 0 t φ y S (t) S,(uv)(t) = c uv y +u v (t), (7) where we have use the notation y +u v (t) := S S:u S,v/ S y S (t). (8) The following notation will be use in the sequel. For any y R S +, S(y) enotes the set of all flui trajectories of the system with initial conition y. Thus it is a subset of C([0,+ ), R S + ), that is the space of continuous, RS + -value functions on [0,+ ). Note that at this stage, neither existence nor uniqueness of flui trajectories has been establishe. The following result shows in what sense such flui trajectories escribe the ynamics of the original Markov process after spatial an temporal rescaling. It implies as a corollary that the set S(y) is nonempty, for any y R S +. However no claim of uniqueness of flui trajectories is mae. Theorem 2: Consier a sequence of initial conitions (X N (0),A N (0)), N > 0, such that for a sequence of positive numbers (z N ) N>0, lim N z N = +, an the limit lim N exists in R S +. Introuce the rescale process z N X N (0) = x(0) Y N S (t) := z N X N S (z N t), S S, where XS N (t) represents the (S-coorinate of the) state of the Markov process with initial conitions (X N (0),A N (0)) at time t. Then for all T > 0, all ɛ > 0, the following convergence takes place: ( ) lim P N inf sup f S(x(0)) t [0,T] Y N (t) f(t) ɛ = 0. (9)

8 8 In wors, the restriction of the rescale process Y N to any compact interval [0,T] converges in probability to the set S(x(0)) of flui trajectories with initial conition x(0), where convergence of processes is for the uniform norm. Proof: It will be more convenient to work with the state variables X S, which count the total number of packets, active or ile, present at noes u S. That is: X S = X S + W=S. We shall thus consier the rescale processes Ỹ N S (W,F) A (t) := z N XN S (z N t) = Y N S (t) + z N W A N W=S. Since they iffer from YS N by at most E /z N, the processes agree in the limit N. Let P uv, (u,v) E, be inepenent unit rate Poisson processes. The Poisson process P uv will be use to etermine the instants at which packet transfers along ege (u, v) complete. Introuce the notation: t Φ N S,(uv) (t) = P uv c uv W=S,(u,v) F s. 0 (W,F) A N (s ) This process keeps track of the number of completions of packet transfers along ege (u, v), for packets that were previously present at noe set S. We thus have the following, for all S S, S {s}: X S N (t) = X S N (0) + Φ N S\{v},(uv) (z Nt) Φ N S,(uv) (z Nt). u S,v S\{u} u S,v/ S We use another unit rate Poisson process P 0 to count fresh arrivals at the source, an write: X N {s} (t) = X N {s} (0) + P 0(λt) v s Φ N {s},(sv) (z Nt). We now show that for any (eterministic) subsequence of the original sequence, there exists a further (eterministic) subsequence f(n) for which the following property hols. Almost surely, the sequence of rescale processes t z f(n) Φ f(n) S,(uv) (z f(n)t) is tight (for the topology of uniform convergence), an any collection of functions φ S,(uv) that are accumulation points of this sequence efine a flui trajectory as per the previous efinition. sup t Φ N S,(uv) t [0,T] z (z Nt) c uv W=S,(u,v) F s N 0 (W,F) A N (z Ns ) sup P uv (z N t) t t [0,c uvt] z N. (0) The following lemma, which is a classical result on the maximal eviation of a Poisson process from its mean is now neee:

9 9 Lemma : Let Ξ be a unit rate Poisson process. Then for all T > 0, N > 0, an all ɛ > 0, it hols that where P(sup 0 t T Ξ(Nt) Nt ɛnt) e NTh(ɛ) + e NTh( ɛ), () h(λ) := ( + λ)log( + λ) λ (2) is the Cramér transform of a unit mean, centere Poisson ranom variable. In the above formula, it is unerstoo that h( λ) = + if λ >. Define the subsequence f(n), together with a sequence ɛ(n) as follows: { f(n) = inf{k > f(n ) : zk N}, Define the event A N as ɛ(n) = N /4. A N = S S,u S,v/ S {sup t [0,T] z f(n) Φ f(n) S,(uv) (z f(n)t) c uv t 0 (W,F) A f(n) (z f(n)s ) W=S,(u,v) Fs } ɛ(n) It is reaily seen, using the above Lemma an the inequality (0), that the following hols: P(A N ) < +. N>0 Thus, by Borel-Cantelli s lemma, with probability only finitely many events A N occur. In the sequel, to lighten notation we write N instea of f(n). To establish the claime convergence of the rescale processes N ΦN S,(uv) (Nt) to Lipschitzcontinuous, non-ecreasing functions φ S,(uv) along subsequences, it is therefore sufficient to establish that such convergence hols for the functions: t c uv t 0 (W,F) A N (Ns ) W=S,(u,v) F s. (3) To this en, we use the following lemma, taken from Ye et al. [3]: Lemma 2: (Lemma 6.3, Ye et al. [3]) Suppose that a sequence of functions f k : [0,T] R has the following properties: (i) {f k (0)} k 0 is boune; (ii) there is a constant M > 0, an a sequence of positive numbers σ k, with σ k 0 as k, such that f k (t) f k (s) M(t s) + σ k, k 0, s,t [0,T]. Then the sequence amits a subsequence that converges uniformly on [0,T] to a Lipschitz continuous function f : [0,T] R with Lipschitz constant M. Clearly the conitions of the Lemma are met for the functions (3), with as a Lipschitz constant M = c uv. Moreover, any limiting function must be non-ecreasing since the functions (3) are all non-ecreasing.

10 0 Note now that for t < t, S S:u S,v/ S c uv t t (W,F) A N (Ns ) W=S,(u,v) F s c uv (t t). This reaily implies that for any given (u,v) E, the limiting functions φ S,(uv) summe over S S such that u S an v / S are c uv -Lipschitz. It now remains to establish the last property in the efinition of flui trajectories, that is: at almost every t, the function φ S,(uv) (t) is ifferentiable, an provie y +u v (t) > 0, then: t φ y S (t) S,(uv)(t) = c uv y +u v (t) By Raemacher s theorem, a Lipschitz-continuous function is ifferentiable almost everywhere. Let thus t be a point where φ S,(uv) (t) is ifferentiable. Consier first the case where y S (t) > 0. Fix some h > 0. We want to evaluate the following quantity: h c uv t+h t (W,F) A N (Ns ) W=S,(u,v) F s. Note that on the interval τ [t,t+h], N XS N(Nτ) equals y S(t)+0(h)+ɛ N, where ɛ N 0 as N, by convergence of the rescale trajectories, an by Lipschitz continuity of the limiting trajectories. Thus, after each completion of a transfer along ege (u,v) uring the interval [Nt,N(t+h)], the probability that the next packet selecte for transmission along ege (u, v) is a previously ile packet, replicate at noes w S is asymptotic to y S (t)/y +u v (t) + 0(h). Furthermore, once such a transfer is starte, the probability that the packet uner consieration is electe for transmission along another ege converges to zero as N, since there are close to Ny S (t) other ile packets that coul alternatively have been selecte for such a transmission. Together these arguments ensure that lim N h c uv t+h t (W,F) A N (Ns ) W=S,(u,v) F s = c uv y S (t) y +u v (t) + O(h) However, the left-han sie of this expression also reas ( φs,(uv) (t + h) φ h S,(uv) (t) ), an thus the erivative of φ S,(uv) at t must equal c uv y +u v(t) as announce. Finally, consier the case where y S (t) = 0, an choose a particular t at which all S with u S, v / S are such that φ S,(uv)(t) are ifferentiable. We know that almost everywhere, the sum of these erivatives can not excee c uv, because it is a Lipschitz constant for the sum of these functions. However, the sum of the erivatives for those S such that y S (t) > 0 equals c uv, therefore the erivatives for those S such that y S (t) = 0 must equal zero. y S(t)

11 B. Flui ynamics: stability In the present section, we establish that any flui trajectories as per Definition satisfy a suitable stability property: Theorem 3: Assume that Conition (2) hols. Let (y S ) S S enote flui trajectories as per Definition. For all S V, efine: y S = y S. S S,S S Then there exist positive parameters β,...,β V, an ɛ > 0 such that the function verifies: L({y S } S S ) := sup β S y S S V L(y(t)) max(0,l(y(0)) ɛt). (4) Denote by K the total number of noes, that is K = V. The proof will rely on the following lemma: Lemma 3: Let α > 0 be fixe. For given δ,a > 0, efine: ɛ K = δ; ɛ K i = δa( + A) i, i =...,K 2, β K = ; β K i = ( ) K j=k i+ ɛ j, i = 2,...,K. Then A an δ can be chosen so that the following properties hol for any (y S ) S S R S +, (y S ) S S 0. For all S V, all u S, v / S: (5) y +u v < ɛ S y S β S y S\{u} > β S y S. (6) Moreover for all S V such that, for all u S, all v / S, y +u v ɛ S y S, assuming there exist u S an v / S such that for some S S: u S, v / S an y S > αy +u v, then it hols that: β S S y S S > β S y S. (7) Proof: (of Lemma 3) Let us first establish sufficient conitions on the parameters ɛ i, β i for the conclusions of the Lemma to hol. Consier the first requirement (6), an let thus S be such that for some u S an v / S, one has y +u v < ɛ S y S. Write now: It thus follows that y S = y S\{u} + S S:u S,S S y S y S\{u} + y +u v < y S\{u} + ɛ S y S. y S\{u} > ( ɛ S )y S.

