Keywords: peer-to-peer, live streaming, bandwidth, seeders, performance

Utilisation de pairs auxiliaires pour de la diffusion pair-à-pair en direct Résumé : Les semeurs (pairs possédant déjà un contenu donné et contribuant à sa dissémination) sont un concept clé du pair-à-pair (P2P). Ils permettent entre autres d'accroître les performances d'un système P2P. Mais alors qu'il est naturel d'avoir des semeurs dans le contexte du partage de fichiers ou de la vidéo-à-la-demande, cela semble incompatible avec de la diffusion en direct. Le but de ce rapport est de montrer que, dans une certaine mesure, cela est réalisable.

Après avoir défini formellement le concept de semeur pour la diffusion en direct, et proposé une définition d'efficacité, nous regardons la performance théorique des semeurs pour des systèmes où le coût de contrôle est négligé. Nous proposons ensuite un modèle avec coût de contrôle affine, et donnons les résultats pour un semeur unique tout comme pour un ensemble de semeurs (un ensemble de semeurs ne se comporte pas nécessairement aussi bien que la somme de ses éléments).

Mots-clés : pair-à-pair, diffusion en direct, bande passante, semeurs, efficacité

Introduction

Upload bandwidth is one of the main bottleneck in peer-to-peer (P2P) content distribution, which relies on the upload capacity of its participants to achieve its purpose. The upload resource is all the more critical since most todays high speed Internet access are asymmetric DSLs connections that are not designed to handle P2P traffic and offer relatively low upload capacity, with typical uplink/downlink ratios between 1/4 and 1/20. On the one hand, the democratization of very high speed, symmetric, Internet access like FTTH is expected to improve the upload capacity of P2P systems, but on the other hand the evolution of content quality standards makes the requirements in terms of content size and rate higher and higher: earlier video feeds on the Internet where low quality, requiring streamrates of a few hundred kbps, whereas HDTV implies rate of up to 20 Mbps, possibly more with the upcoming of 3D video content. It is therefore likely possible that upload will still be a major bottleneck of tomorrow's P2P content distribution.

Motivation

In order to increase the available resources, a standard P2P technique is to leverage the capacity of the system by using seeders, i.e. peers that contribute to the system but are (currently) not needing anything. Using seeders is quite natural for file-sharing or Video-on-Demand: after a peer has downloaded its file or video, it becomes a potential seeder for that content. However, it is counter-intuitive live streaming systems: "live" content is created on the fly, so it cannot be pro-actively possessed by peers. Therefore, for a peer to act as a seeder, it has to receive at least a part of the corresponding content, which it does not want to watch by definition.

Scope and contribution

The goal of this paper is to describe the feasibility and performance one can expect from P2P live seeding from a bandwidth budget perspective. This generic theoretical framework can be used to derive simple dimensioning rules and recommendations for the design of P2P live streaming with seeders.

In details, we analyze the seeders' efficiency, which is the useful throughput (goodput) they add to the system, compared to their upload capacity. We provide explicit, tight, upper bounds for efficiency, taking the overhead explicitly into account. We also address the aggregation issues that come from using several seeders. We give conditions and simple diffusion schemes that allow to nearly achieve the theoretical bounds, and provide a few simple examples that illustrate the potential of our findings.

Remark focusing on a single scenario (live streaming) and a single type of peer (seeders) was a deliberate choice, in order to get a clean framework for investigating theoretical performance, especially with regards to the overhead modeling aspects. This does not preclude of possible extensions of the approach presented here to other use cases.

Roadmap

The next Section introduces the models that we use to derive our results. The related work with respect to P2P bandwidth dimensioning is briefly exposed in Section 3. In Section 4, a formal definition of seeders' efficiency is proposed. Section 5 proposes a preliminary study of efficiency for two overhead-free models. This study is a starting point for the main results of this paper, which derive the efficiency of seeders in a model with explicit overhead (Section 6). The validity conditions and applications of the results are discussed in Section 7. Section 8 concludes.

Model

We consider a live content that needs to be streamed to a set of users at a constant rate r. The delivery is handled by a P2P live streaming system. The specificity of live streaming is that the content cannot be prefetched. A play-out buffer may tolerate some jitter, but the live constraints usually limit the size of that buffer to less than a few seconds, so a conservative, yet realistic assumption is that content must received at exactly the rate r during the whole watching experience. To compare with, filesharing usually requires no minimal rate, while in the case of Video-on-Demand, content may be prefetched at a rate greater than r.

C/S/L systems

We classify the nodes of the system into three categories:

• Central servers are in charge of injecting initial copies of the stream into the system. We assume they have a cumulated bandwidth capacity that allows to inject N C copies of the stream, with N C ≥ 1.

• Leechers are peers that want to watch the live content.

• Seeders1 are peers that do not want to watch the live content, but can provide bandwidth to the system.

Remark we do not focus on the way seeders could be enforced in a real live streaming system. However, most of the ideas from P2P filesharing or VoD systems should apply to P2P live streaming. For instance:

• Some peers may remain connected to the system even when idle.

• In a multi-channel system, leechers from an overprovisioned channel may act as seeders for another channel that lacks resources.

• A share-ratio policy can encourage the peers to seed: peers that do not offer enough instant bandwidth may have to act as seeders for a while in order to "pay" their bandwidth debt. That kind of policy can be enforced through penalties (no service guarantee, reduced catalog) and rewards (higher QoS, access to premium content).

• In the case of networks managed by some ISP or content provider, managed seeders may be deployed by the provider to enhance the system performance.

We denote by C, L and S the sets of servers, leechers and seeders respectively. The number of leechers (resp. seeders) is denoted by N L (resp. N S ). Every peer p in L or S has an upload capacity u p ≥ 0 devoted to the service. We assume that the download capacity is always sufficient to support the content rate r and a possible overhead. U X and ūX are respectively the total and average upload bandwidths of set X (ū X = UX NX ). Note that the bandwidth distribution of the seeders may differ from the one of the leechers. For instance, if seeders are former leechers forced to remain because of some share-ratio policy, low bandwidth peers will have to seed longer [START_REF] Benbadis | Playing with the bandwidth conservation law[END_REF], so the average seeders' bandwidth will be lower than the leecher's one. One the other hand, seeders deployed by some content provider should probably have higher bandwidths.

A diffusion scheme for the system is a policy that describes how the content is distributed. We assume here static diffusion schemes: between any two peers (or servers) p and q, the scheme gives a stream of goodput 0 ≤ r p,q ≤ r that is sent from p to q. If 0 < r p,q < r, r p,q is called a substream. For convenience, we consider that the substreams received by a given peer are non-overlapping, so a peer p receives an input of rate i p = q∈{L,S,C} r q,p .

(

Remark overlapping substreams can always be seen as non-overlapping ones: if r p,q and r s,q are overlapping, with redundant data of rate r p∩s,q , we just have to consider rp,q := r p,qr p∩s,q and see a rate r p∩s,q from p to q as overhead.

Of course, choosing which redundant data is treated as overhead is arbitrary. Servers apart, a node cannot send something it doesn't possess, so a diffusion scheme verifies the condition ∀p, q ∈ {L, S}, r p,q ≤ i p .

(

A scheme is a solution of the live diffusion if it ensures that all leechers can view the content, i.e.

∀p ∈ {L}, i p = r.

(3)

Connectivity

In this work, we use an explicit linear overhead to account for connectivity constraints. We also propose two simpler models that will serve for didactic purposes: perfect systems and limited fanout systems.

Perfect systems

In perfect systems, peers can arbitrarily use the upload capacity devoted to the service at no cost [START_REF] Liu | Performance bounds for peer-assisted live streaming[END_REF]. In particular, a perfect system possesses the following properties:

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• No overhead : all the bandwidth capacity can be used to effective data transfer (goodput);

• Unlimited fanout : one peer can send data to an arbitrary numbers of other peers simultaneously;

• Stream continuity: the live stream can be divided into arbitrary small substreams of constant rate.

Limited fanout

As we will see in Section 5, optimizing perfect systems often leads to full mesh solutions, which are not very practical. A first idea to make the model more realistic, without explicitly considering the overhead, is to assume that the number of non-null substreams r p,q is limited: each peer p has a limit c p on the number of outgoing connections it can sustain. This limited fanout implicitly acknowledges the fact that managing a connection has a cost. Perfect systems correspond to the extreme case c p = ∞, for all p ∈ {L, S}).

Explicit linear overhead

In order to get a more realistic and flexible model of real systems, we propose to assume that the overhead is linear: the actual bandwidth used for sending some content at rate e from one peer to another is (1 + a)e + b, for some constants a, b ≥ 0. a is the proportional cost and b the additive cost. For simplicity, we consider that the overhead cost is supported by the sender only (this assumption will be discussed in 6.2.5).

The motivation for this model is that most existing sources of overhead are, at least in a rough approximation, proportional or additive:

• Periodic signaling messages (keep-alive, overlay maintenance) are additive;

• In chunk-based systems, the stream is split into atomic units of data (the chunks) that are distributed independently. For a constant chunk size, the signaling for sending one chunk is expected to induce a proportional overhead;

• The cost for initiating a connection, averaged over the lifetime of that connection, can be considered as additive;

• Some randomized diffusion scheme can have a non-null probability to to send useless data, because it is outdated or redundant [START_REF] Bonald | Epidemic live streaming: optimal performance trade-offs[END_REF]. This can be considered as proportional overhead.

Under the linear overhead model, a peer of bandwidth u maintaining c outgoing connections has a useful output limited to u-bc 1+a . For b > 0, ⌊ u b ⌋ is the maximal fanout sustainable by that peer. For b = 0, the model is indeed equivalent to perfect systems, except that all bandwidth capacities have to be normalized by 1 1+a . The notation used is summarized in Table 1.

Table 1: Table of notation r Streamrate of the content (constant) u p Available upload bandwidth of peer p U X /ū X Total/average upload capacity of population X N X Number of nodes in X N C Normalized capacity of servers (U C = N C R) i p Input rate of node p r p,q Substream from p to q η d (X) Efficiency of set X in diffusion scheme d c p Fanout of peer p a Proportional cost of a connection b Constant cost of a connection R := (1 + a)r + b Bandwidth consummed by goodput r

Related work

Understanding the bandwidth dimensioning is a crucial question in P2P systems, as upload bandwidth is a scarce resource. The bandwidth conservation law [START_REF] Benbadis | Playing with the bandwidth conservation law[END_REF] tells that, if all available bandwidth resources can be used to useful content transfer, then the condition for a live streaming system to admit a solution is

α L + βα S + N C N L ≥ 1, with α X = ūX r , β = NS NL . (4)
In reality, not all bandwidth can be used all the time. Of course, there is the issue of overhead, but other phenomena can prevent from using all available bandwidth. For instance, a peer may have nothing to give at a given time; or some bandwidth may be required for other purposes than feeding the leechers. This explains the concept of efficiency. Taking efficiency into account, equation (4) becomes η(L)α L + η(S)βα S + η(C) NC NL ≥ 1, where η(X) is the efficiency of set X.

(

Efficiency was introduced by Qiu and Srikant [START_REF] Qiu | Modeling and performance analysis of bittorrent-like peer-to-peer networks[END_REF] for BitTorrent-like filesharing systems [START_REF] Cohen | Incentives Build Robustness in BitTorrent[END_REF]. Its role was to quantify the fact that leechers cannot always upload at full bandwidth capacity, as they may lack the content required by others.

In the case of standard peer-assisted live streaming, with no seeders (S = ∅), Liu et al. have shown that one can reach η(L) = η(C) = 1 for perfect and limited fanout systems. In other words, a perfect use of the available bandwidth can be achieved [START_REF] Liu | Performance bounds for peer-assisted live streaming[END_REF].

Defining seeders' efficiencies

We propose to extend the concept of efficiency to seeders as follows: in a given diffusion scheme d, the efficiency η d (s) of a seeder s is the ratio between the data bandwidth it adds to the system and its upload bandwidth u s . In the bandwidth budget, we need to acknowledge that the input rate i s received by s is "wasted" : the rate i s could have been directly sent to some leechers, but instead it is sent to peer s, which does not want to watch the content. We say that s "removes" i s from the pool of useful resources, in a matter of speaking2 . So if in d, s transmits at rates r s,p1 , . . . , r s,pc to c other peers (Figure 1), its efficiency is The efficiency of a set X ⊆ S is defined the same way: we consider the difference between what comes out of X and what enters, all reported to capacity:

η d (s) := c k=1 r s,p k -i s u s . ( 6
)
WVUT PQRS Seeder/ Leecher WVUT PQRS Seeder/ Leecher _^]\ XYZ[ Seeder/ Leecher/ Server is / / ONML HIJK Seeder s rs,p 1 @ @ Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð Ð
η d (X) := s∈X,q∈{L,S\X} r s,q -p∈{C,L,S\X},s∈X r p,s U X . ( 7
)
If we add and subtract the term s,t∈X r s,t in the numerator in (7), we obtain a more compact expression for η d :

η d (X) = s∈X η d (s)u s U X . ( 8
)
Equation (8) tells that η d (X) is also the weighted average of the seeders individual efficiencies.

Optimal efficiency

The optimal efficiency η OP T (s) of a seeder s in a given system is defined as the supremum of the efficiencies it can get over all possible diffusion schemes.

η OP T (s) = sup d (η d (s)) (9)
η OP T (s) is an upper bound for the proportion of the upload bandwidth that can be useful for that system. The same definition stands for the optimal efficiency of any subset X ⊆ S:

η OP T (X) = sup d (η d (X)) (10)
However, there is no guarantee that the individual optimal efficiencies of seeders can be aggregated, because they may correspond to distinct schemes (a counterexample is given in Section 5). As a consequence, Equation (8) becomes an inequality when considering optimal efficiency:

η OP T (X) ≤ s∈X η OP T (s)u s U X . (11)
For convenience, subscripts may be omitted when there is no ambiguity. We may also use metonymic notation in order not to clutter notation: η(y) may denote the efficiency of a seeder characterised by some property y (like the input rate, upload bandwidth, fanout, . . . ).