12 2 Thus the esire conclusion (6) will follow provie: β i ( ɛ i ) β i, i = 2,...,K. (8) Clearly, this conition will be satisfie with the particular choice of coefficients β i as in (5), provie the ɛ i lie in the interval (0,), which will be ensure by taking δ > 0 sufficiently small. Let us now turn to Conition (7). Let thus S V be such that for all u S an v / S, ɛ S y S y +u v. Assume moreover the existence of u S, v / S, an S S such that u S, v / S, an satisfying in aition: Then necessarily, one has: y S > αy +u v. y S > αɛ S y S. The left-han sie of Conition (7) then verifies: Therefore, (7) will hol provie β S S y S S β S S (y S + y S ) > β S S ( + αɛ S )y S. β S S ( + αɛ S ) β S. For sufficiently small δ > 0, the coefficients ɛ i as in (5) will be strictly less than, an hence the coefficients β i as in (5) will be ecreasing with i. Thus, the above conition will be satisfie provie: β K ( + αɛ i ) β i, i =,...,K 2. (9) For i = K 2, this conition reas + αɛ K 2 /( ɛ K ). Recalling from (5) that ɛ K = δ, the right-han sie reas + δ + o(δ), while the left-han sie reas + αδa. Thus, this particular conition is met provie Aα >, an δ > 0 is small enough. Let us now consier i {,...,K 3}. Note that the right-han sie of (9) is equivalent to, for small δ > 0: β i = ( ) K j=i+ ɛ j = + K j=i+ ɛ j + o(δ) = + δ + K 2 j=i+ δa( + A)K 2 j + o(δ) = + δ + δa K 3 i j=0 ( + A) j + o(δ) = + δ + δa (+A)K 3 i+ A + o(δ) = + δ( + A) K 2 i + o(δ). On the other han, the left-han sie of (9) equals + αδa( + A) K 2 j. Thus, provie αa >, an δ > 0 is small enough, the announce properties hol.

13 3 Proof: (of Theorem 3) Consier the particular parameters β i, ɛ i as in Lemma 3. Clearly, for a vector y R S + that is non-zero, any set S V achieving the maximum in max S V β S y S is such that y S > 0. Moreover, for all u S, v / S, one must have: y +u v ɛ S y S > 0, for otherwise optimality of the set S woul be contraicte by Conition (6). In aition, for all u S, v / S, an all S S such that u S, v / S, necessarily y S αy +u v, for otherwise optimality of S woul be contraicte by (7). One thus has the following evaluation: t y S = S S t y S = λ u S,v/ S S S,u S t φ S,(uv) = λ [ u S,v/ S c uv S S,u S,v/ S y S λ u S,v/ S c uv + u S,v/ S c uv λ u S,v/ S c uv + max e E c e E 2 K α. y +u v ] S S,u S,v/ S α In the above, we have use the expression (7) for the erivative of the functions φ S,e, an the boun of α on the ratio y S /y +u v previously establishe. Furthermore, the conitions (8) an (9) use in the proof of Lemma 3 can be shown to imply the following. For a set S such that β S y S ( r)β S y S, where r > 0 is some small positive constant, necessarily for all u S, v / S, y +u v ( ɛ S r ) y S. In aition, for u S, v / S an S S such that u S, v / S, then one has: ( ) + αɛ S y S y +u v = (α + O(r)) y +u v. r ɛ S r Thus, for such S, one has the similar evaluation t y S λ c uv + max c e E 2 K α ( + O(r)). (20) e E u S,v/ S Note that the choice of α > 0 in Lemma 3 was arbitrary. For efiniteness, set α = min S V u S,v/ S c uv λ 2 E 2 K max e E c e This is positive, uner the stability conition (2). Then from the above evaluation (20), it follows that necessarily, almost everywhere the Lipschitz continuous function L(y(t)) must satisfy: t L(y(t)) ɛ y(t) 0, where ɛ := min c uv λ. 2 S V The result of Theorem 3 follows. u S,v/ S