Perfect and limited fanout systems

In this section, we derive the optimal efficiency of seeders when there is no explicit overhead.

Perfect systems

The optimal performance of seeders in a perfect system is given by the following theorem:

Theorem 1. The optimal efficiency of a subset X ⊆ S of seeders is

η(X) = (1 - 1 N L ) min(1, N L r U X ). ( 12
)
Proof. First we give a scheme that achieves the efficiency given by (12). The scheme is the following: each seeder s ∈ X receives from the servers a distinct substream of rate us NL (if U X ≤ N L r) or us US r (otherwise), and broadcasts that substream to the N L leechers. Under that scheme, the input received by X from nodes outside X is min( UX NL , r), and the output given to leechers is min(U X , N L r). Subtracting the input from the output and dividing by U X gives the efficiency η(X) from (12).

Then, we need to prove that η(X) cannot be greater than (1-1 NL ) min(1, NLr UX ). If I X is the input received by X in a given scheme, the corresponding useful output cannot be more that min(U X , min(I X , r)N L ) because :

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• U X is the capacity of X;

• min(I, r) is the maximal rate of information that X can get. The best it can achieve is to send that rate to the N L leechers: sending it to more peers, for instance seeders outside X, would be ineffective because all leechers already get the information received by X.

Given the input and output rates, and according to Equation (7), the efficiency of X for a given input

I X is bounded by min(1, I X N L U X , r N L U X ) - I X U X .
We deduce that the optimal efficiency is bounded by sup

IX ≥0 min(1, I X N L U X , r N L U X ) - I X U X .
If U X ≤ N L r, we get a maximal efficiency 1 -1 NL for I X = UX NL , and if U X ≥ N L r, we get r(NL-1) UX for I X = r. Therefore the efficency is never more than (1 -1 NL ) min(1, NLr UX ). This concludes the proof. Note that the condition U X > N L r corresponds to an overprovisioned system, as the seeders from X have more bandwidth than required to feed the stream r to all leechers by themselves. In the definition of efficiency we proposed, it is normalized by the dedicated upload bandwidth, so overprovisioned systems naturally have lower efficiencies. On the other hand, for any nonoverprovisioned system, Equation (12) simplifies to

η(X) = 1 - 1 N L . ( 13
)
In other words, seeders are asymptotically optimal in a perfect P2P live streaming system. The explanation is that the only bandwidth waste boils down to at most one streamrate redirected to them for replication.

Limited fanout

Each seeder s has now a limited fanout c s . Without loss of generality, we assume that ∀s ∈ S, c s ≤ N L .

Theorem 2. The optimal efficiency of a single seeder s ∈ S with limited connections c s is

η(s) = (1 - 1 c s ) min(1, rc s u s ). ( 14
)
In particular, if rc s ≥ u s (the fanout is high enough for allowing to use all the upload of s), we just have

η(s) = 1 - 1 c s . ( 15
)
Proof. As s cannot reach more than c s peers, we just consider a sub-system made of C, s and c s leechers, and we conclude by applying Theorem 1, with c s instead of N L .

The bad news is that this result stands for a single seeder, and is not easy to extend to a set of seeders. Equation (11) can be a strict inequality, meaning that efficiency is lost in the process of making multiple seeders work together. Consider for instance a toy system made of N L = 3 leechers and two seeders s 1 and s 2 with parameters u 1 = 3 2 r, c 1 = 2, u 2 = r, c 2 = 3. Using Equation ( 14), we get η OP T (s 1 ) = 1 2 and η OP T (s 2 ) = 2 3 , so

η OP T (s 1 )u 1 + η OP T (s 2 )u 2 u 1 + u 2 = 17 30 .
But if we try to find a scheme that maximize the efficiency of {s 1 , s 2 }, the best solution leads to

η OP T ({s 1 , s 2 }) = 8 15 < 17 30 .
The good news is that for specific scenarios, we can have η OP T (X) = s∈X ηOP T (s)us UX

. This is for instance the case when X is proportionally homogeneous.

Theorem 3. Consider a set X ⊆ S that is proportionally homogeneous, i.e. there is a rate e so that u s = ec s for all s ∈ X. Then, for

N X ≤ ⌊ NL-1 maxs∈X (cs)-1 ⌋⌊ r e ⌋ η OP T (X) = s∈X (1 -1 cs )u s U X = s∈X η OP T (s)u s U X . (16)
Note that although we did not precise e ≤ r, it is an implicit condition: otherwise, the result only apply for N X ≤ 0, or in other words, the empty set.

Corollary 1. If all seeders in X have the same upload u, maximal fanout c, and if

N X ≤ ⌊ NL-1 c-1 ⌋⌊ cr u ⌋, then η OP T (X) = 1 - 1 c . ( 17
)
Remark In the homogeneous case, if we neglect truncation effects, the condition of Corollary 1 corresponds to U X ≤ (N L -1)r c c-1 . As (N L -1) c c-1 ≥ N L (because c ≤ N L ), we get the sufficient condition U X ≤ rN L . Therefore, Corollary 1 can be interpreted as follows: in the homogeneous limited fanout model, up to truncation effects, efficiencies can be aggregated without loss for any nonoverprovisionned subset X.

Proof. Given Equations ( 11) and (15), we just need to give a diffusion scheme d such that η d (X) = s∈X ηOP T (s)us UX . That diffusion scheme is the following: the streamrate r is divided into ⌊ r e ⌋ distinct substreams of rate e. We then build up to ⌊ r e ⌋ trees such that: each seeder s in X is an internal node for exactly one tree, having exactly c s = us e children; the leaves are taken among the leechers; a leecher can belong to several trees, but is contained at most once per tree.

A given tree can have up to N L leaves, but no more. We deduce that one tree can contain ⌊ NL-1 maxs∈X (cs)-1 ⌋ seeders, because a tree with k internal nodes (from X) has at most k(max s∈X (c s ) -1) + 1 leaves. So the rules of the scheme can be respected if N X ≤ ⌊ NL-1 c-1 ⌋⌊ cr u ⌋.

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In the corresponding diffusion scheme, where each tree is used to transmit one of the ⌊ r e ⌋ distinct substreams of rate e, we verify that each seeder s works at optimal efficiency 1 -1 cs . Equation (8) concludes the theorem. The corollary is just a special case where e = u c and max s∈X (c s ) = c.

Remark We can see in the proof that the bound on N X is actually related to the numbers of seeders that can fit in a tree with the constraints that each seeder s is an internal node with exactly c s children and there are no more than N L leaves. The bound we gave is very conservative, because it assumes max s∈X (c s ) children for all seeders. It may not be tight, especially if c s spans a wide range. However, finding out the optimal number of seeders that can collaborate at optimal efficiency is difficult, as it is equivalent to solving a multiple knapsack problem.

Explicit overhead

From now on, we will focus on the explicit linear overhead model, with proportional cost a and additive cost b. Under this model, the bandwidth required for sending one copy of the stream through a single connection is R := (1 + a)r + b.

One easily checks that η max := r R is the maximal efficiency achievable in our model for any peer (leecher or seeder).

When illustrating our results with numerical example, we consider a live streaming system with r = 100 KBytes/s, a proportional overhead of 10% (a = 0.1), and two possible additive costs, small (b = 1.7 KBytes/s) and large (b = 25 KBytes/s). In the figure, we use the relative efficiency η/η max instead of η, in order to facilitate the comparison between the two overhead settings.

Efficiency of a single seeder: main theorem

The following theorem gives the optimal efficiency of one single seeder when the overhead is linear. Theorem 4. If the overhead follows a linear function, then the optimal efficiency of a seeder s is η(s)

= (NL-1)r us if u s ≥ N L R. If u s < N L R, then we have η(s) =        0 if 0 ≤ u s ≤ 2b, (1- b us ) 2 1+a -ǫ 1 (u s ) if 2b ≤ u s ≤ R 2 b , r R -r us -ǫ 2 (u s ) if u s ≥ R 2 b , with (18)
   0 ≤ ǫ 1 (u s ) ≤ 1 1+a b us 3 2 , 0 ≤ ǫ 2 (u s ) ≤ 1 1+a b us ≤ 1 1+a b R 2 .
Proof. The easy part of the proof is for u s ≥ N L R. This corresponds to an overprovisioned situation where s alone can provide the live content to all leechers. This is the optimal scheme for s, so it is straightforward that η(s) = (NL-1)r us . Equation (18), which corresponds to the case u s < N L R, can be proved in three steps:

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• η 2 goes to -∞ for c going to 0 and +∞. It has a unique maximum

(1- b us ) 2 1+a
, which is reached for c = u b • η 1 = η 2 for c = 1 (corresponding efficiency is 0) and c = u R (corresponding efficiency is r R -r u ). We deduce that the optimal efficiency for given c, η = min(η 1 , η 2 ) is equal to η 1 for 1 ≤ c ≤ u R and η 2 for c ≥ u R . Two cases are then to be considered:

• if u b ≤ u R (that is u ≥ R 2 b ), then η is increasing for 1 ≤ c ≤ u R , decreasing for c ≥ u R . The maximal efficiency is therefore r R -r u , reached for c = u R ; • if u b ≥ u R (that is u ≤ R 2 b
), then the maximal efficiency is the one of η 2 ,

(1- b us ) 2 1+a
, reached for c = u b .

Bounding the quantification gap

While the optimal value c OP T we found is a real number, only integer value are eligible. However, as the function η = min(η 1 , η 2 ) always admits a unique maximum, the effective optimal efficiency η(s) is necessarily max(η(⌊c OP T ⌋), η(⌈c OP T ⌉)).

In particular, we have η(c OP T + 1) ≤ η(s) ≤ η(c OP T ), from which we deduce

η(s) = η(c OP T ) -ǫ, with 0 ≤ ǫ ≤ η(c OP T ) -η(c OP T + 1)
From there, noticing that η = η 2 for c ≥ c OP T , we get

η(c OP T ) -η(c OP T + 1) = b u - 1 cOP T (cOP T +1) 1 + a . • If u ≤ R 2 b , then c OP T = u b , so we get η(c OP T ) -η(c OP T + 1) = b u (1 -1 1+ √ b u ) (1 + a) ≤ ( b u ) 3 2
(1 + a) ;

• if u ≥ R 2 b , we just use η(c OP T ) -η(c OP T + 1) ≤ b u(1 + a)
, and note that b u ≤ ( b R ) 2 . This concludes the proof.

Efficiency of a single seeder: discussion

Following theorem 4 and proof, the following remarks can be made. In other words,

η(s) ≈    (1- b us ) 2 1+a if 2b ≤ u s ≤ R 2 b , η max -r us if u s ≥ R 2 b . ( 20
)
To illustrate the validity of this approximation, Figure 2 compares it to the exact efficiency for the two numerical settings we proposed at the beginning of this Section. We can see that the difference is barely noticeable for a large additive overhead, and invisible for a small one.

Low/medium bandwidth

The case u s ≤ R 2 b can be interpreted as the upload bandwidth is no more than R b times the rate R. In most practical situations, one would expect b ≪ R, so most seeders would probably fall in this case, which corresponds to low, medium and reasonably high bandwidths. Within this range, it is interesting to note that both the optimal number of connection and corresponding efficiency are independent of r. Moreover, one can note that the number of connections, us b , is quite similar to the empirical formula used in the current BitTorrent mainline client, √ 0.6u [START_REF] Carra | On the impact of greedy strategies in bittorrent networks: The case of bittyrant[END_REF]. This makes

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us think that the results given here could be adapted to other scenarios than live seeding (this would need to be further investigated in a future work). The 0.6 factor would corresponds to an additive connection cost b ≈ 1.7 KBytes/s, which explains why we use this value as one of our numerical settings (the other value, b = 25 KBytes/s, is totally arbitrary).

(Very) high bandwidth

For very high bandwidths (corresponding for instance to seeders managed by some provider), the efficiency tends to η max as u s goes to infinity (under the assumption that the scenario is not overprovisioned, i.e. us NL < R): super-seeders can asymptotically reach the best achievable efficiency given the overhead constraints.

Importance of input shaping

Seeders do not need to get the whole streamrate. This fact allows to adjust their input rate as desired, which is a key to achieve optimal efficiency.

For instance, under the assumption that the input rate of a seeder s is r, one easily checks that its best achievable efficiency is

η r (s) = max(0, r(⌊ us R ⌋ -1) u s , 1 -b us ⌈ us R ⌉ 1 + a - r u s ) (21)
(the case 0 corresponds to situations where the best choice is not to use s, saving the input rate). 3 gives a graphical comparison of η OP T and η r . While seeders with optimized input rates can get a decent efficiency starting from a few b's of upload bandwidth, if the input is r, seeders with an upload bandwidth less than R are totally inefficient (they cannot give more than they receive, so the best choice is not to use them). We also notice that the difference remains important even for higher upload bandwidth, especially if the additive overhead is small.

About receiver-side overhead

In our model, we made the assumption that the burden of the overhead was only on the sender. A more general model would consist in assuming that in addition to the sender overhead of parameters (a, b), there is a receiver overhead of parameters (a r , b r ) (if p receives a streamrate r q,p from q, it has to use an upload bandwidth a r r q,p + b r ).

Theorem 4 and proof can be adapted to the general model, at the price of increased complexity. For instance, in the medium range scenario (2b < u s ≤ R 2 b ), we have an optimal (continuous) number of connections

c OP T = u s b . (22)
In the general model, this would become

c OP T = -a r b + b (a + a r + 1) (u -b d -ab r + a r b + au) b (a + 1) . ( 23
)
We see that formulas get much more complex in the general model. However, if one compares the practical values given by ( 22) and (23), we see that the general behavior remains practically the same. This is depicted in Figure 4 (receiver overhead is assumed to be the same that the sender overhead, i.e. a r := a and b r := b).