14 4 C. Proof of Theorem The proof of Theorem will require to combine Theorems 2, 3 an the following ergoicity criterion, which is a irect consequence of Theorem 8.3, p.224 in Robert [9]: Theorem 4: Let Z(t) be a Markov jump process on a countable state space Z. Assume there exists a function L : Z R + an constants M, ɛ, τ > 0 such that for all z Z: L(z) > M L(z) E zl(z(l(z)τ)) ɛ. (2) If in aition the set {z : L(z) M} is finite, an E z L(Z()) < + for all z Z, then the process Z(t) is ergoic. Let us show how this result applies in the present context. Here we have Z(t) = (X(t),A(t)), an our caniate Lyapunov function takes as argument the X-component only, an reas L(Z) = sup β S X S. S V Let us set τ =, where ɛ is as in Theorem 3, an establish that (2) hols by contraiction. Assuming it fails, there must exist a sequence of initial conitions Z N (0) such that L(Z N (0)), an such that lim N L(Z N (0)) EL( Z N (L(Z N (0))τ) ) > ɛ. (22) However, by Theorem, any accumulation point of the sequence L(Z N (0)) XN (L(Z N (0))τ) must be equal to y(τ) for some flui trajectory y issue from an initial conition y(0) such that L(y(0)) =. Furthermore, this family of ranom vectors is uniformly integrable: inee, writing L(Z N (0)) XN S (L(Z N (0))τ) XN S (0) β S XS N(0) + L(Z N (0)) P e (L(Z N (0))c e τ), where the P e are the Poisson processes previously introuce, uniform integrability can be reaily checke. Since the function L grows not faster than linearly, the family of ranom variables L(Z N (0)) L( X N (L(Z N (0))τ) ) is also uniformly integrable. Since the function L is continuous, accumulation points of this sequence must be of the form L(y(τ)), for some flui trajectory y issue from an initial conition y(0) such that y(0) =. By Theorem 3, all such accumulation points are less than, or equal to ɛ. This together with uniform integrability ensures that lim sup N e E L(Z N (0)) EL( Z N (L(Z N (0))τ) ) ɛ, which contraicts (22). The proof is conclue by verifying the other assumptions of Theorem 4, i.e. that {z : L(z) M} is finite for sufficiently large M. This hols trivially, because

15 5 for any X-component the number of potential A-components is boune (say by E times the number of subgraphs of G). Finally, one must check that E z L(Z()) < + for all z; this is easily verifie, once more by bouning X S () by its initial value plus increments of Poisson processes. A. Moel an Algorithm IV. NODE-CAPACITATED NETWORKS Neighbour selection: Here, the system is also escribe by a a graph G = (V,E). However, the capacities are now associate with noes rather than with eges. We shall enote by c u the capacity of noe u, an assume that each noe evotes its capacity to one of its most eprive neighbours. By this, the following is meant. For each of its neighbours v, noe u evaluates the number Z +u v of packets that it coul usefully forwar to noe v. Using the same notation as before, this reas: Z +u v = X +u v + X a +u v. It then elects one neighbour v for which the corresponing quantity Z +u v is maximal. Ties can be broken either at ranom, or in a systematic manner. Once the target neighbour v is chosen, then one of the Z +u v packets hel by u an useful to v is chosen, an forware from u to v, at rate c u. Packet selection: We now escribe how packets are electe for transmission once a noe s capacity becomes available. For non-source noes u, who have chosen to transmit to some most eprive neighbour v, then the packet to be transmitte is selecte at ranom among all the possible Z +u v possible choices. For the source noe s, having chosen to transmit to some most eprive neighbour v, the following strategy is use: if the source has a packet that it has not sent to anyone before (a fresh packet), that is if X {s} > 0, then one such fresh packet is forware to noe v; if no such fresh packet is available, then the packet to be forware is selecte uniformly at ranom from the Z +s v possible choices. As in the ege capacitate case, the state space consists in the collection of variables X S, for all S S, an the collection of active packet states A = ((W,F ),...,(W m,f m )). The constraints on these active packet states are ifferent though: we now assume that each noe forwars a packet to only one of its neighbours at a given time. Thus for each noe u, there is at most one ege (u,w) appearing in the sets F i, i =,...,m. Otherwise the same constraints apply: for a given active packet (W,F), an each ege (u,v) F, necessarily, u W an v / W ; also, there is no other ege (u,v) pointing towars v in F. We shall assume that packet transmissions are not preempte, even if a neighbour of some noe u becomes more eprive than the neighbour v to which noe u is currently transmitting. As in the ege-capacitate case, we assume Exponentially istribute inter-event timers. Specifically, the time for transmission of a packet from some noe u is exponentially istribute with mean /c u, an fresh packets arrive at the source noe s at the instants of a Poisson process with rate λ.

16 6 B. Flui limits We first efine the caniate flui trajectories for the system uner consieration: Definition 2: The real-value, non-negative functions (y S ) S S are calle flui trajectories of the noe-capacitate system if the following properties hol. For all S S, u S, v / S such that (u,v) E, there exist non-ecreasing, Lipschitzcontinuous functions φ S,(uv) with Lipschitz constant c u, such that Equations (6) hol. Furthermore, using notation y +u v := y S, S S:u S,v/ S for all S S, u S, the functions {φ S,(uv) } v/ S,(uv) E are ifferentiable at almost every t, an if v:(u,v) E y +u v(t) > 0, their erivatives satisfy: t φ S,(uv)(t) = 0 if y +u v (t) < max (y +u v (t)), (23) v :(u,v ) E t φ S,(uv)(t) = c u. (24) v:(uv) E S:u S,v/ S If u s, that is for a non-source noe, one also has, for all v such that (uv) E an assuming the conition t φ S,(uv)(t) > 0 hols, the following equation: S:u S,v/ S S/u S,v / S, t φ y S (t) S,(uv)(t) = S :u S,v/ S y S (t) For the source noe s, one has the following: y {s} > 0 v s S :u S,v/ S t φ S,(uv)(t) (25) t φ {s},(sv)(t) = c s. (26) In the case where y {s} = 0, one then has for all v such that (sv) E, assuming the conition hols, the following: S S/S {s},v / S : S S:S {s},v/ S t φ S,(sv)(t) = t φ S,(sv)(t) > 0 y S (t) S S:S {s},v/ S y S (t) S S:S {s},v/ S t φ S,(sv)(t) (27) We now establish the following Theorem 5: The statement of Theorem 2 hols true with (X N (t),a N (t)) enoting the state of the process corresponing to the noe-capacitate system, an with S(x) enoting

17 7 the set of flui trajectories efine in Definition 2. That is, rescale trajectories converge in probability to the set of flui trajectories. Proof: Introuce the functions t Φ N S,(uv) (t) := P u c u W=S,(u,v) F s, 0 (W,F) A N (s ) where P u are inepenent, unit rate Poisson processes. The existence of functions φ S,(uv) that are non-increasing an Lipschitz continuous with Lipschitz constant c u, an such that for functions y S given by (6), the claime convergence in probability hols, is establishe exactly as in the proof of Theorem 2, an hence the etaile argument is omitte. It only remains to establish properties (23 27) of the erivatives t φ S,(uv)(t). Fix thus h > 0, an consier the quantity h ( N ΦN S,(uv) (N(t + h)) N ΦN S,(uv) (Nt) ). (28) Assume that the noe u is such that the limiting processes (y) satisfy y +u v (t) > 0. (29) v u Then, provie y +u v (t) < max v v y +u v (t), by Lipschitz continuity of the limiting trajectories, the same inequality hols throughout the interval [t, t + h]. Thus, by convergence of the rescale trajectories to the flui limits, for large enough N, neighbour v is never selecte for transmission by noe u over the whole interval [Nt,N(t + h)]. It then follows that the term (28) converges to 0 as N. This establishes (23). Note next that, when (29) hols, for large enough N one has the following equality: v u,s S:u S,v/ S h ( N ΦN S,(uv) (N(t + h)) N ΦN S,(uv) (Nt) N v u,s S:u S,v/ S ) = N (P u(n(t + h)) P u (Nt)). This is because noe u s capacity is always use when there are packets that noe u can usefully transmit. This ientity guarantees that ( lim h N ΦN S,(uv) (N(t + h)) ) N ΦN S,(uv) (Nt) = c u, from which (24) follows. Assume now that for non-source noe u, noe v is such that t φ S,(uv)(t) > 0. S:u S,v/ S Then for all S such that u S, v / S, of all the instants uring the interval [Nt,N(t + h)] at which noe u chooses to sen a packet to noe v, a fraction y S (t)/y +u v (t) + 0(h) + 0(/N) of these choices is towars an ile packet previously replicate at all noes in S.