As the added complexity does not seem to bring lot of practical difference, we choose to discard the receiver overhead in our model. However, the reason we can do that is probably that the natural use of live seeders is to feed them with a single input rate, which reduce the impact of receiver overhead. If we want to extend our framework to leechers, which usually receive multiple substreams from multiple sources, a proper modeling of the receiver overhead may become mandatory.

Efficiency of a set of seeders

Like for the limited fanout model, there is no guarantee that the optimal single efficiencies of seeder can be aggregated in a common scheme. In the following, we propose two heuristics that allow to somehow adapt Theorem 3 to the overhead model: the mono-rate and dichotomic rates diffusion schemes.

Mono-rate scheme

The idea of the mono-rate approach is somehow simple: if a set of seeders agree to a common substream rate e, they can behave as a proportionally heterogeneous set. Their efficiency obeys to the following theorem: Theorem 5. Consider a set X ⊆ S that verifies:

• ūX ≤ 2R 2 b ; • N X ≤ ⌊ NL-1 ⌊ max s∈X (us ) E ⌋-1 ⌋⌊ R-b E-b ⌋, with E = būX 2 .
Then, if all seeders on X agree on a common rate e := E-b 1+a used for all inputs and outputs, the efficiency η e (X) of the corresponding scheme verifies

(1 - 2b ūX ) 2 1 + a < η e (X) ≤ (1 - b ūX ) 2 1 + a (24)
Proof. Consider a given rate e ≤ r. Call E := (1 + a)e + b the corresponding rate with overhead. The maximal efficiency of a seeder s having e as input and ouputs is reached when s opens the maximal number of outgoing connections allowing to stream e. This leads to

η e (s) = (⌊ us E ⌋ -1)e u s . In particular, e E -2 e u s < η e (s) ≤ e E - e u s .
Assume that the number of seeders in X is small enough to allow perfect aggregation of efficiencies, like for Theorem 3 (the corresponding condition will be derived later). We then have η e (X) = s∈X ηe(s)us

UX

, therefore

e E -2 e ūX < η e (X) ≤ e E - e ūX .
The maximal value of e E -e ūX is

(1- b ūX ) 2 1+a
, proving the right part of (24). The maximal value of e E -2 e ūX is

(1- 2b ūX ) 2 1+a
, and it is reached for E = būX 2 . As we have E ≤ R, this implies ūX ≤ 2R 2 b . We then need to give a sufficient condition for aggregating the efficiencies without losses. We can use the condition from Theorem 3, N X ≤ ⌊ NL-1 maxs∈X (cs)-1 ⌋⌊ r e ⌋. Noticing that c s = ⌊ us E ⌋ allows to conclude.

Dichotomic scheme

The dichotomic approach consists in the diffusion of several substreams whose rates are dividers of r, instead of using a single rate e. In details, the predetermined substreams are:

• The video stream of rate r, which can be split into

• 2 non-overlapping substreams of rate r 2 , each of which can be split into 2 substreams • . . .

• 2 kmax non-overlapping substreams of rate r 2 kmax , for some k max ≥ 0. k is called the level of a substream of rate r 2 k A seeder s is said to operate at level k if it behaves as follows:

• it receives as input a level k substream; let l := k be his working level;

• As long as s has a residual upload bandwidth greater than b and l ≤ k max , do:

if there is not enough residual upload bandwidth to establish a new output of level l,

then l = l + 1 (a children substream of the current level l substream is chosen),

else create a new output of level l.

The corresponding efficiency is denoted η k (s). In order to optimize the dichotomic approach, each seeder operates at a level that maximizes its single efficiency, i.e. chooses a level k s such that η ks (s) = max 0≤k≤kmax η k (s). The corresponding efficiency is denoted η Bin (s).

As the operating rate is necessarily a divider of r, η Bin (s) is necessarily suboptimal. However, the different levels allow enough freedom to get an efficiency close enough to be optimal. For instance, Figure 5 gives a graphical comparison of η Bin (s) and η OP T (s), using k max = ⌊log 2 ( r b )⌋ (this is an arbitrary choice that corresponds to stopping the subdivision when substreams need more overhead that their actual goodput). One observes that the individual efficiency loss is quite sustainable, especially for a low additive overhead.

For a given set X of seeders, the construction of a dichotomic diffusion scheme is rather simple: • all seeders operating at level k organize to achieve up to 2 k diffusions tree for the level k; each seeder try to join the level k diffusion tree which currently possesses less leaves.

• if a level k seeder has outputs of level k ′ > k, they can either be directly transmitted to leechers or serve as root for a level k ′ diffusion tree;

• if some seeders at level k miss the input streamrate to build their diffusion scheme, they may use a leaf from a parent substream diffusion tree (some of parent rate will be wasted).

Under some conditions, we can evaluate the efficiency of X under a dichotomic diffusion. Theorem 6. If, for a given set X ⊆ S, we have U X ≤ N L R, and if all nonempty diffusion trees can be rooted with proper input, then the efficiency η Bin (X) of X under a dichotomic diffusion verifies

s∈X η Bin (s)u s U X - rk max U X ≤ η Bin (X) ≤ s∈X η Bin (s)u s U X ( 25
)
The interpretation is the following: up to a term rkmax UX , which is small if U X is big enough, the individual dichotomic efficiencies, which are close to the optimal individual efficiencies, can be aggregated without loss.

Proof. The condition U X ≤ N L R ensures that no diffusion tree has more leaves than there are leechers in need of the corresponding substream. This can be shown by induction:

• at level 0, the diffusion tree cannot have more than ⌊ UX R ⌋ leaves, which is smaller than N L .

• at level k, let U k denote the bandwidth that remains after the bandwidth consumed from lower level is substracted; let N k the maximal number of leechers that can be leaves at that level (a given leecher is counted with multiplicity equal to the number the level k substream it needs; let M k the number of leechers that get a level k substream (with multiplicity). Note the relation

N k = 2(N k-1 -M k-1
), i.e. the maximal number at a given level is twice the slots that have not been filled in the previous level. Assume that

U k-1 ≤ N k-1 ((1 + a) r 2 k-1 + b)
, that is at level k -1, the residual bandwidth is not overprovisioned compared to the number of possible leaves Then we have

U k ≤ U k-1 -M k-1 ((1 + a) r 2 k-1 + b) ≤ (N k-1 -M k-1 )((1 + a) r 2 k-1 + b) ≤ N k ((1 + a) r 2 k + b 2 ) ≤ N k ((1 + a) r 2 k + b)
So at any given level, a diffusion tree can always find a leecher to give its output to. Therefore the only waste compared with individual efficiencies lies when the root input of a tree comes from a parent substream. This is bounded by r when considering all roots at a given level k > 0, leading to a total waste bounded by rk max . Normalizing by U X concludes the proof.