18 8 Furthermore, once transfer of such previously ile packets has starte, such a packet is electe for transmission by some other noe with probability 0(/N). This thus shows that ] ( ) lim N N [Φ N S,(uv) (N(t + h)) ΦN S,(uv) (Nt) = ys(t) y + 0(h) +u v(t) [ φs,(uv)(t + h) φ S,(uv)(t) ]. S :u S,v/ S Diviing by h an letting h ten to zero establishes (25). Equation (26) follows by similar arguments, relying on the fact that the source noe s forwars fresh packets, whenever there are some available. Equation (27) is also establishe by similar arguments, now relying on the fact that the source, when sening non-fresh packets, selects such packets uniformly at ranom. C. Stability for the complete graph The main result we shall establish is in the case of the complete graph, that is all eges (u,v), u v, are present in E. We then have the following Theorem 6: Assume that the graph G = (V,E) is complete, an that the injection rate λ verifies: λ < min ( c s, u V c u K ), (30) where K = V. Then the Markov process keeping track of the system state uner ranom useful to most eprive neighbour scheuling strategy is ergoic. The proof of Theorem 6 parallels exactly that of Theorem, relying on a combination of Theorem 4 with Theorem 5 (taking the role playe by Theorem 2 in the proof of Theorem ) an of Theorem 7 below (taking the role playe by Theorem 3 in the proof of Theorem ). We shall not reprouce the whole argument, but shall instea only etail the proof of the following result on stability of flui trajectories: Theorem 7: For any y = (y S ) S S R S +, efine the workloa function w(y) as: w(y) = S S y S (K S ), (3) where K = V. Uner the assumption (30), when the graph G is complete, any flui trajectory y as per Definition 2 is such that, for some ɛ > 0, w(y(t)) max(0,w(y(0)) ɛt). (32) Proof: To establish (32), it suffices to show that, for all flui trajectory y, at a point t where y(t) is ifferentiable an y(t) 0, one has t w(y(t)) ɛ. This is true because the function t w(y(t)) is Lipschitz-continuous, which follows from Lipschitz continuity of the iniviual functions t y S (t).

19 9 We istinguish two cases. First, consier the case where at t, for all u V, one has y +u v (t) > 0. (33) Write then, using (6): v u t w(y(t)) = S S (K S ) t y S(t) = λ(k ) S S = λ(k ) u V = λ(k ) u V c u, v:(uv) E S:u S,v/ S t φ S,(uv)(t) u S,v/ S t φ S,(uv)(t) where the last equality follows from (24), which is applicable in view of Assumption (33). Thus in the present case, uner Assumption (30), the time erivative (/t)w(y(t)) ecreases at a constant spee as esire. Consier now the case where for a non-empty set S, all u S are such that y +u v (t) = 0. v u Equivalently, for all S S such that u S, one has y S (t) = 0. It reaily follows that for any noe u V, the set of most eprive neighbours consists precisely of those noes v S. Distinguish now accoring to whether y {s} (t) = 0 or not. In the first case where y {s} (t) = 0, necessarily there must exist T S, T {s} for which y T (t) > 0, by the assumption that y(t) 0. Note now that, by non-negativity of the function t y {s} (t), one must necessarily have: t y {s}(t) = 0, (34) an by the same argument, for all S such that S S, one also has t y S(t) = 0. (35) On the other han, it follows from Equation (23) that the left-han sie of (34) also reas It thus follows from (24) that Using (34 35), write then λ v S S S,S {s} v S t φ {s},(sv)(t) t φ S,(sv)(t) = c s λ > 0. t w(y(t)) = S S:S {s},s S = (K S ) t y S(t) = S S:S {s},s S = (K S ) u S,v S S S:S S = (K S ) v S = (c s λ). t φ S,(sv)(t) t φ S,(uv)(t)