Comparison of the two methods

The mono-rate approach is simple to describe, which makes it a good proof of concept of using multiple seeders in a system with overhead. However, the dichotomic approach, although more complex, has many advantages over the mono-rate approach that make it more suitable for a practical use.

Firstly, the substreams are pre-determined, while mono-rate requires to determine the proper input rate e, which depends on ūX . Among other things, this facilitate considerably the interaction with the leechers' diffusion process. Furthermore, under the dichotomic approach, a seeder s can determine its operating level by itself (it is just a function of u s ) while in the mono-rate approach, knowing ūX implies some knowledge of the whole set X. This is even worse when considering dynamics in X: A change in e = f (ū X ) requires a complete upset of the diffusion trees in the mono-rate approach, while changes are expected to be mostly local in the dichotomic approach.

Also note that as streamrate are dividers of r, the quantification effect ⌊ r e ⌋ that may limit the mono-rate approach (cf Theorem 5) has no equivalent in the dichotomic approach.

Finally, the mono-rate approach can force lot of seeders to use an input rate that is far from the single seeder optimal. This impact is bounded (cf Theorem 5), but can be non negligible, especially if the seeders' bandwidths are highly heterogeneous. In contrast, the dichotomic approach adjusts afor each seeder s a level k s such that the input rate is to far from the optimal.

Discussion

Leecher diffusion process

We did not consider in details the way to make the diffusion processes of leechers and seeders work together. This is a problem in itself, which deserves a separate study. The study performed in [START_REF] Liu | Performance bounds for peer-assisted live streaming[END_REF] seems to be adaptable to the case with seeders, at least for the limited fanout model, but a further work is required to transpose the results to the overhead model (including keeping in mind the existence of receiver-side overhead).

However, we argue that knowing how to optimize the diffusion process of seeders alone is not a bad starting point.

Make a minimal use of seeders

While all this paper is devoted to make the best possible use of seeders, we should recall that in the design of a real system, targeting the maximal seeder efficiency is not necessarily the smartest thing to do.

In fact, seeders "waste" their input rate by design, which makes them inherently less efficient that leechers. Therefore, one should use seeders as minimally as possible. The proper way to use seeders is:

• Try to achieve the most of the content diffusion by using the servers and leechers alone. If possible, the leechers should perform a lossless diffusion of a common substream of rate r ′ ≤ r among all of them instead of a partial or lossy diffusion of rate r;

• if r ′ < r, use seeders to finish the job. This is were the results of this paper apply, which describe the best one can expect from seeders and how to achieve it.

Application: dimensioning a scalable live streaming system

Many dimensioning rules can be derived by using the formulas we proposed. For instance, determining if the system is scalable would consist in checking if η(L)α L + η(S)βα S ≥ 1 [START_REF] Benbadis | Playing with the bandwidth conservation law[END_REF]. If we assume here for simplicity homogeneous bandwidth u, η(S) = η OP T (u) (neglecting aggregation issues), and optimal leechers' efficiency η L = η max3 , one can derive the relationship that u and β must verify for the system to be scalable:

βη OP T (u) ≥ r u -η max . ( 26
)
If β, which indicates the ratio between idle (seeders) and active (leechers) users, is a given parameter of the system, Equation (26) can be used to derive the bandwidth u that is required for the system to be scalable. This is illustrated by Figure 6 (the performance of the perfect system, i.e. a = b = 0, is also plotted for comparison). Notice how even little values of β (less than 1) can give significant decrease of the required bandwidth, which is R for a seedless system with perfectly efficient leechers.

About delays

We do not have taken delay issues into account. The diffusion delay is obviously a major concern in the design of a live streaming system. However, it should be noted that the two heuristics we proposed are based on diffusion trees. Therefore the induced delay is at most equal to the delay of a single connection times a logarithm of N L . This is exactly the same type of delay that is experienced for diffusion based on leechers only, so we argue that using seeders should not impact the delay performance of a P2P live streaming system.

Conclusion

In this paper, we gave the keys to understand how seeders could be used in P2P live streaming if servers and leechers do not suffice. After a preliminary work on perfect and limited-fanout systems, we conducted our study on a model with linear overhead. Although this is a preliminary study, with results that are more theoretical than practical, we believe that the present work may have a significant impact in the design and dimensioning of live streaming systems using seeders.

In a future work, we plan to pursue the matter of leechers/seeders interaction in the general overhead model. We also think that the concept of live seeders introduced here could be extended to a more general concept of half-seeders,

1

  Upload bandwidth u (in KBytes/s) Relative efficiency η η max b = 1.7 KBytes/s, continuous b = 1.7 KBytes/s, discrete b = 25 KBytes/s, continuous b = 25 KBytes/s, discrete

Figure 2 :

 2 Figure 2: Validity of the continuous approximation of the optimal efficiency

1Figure 3 :

 3 Figure 3: Impact of a badly shaped input rate

Figure 4 :

 4 Figure 4: General overhead model vs simple overhead model

Figure

  Figure3gives a graphical comparison of η OP T and η r . While seeders with optimized input rates can get a decent efficiency starting from a few b's of upload bandwidth, if the input is r, seeders with an upload bandwidth less than R are totally inefficient (they cannot give more than they receive, so the best choice is not to use them). We also notice that the difference remains important even for higher upload bandwidth, especially if the additive overhead is small.

Figure 5 :

 5 Figure 5: Dichotomic vs optimal individual efficiencies

  ratio β Required bandwidth u (in KBytes/s) b = 1.7 KBytes/s b = 25 KBytes/s Ideal system (a = b = 0)

Figure 6 :

 6 Figure 6: Average bandwidth required for scalability

RR n°7608

The terms leecher and seeder comes from the BitTorrent vocabulary[START_REF] Cohen | Incentives Build Robustness in BitTorrent[END_REF].RR n°7608

In fact, deciding whose peer is responsible for the "waste" of is is arbitrary, and one could decide to substract is from the bandwidth of the senders. However, making the seeders responsible for their own input rates make the analysis simpler.RR n°7608

The efficiency of leechers should take into account the number of outgoing connections like we did for the seeders. However, η L is not the main matter of this paper, so we assume without remorse perfect efficiency ηmax.RR n°7608

i.e. seeders with not all resources expected from a traditional seeder. Studying half-seeders could allow to extend our results to all P2P content distribution systems, including file-sharing and Video-on-Demand systems.

• Finding the maximal efficiency for a given bandwidth u and fanout c;

• Maximizing the corresponding equations for a continuous c;

• Bounding the gap induced by the fact that c has to be an integer.

Maximizing η for given u, c

we first notice that for achieving maximal efficiency, all output rates have to be equal to the input rate: if it is not the case in a given scheme, replacing all output rates by their average value allows to reduce the input rate to that average value (it had to be greater than the maximal output in the original case), increasing efficiency. Therefore the optimal efficiency must be of the form η(s) = (c-1)e u , for some rate 0 ≤ e ≤ r. It is then obvious that one have interest to choose the highest value of e that is feasible.

Note that if c = 1, the seeder can only replicate its input and has null efficiency; the seeder needs to maintain at least 2 connections with spared bandwidth to have a non-null efficiency. This settles that η = 0 for u ≤ 2b. Otherwise, two cases are to be considered:

• if c is the bottleneck (this happens for u ≥ Rc), then s has enough bandwidth to broadcast the whole stream r to c targets, achieving efficiency

For u < RN L , the bottleneck is necessary one of the above, so we deduce that the optimal efficiency for given u and c is The two functions have the following properties:

• η 1 is always increasing, and positive for c ≥ 1;

RR n°7608


Pascal Bador
email: pascal.bador@univ-lyon1.fr

Thierry Lafouge
email: thierry.lafouge@univ-lyon1.fr

Comparative analysis between impact factor and h-index for pharmacology and psychiatry journals

Keywords: Journal Impact Factor, Hirsch index, Journal ranking

)), IF 2006 was compared with h-index 2006 for two samples of "Pharmacology and Pharmacy" and "Psychiatry" journals computed from the ISI Web of Science. For the two samples, the IF and the h-index rankings of the journals are very different.

The correlation coefficient between the IF and the h-index is high for Psychiatry but lower for Pharmacology.

The linearity test performed between the h-index and

showed the great sensitivity of the model compared with  .

The IF and h-index can be completely complementary when evaluating journals of the same scientific discipline.

Introduction

The Thomson Scientific journal Impact Factor (IF) [START_REF]Citation indexes to science: a new dimension in documentation through association of ideas[END_REF] is well known for being the document measure of journal impact. IF is often used to rank scientific journals, despite several recognised limitations well summarized by [START_REF] Hunter | What the impact factor means for surgery journals[END_REF], [START_REF] Delavalle R | Refining dermatology journal impact factors using pagerank[END_REF], [START_REF] Dong | The "impact factor" revisited[END_REF], [START_REF] Hecht | The journal impact factor: A misnamed, misleading, misused measure[END_REF].

HIRSCH [2005] recently suggested a new research performance indicator that is designed for application at the micro level. The Hirsch-Index, or h-index, quantifies as a single-number criterion the scientific output of a single researcher. The h-index is a very simple new measure incorporating both quantity and visibility of publications [START_REF] Bornmann | What do we know about the h index[END_REF]: "A scientist has index h if h of his or her N p papers have at least h citations each and the other (N p -h) papers have fewer than ≤ h citations each" * [START_REF] Hirsch | An index to quantify an individual's scientific research output[END_REF]. For example h-index of 20 means that the scientist has published 20 papers that each had at least 20 citations. BRAUN et al. [200"6] proposed that the h-index could be usefully applied to the citation analysis of journals, as well. The h-index for evaluating the scientific impact of journals as a robust alternative indicator can be an advantageous complement to journal IF. The journal h-index is calculated as follows: "Retrieving all source items of a given journal from a given year and sorting them by the number of times cited, it is easy to find the highest rank number which is still lower than the corresponding 'Times Cited' value. This is exactly the h-index of the journals for the given year" [START_REF] Braun | A Hirsch-type index for journals[END_REF].

In order to progress in the IF/h-index comparison analysis, this study compares IF and h-index using exactly and strictly the same parameters (identical two publication years (2004)(2005) and identical one-year citation window ( 2006)). Hence, we propose here to compare IF 2006 and what we call h-index 2006 for two samples of "Pharmacology and Pharmacy" and "Psychiatry" journals taken from the Journal Citation Reports (JCR) 2006.

Methods

Constitution of the two samples

We ranked the 199 journals of the "Pharmacology and Pharmacy" section of the JCR 2006 drawn from the Web of Science in descending order of IF and we took the first 50 journals to constitute our first sample. We proceeded in the same way to constitute our second sample of 50 journals coming from the "Psychiatry" section (94 journals) of the JCR 2006.

In order to strictly compare their IF and h-index using the same data, we wanted to work on all the citations obtained in 2006 by the articles published in 2004-2005 in the 50 journals of our two samples of "Pharmacology and Psychiatry" and "Psychiatry".

Calculation of h-index 2006

The IF was easily extracted from the JCR 2006 in June 2008 whereas h-index 2006 was calculated manually in July 2008 for each of the 50 journals of our two samples in the following way:

-1) Search for articles of 2004 -2) We displayed the references citing each article obtained ("Times Cited" link) -3) Using the window obtained and the "Refine Results" function, we extracted the number of articles of 2006 (Citations 2006) from the "Publication Years" menu -4) The same procedure for the articles of 2005 Thus, for each of the 100 journals, we compiled a table similar to the one shown in table 1 for the journal "Annual Review of Pharmacology and Toxicology". We then identified the h-index 2006 corresponding to the number h of articles published in 2004-2005 and cited at least h times during 2006.

Please insert Table 1: Calculation of the h-index 2006

Also, for all the articles of 2004-2005 published in the 100 journals, we identified the number of "Reviews" using the "Refine Results" function and the "Document Types" menu. We thus calculated the percentage of Reviews compared with the total number of articles published in 2004-2005.

Correlation between the h-index and the impact factor

We studied the statistical correlation between the IF and h-index. For this, we calculated Pearson's correlation coefficient. If the result of the test was negative, we classified journals characterised (IF, h-index)

N i H IF i i , 1  N i IF IF Z IF i IF i , 1     N i H H Z H i i H , 1    

Let axis

x , impact factor, and axis y ,h-index

The four groups are:

-First quarter:

0 0   y x
high h-index, high impact factor -Second quarter:

0 0   y x
high h-index, low impact factor -Third quarter:

0 0   y x
low h-index, low impact factor -Fourth quarter:

0 0   y x
low h-index, high impact factor 2.4. The laws of information and the h-index We recall the theoretical background obtained in recent papers [START_REF] Glanzel | On the h-index -A mathematical approach to a new measure of publication activity and citation impact[END_REF] and [START_REF] Schubert | A systematic analyse of Hirsch-type indices for journals[END_REF]). The authors interpret, theoretically, relationships between the h-index and IF, given the underlying citation distribution, on the basis of extreme values statistics.

Let X be a random variable.