20 20 In the above, we have use the fact that the most eprive noes are those in S, an hence by (23), for all S such that S S =, all u S, v S \ {u}, necessarily t φ S\{v},(uv)(t) = 0, for the capacity of noe u is fully targete towars noes in S. The last case to consier is when y {s} (t) > 0. Then in view of (26), This entails that t y {s}(t) = λ c s. t w(y(t)) = (K )(c s λ) + S S:S {s},s S = (K S ) t y S(t) = (K )(c s λ) S S:S {s},s S = (K S ) u S,v S t φ S,(uv)(t) (c s λ)(k ). Thus, it follows that (32) hols, with ɛ = min(c s λ, u V c u (K )λ). V. LIMITED NEIGHBOURHOODS AND MULTIPLE COMMODITIES We now exten the results of the previous section to limite neighbourhoos an multiple commoities. A. Limite neighbourhoos Given a set V of noes u with associate capacities c u, we assume that each noe has at any given time a finite set of neighbours N(u), that it can sen to. We aapt the most eprive neighbour selection rule to this context, by requiring that each noe u sens to the most eprive noe v from its limite neighbourhoo N(u). In aition, we assume that for any v V \ {u}, noe u contacts noe v at the instants of some Poisson process S uv with intensity σ uv. It then upates its neighbourhoo as follows. It first as v to it, an then removes a least eprive peer from the resulting set (breaking ties at ranom). Thus, the neighbourhoo is eventually moifie, but remains of constant size, say u. In this context, we have the following: Theorem 8: Consier the most eprive ranom useful selection mechanism with aaptive neighbourhoos as previously efine. Assume that the sampling rates s uv an the neighbourhoo sizes u are positive. Then the resulting Markov process is ergoic uner the conition (30) on the injection rate λ. Proof: The result follows from the fact that the Markov process keeping track of the states of packets, as well as the composition of the ynamic neighbourhoos, amits exactly the same flui limits as the process with full neighbourhoos consiere in the previous section. Ergoicity then follows exactly as in the proof of Theorem 6. We shall only show that caniate flui trajectories must satisfy (23); we omit the etaile arguments for the other equations (24-27), since these consist in very similar aaptations of their proofs for the complete graph scenario.

21 2 For fixe h > 0, consier the quantity h ( N ΦN S,(uv) (N(t + h)) N ΦN S,(uv) (Nt) Assume that noe u is such that the limiting processes (y) satisfy y +u v (y) > 0. v u ). (36) Then, provie y +u v (t) < max v u y +u v (t), by Lipschitz continuity of the limiting trajectories, the same inequality hols throughout the interval [t, t + h]. Let w be some arbitrary noe that achieves the maximum in y +u v (t) over v. Let now ɛ (0,h) be some fixe, arbitrary number. Then necessarily, there exists a subsequence (still enote by N) uner which the sampling process S uw is such that S uw (N(t + ɛ)) S uw (Nt) > 0 for large enough N. For large enough N, no transfer from u to v can take place uring [N(t+ɛ), N(t+h)]. Inee, as previously explaine, v is strictly less eprive than w over this interval; however, since w has been consiere for inclusion in the neighbourhoo of u uring [Nt,N(t + ɛ)], either it has been ae, or an even more eprive noe was present in the neighbourhoo at time N(t+ɛ). Thus, throughout the remaining interval [N(t+ɛ),N(t+h)], noe u always has a neighbour that is more eprive than v, an never sens to v throughout this interval. Thus the quantity (36) is asymptotically no larger than c u ɛ. Since ɛ was arbitrary, then it must converge to zero. This establishes (23). Remark : The mechanism we have consiere for sampling new caniate neighbours consiste in noe u contacting any noe v at the instants of a Poisson process S uv. The ergoicity result oes not epen on the corresponing rate σ uv. It is not har to see that the ergoicity result oes not epen either on the statistical etails of the sampling mechanism. For instance, it woul still hol uner the following neighbour sampling mechanism. Assume a connecte, unirecte graph G S = (V,E S ) is given, an further assume that each noe u V can communicate irectly, at all times, with the noes v such that (u,v) E S. Graph G S is use only for sampling purposes, an this in the following manner. At the instants of a Poisson process S u, noe u picks uniformly at ranom one noe say v in the set N(u), an then picks uniformly at ranom a neighbour of v accoring to the neighbourhoo structure of the sampling graph G S, say noe w. This is the noe that is consiere for inclusion in the neighbourhoo N(u). B. Multiple commoities We now assume that there are multiple commoities, each with a eicate source noe. We let I V enote the collection of such source noes. For each source noe i I, we let V i V enote the set of receivers of the corresponing commoity, incluing the source noe i. Orinary noes can be receivers of several commoities. However, we o not allow source noes to be receivers of any commoities, except the ones they are sources of. Formally, this reas i,j I, i j i / V j.

22 22 We shall also make use of the notation I(v) to represent the set of source noes i that v is a receiver of, i.e. v V i. By convention, for source noes i I, we let I(i) = {i}. In this setup, we enote by λ i the packet creation rate at source noe i an c u the capacity of any noe u V. We still assume exponentially istribute ranom variables for the intertimes between both packet creations at any source noe, an packet transmissions from any noe. In the sequel, we shall assume that noes are only willing to relay content that they are themselves intereste in. This constraint can be thought of as moeling a form of selfishness of users, who will not participate in the elivery of ata they o not nee. This can be relate to the so-calle Tit-for-tat mechanism implemente in BitTorrent, with two important istinctions. Our constraint is impose at the level of commoities, while BitTorrent s Titfor-tat is use to etermine which noe to sen to, in a group of users intereste in the same content. Also, we o not put constraints on reciprocation, i.e. users are not prevente from sening ata to others who o not provie them service in return. Uner these assumptions, the natural necessary conitions on the injection rates λ i of commoities i I for feasibility of broacast are the following: ( V i )λ i i J λ i c i, i I. (37) u i JV i c u, J I. (38) Inee, the first inequality (37) states that each source noe i must have a capacity larger than the rate λ i at which it receives fresh ata. Inequality (38) states that for each set of sources J, the total capacity require to forwar the corresponing commoities to their receivers is no larger than the total capacity of users that can take part in their transmissions. Let us now escribe the scheuling policy we shall consier in the present setup. To etermine who to sen to, any noe u evaluates the overall eprivation of potential receivers v as the total number of packets that it can sen to v, an that v has neither receive, nor is currently receiving. This evaluation is one over all commoities that both noes are receivers of, i.e. over packets generate by all sources i I(u) I(v). Noe u then sens to its most eprive neighbour, where eprivation is measure as just ecribe. Ties between equally eprive noes are broken uniformly at ranom. The ecision of which packet to sen is one as follows. If u is a source noe, it sens a fresh packet if it has any. In any other event, noe u chooses which packet to sen uniformly at ranom among the set of useful packets, i.e. packets originating from sources i I(u) I(v). We refer to these rules as the bunle most eprive - ranom useful strategy, where bunling is over commoities. In this context, we have the following Theorem 9: Assume that conitions (37 38) hol, with strict inequalities. Assume further a complete communication graph, i.e. selection of the most eprive neighbour is mae from the whole collection of noes V. Then the Markov process escribing the state of the system is ergoic.

23 23 Remark 2: The proof will again rely on flui limit techniques. In fact, the flui limits will be the same if we assume as in the previous subsection ynamic, restricte neighbourhoos instea of a complete communication graph. Thus, the above theorem remains true uner the relaxe assumption on the neighbourhoos. We now provie the main lines of the proof. We omit the parts that can reaily be fille in from the previous proofs, an only escribe the flui trajectories in the present setup, the Lyapunov function an the proof that it ecreases along these flui trajectories. C. Flui Trajectories In the multicommoity setup, we let S i enote the collection of subsets of V i that inclue noe i, an ys i (t) enote the quantity of packets generate by source i, an currently replicate at noes u S, at time t. The cumulative number of packets originate from source i, which noe u starte to forwar to noe v while they were replicate at noes in set S over the time interval [0,t] is enote by φ i S,(uv) (t). We now escribe the flui trajectories of the multicommoity system. Definition 3: The real-value, non-negative functions (ys i ) i I,S Si are calle flui trajectories of the multi-commmoities noe-capacitate system if the following properties hol. For all i I, S S i, u S, v / S such that (u,v) V i, there exist non-ecreasing, Lipschitz-continuous functions φ i S,(uv) with Lipschitz constant c u, such that Equations (6) hol. Furthermore, using notation y+u v i := ys, i S S:u S,v/ S for all i I, S S i, u S, the functions {φ i S,(uv) } v V i\s are ifferentiable at almost every t, an if j I v V yj +u v t φi S,(uv) (t) = 0 if (t) > 0, their erivatives satisfy: y j +u v (t) < max v V j I(u) I(v) j I(u) I(v ) i I v V S:u S,v/ S y j +u v (t), (39) t φi S,(uv) (t) = c u. (40) If u / I, that is for a non-source noe, one also has, for all v V, an assuming the conition t φi S,(uv) (t) > 0 i I S:u S,v/ S hols, for all i I(u) I(v) an all S S i such that u S, v / S, the following equation: t φi S,(uv) (t) = ys i (t) j I(u) I(v) yj +u v (t) j I(u) I(v) S S j :u S,v/ S t φj S,(uv)(t) (4)