X represents the citation rate of a paper. n is a given sample with distribution X . The question is how the h-index of a journal is determined by the parameters of X , its expected value (IF) and n , the number of papers published in the journal. (1) hence

    ck k X G ) (
where c is a positive constant (2) Using Gumbel's r-th characteristic extreme value

r u Schubert et al. define the theoretical h-index H :   r u r Max H r   :
where

          n r k G k Max n r G u r ) ( : ) ( 1 (3)

Since

X is Paretian (2) and by elementary manipulations of (3) hence we have:

 1 ) .( r n m u H H   (4)
Applying the Hirsh condition r H  to (4):

 1 ) ( H n m H   m is a positive constant (5)
The expected value of the Pareto distribution (1) (impact factor) IF is:

1    d IF d is a positive constant (6)
while the constant m d  hence (5) (6) implies the principal result of their theoretical background Schubert et al. use (6) in the special case of 2   . We do not support this hypothesis. Hence (6) (7) implies

1 1 1 .       n d H (7)
1 1 1 .       n IF a H
where a is a positive constant (8)

Results

Tables 2A and2B present the data obtained (IF 2006(IF , h-index 2006, number of articles published in 2004-2005, percentage of articles that are reviews), respectively for our sample of 50 journals from "Pharmacology and Pharmacy" and for our sample of 50 journals from "Psychiatry", both coming from JCR 2006 and ranked in descending order of IF 2006. We see that the ranking would obviously be different if we took into account the descending order of h-index 2006. For example, in table 2A,

Please insert Table 2A: Impact Factor 2006 ranked list of the 50 first Pharmacology and Pharmacy journals

Please insert Table 2B: Impact Factor 2006 ranked list of the 50 first Psychiatry journals

One can see that 7 journals are common to the 2 samples studied: Neuropsychopharmacology, International Journal of Neuropsychopharmacology, Journal of Clinical Psychopharmacology, CNS Drugs, European Neuropsychopharmacology, Psychopharmacology, Journal of Psychopharmacology ranked with IF 2006 respectively at 12 th , 20 th , 24 th , 29 th , 39 th , 42 nd and 47 th in the "Pharmacology and Pharmacy" section, were ranked respectively 5 th , 8 th , 10 th , 15 th , 21 st , 24 th and 26 th in the "Psychiatry" section.

As a complement, to illustrate the data in tables 2A and 2B in graph form, we present figures 1A and 1B, which show the ranking of the two samples in descending order of h-index 2006.

Please insert Figure 1A: Comparison of IF 2006 and h-index 2006 for Pharmacology and Pharmacy journals (h-index ranking)

Please insert Figure 1B: Comparison of IF 2006 and h-index 2006 for Psychiatry journals (h-index ranking)

Figure 1A shows a decrease in h-index 2006, revealing 16 steps and therefore 16 groups of "Pharmacology and Pharmacy" journals whose h-index varies from 30 to 5. The h-index 2006 is much higher than the IF 2006 (up to 4 times higher), but with four exceptions: Annual Review of Pharmacology and Toxicology, Pharmacological Reviews, Reviews of Physiology Biochemistry and Pharmacology, and Critical Reviews in Therapeutic Drug Carrier Systems. These four journals have the characteristic that they publish very few articles (respectively 52, 48, 32, and 21 articles in 2 years) but only reviews. We can see that in our sample of 50 journals, 15 journals (meaning 30%) have published between 89% and 100% of articles in the form of reviews.

Figure 1B also reveals 16 steps and therefore 16 groups of "Psychiatry" journals whose hindex varies from 27 to 5. In this second sample, the h-index 2006 is always, and without exception, much higher than the IF 2006 (here also up to 4 times higher for the Psychopharmacology journal). The "Psychiatry" journals publish many less articles in the form of reviews: only two journals (4%), Mental Retardation and Developmental Disabilities Research Reviews, and Progress in Neuropsychopharmacology and Biological Psychiatry, respectively published 100% and 94% of reviews.

For the "Pharmacology and Pharmacy" journals, we obtained a low correlation coefficient of 0.59, which we considered as insignificant. We then categorized the journals and rated them using the method based on the reduced centred coordinates. This way of categorizing is shown in figure 2. Thus, the journals were spread over four quarters, the composition of which is given in table 3. Quarter 1 of the journals, showing the highest IF 2006 and h-index 2006, represents the journals that published the most reviews (on average 71% of the articles) against respectively an average of 22% for quarter 2, 50% for quarter 3 and 35% for quarter 4.

Please insert Figure 2: Standard scores of IF 2006 and h-index 2006 for Pharmacology and Pharmacy journals

Please insert Table 3: Pharmacy and Pharmacology journals

For the "Psychiatry" journals, we obtained a high correlation coefficient of 0.88. In figure 3, we show the scatter plot and the associated linear regression line.

Please insert Figure 3: Correlation of IF 2006 and h-index 2006 for Psychiatry journals

In order to systematically test the validity of Eq (8), H ) ( y was plotted against the product

1 1 1 .      n IF
) ( x using data from the two collections.

To calculate the parameter a of (8) we consider the linear regression line for different values of  close to 2. We choose the value of  that gives the best adjustment (Figure 4A, Figure 4B).

Pharmacy journals

2   1187 . 1 6508 . 0   x y 9157 . 0 2  R 2 . 2   0384 . 1 7055 . 0   x y 9219 . 0 2  R 3 . 2    y 1386 . 1 7273 . 0  x 9196 . 0 2  R Psychiatry journals 4 . 2   9709 . 0 7715 . 0   x y 2 R = 0.9577 5 . 2   9981 . 0 7962 . 0   x y 9577 . 0 2  R 6 . 2   0278 . 1 82 . 0   x y 2 R = 0.9573
Please insert Figure 4A: Correlation of the h-index with n 1/(2.2+1) IF 2.2/(2.2+1) for Pharmacology and Pharmacy journals Please insert Figure 4B: Correlation of the h-index with n 1/(2.5+1) IF 2.5/(2.5+1) for Psychiatry journals For psychiatry, we studied the variation of a according to  with a coefficient of determination remaining higher than 0.93 (Figure 4C). 4C: Variation of a and R 2 according to 

Please insert Figure

Discussion

The interest and originality of our study was, to our knowledge for the first time, to compare rankings based on IF 2006 and h-index 2006, using strictly the same data based on the usual definition of IF (identical two publication years 2004-2005 and identical one-year citation window 2006) for two samples of journals of the health field. The study by [START_REF] Schubert | A systematic analyse of Hirsch-type indices for journals[END_REF] is based on strictly the same parameters as well, but particularly on one publication year and on a three-year citation window beginning with the publication year for both the journal impact measure and the hindex. The results given in tables 2A and 2B show that, for the "Pharmacology and Pharmacy" journals and the "Psychiatry" journals, the two rankings are completely different. The IF allows a ranking using a customary descending order starting from the values of the JCR given to the nearest thousandth. However, the h-index offers a decreasing ranking starting from values that are integers. Therefore, the h-index ranking is much less fine and precise and reveals steps (figures 1A and 1B) each corresponding to a group of journals with the same h-index. We thus reveal 16 groups for our two samples. Also, the amplitude of the h-index values is higher (25 for "Pharmacology and Pharmacy", 22 for "Psychiatry") than that for the IF values (approximately 20 for "Pharmacology and Pharmacy", approximately 12 for "Psychiatry"). Furthermore, we must note that for a given journal the h-index is generally higher than the IF (up to 4 times higher), which was seen in our two samples. Only four journals of the "Pharmacology and Pharmacy" section have an IF higher or equal to the hindex. Moreover, these four journals specializing exclusively in the publication of reviews have the characteristic of publishing a very low annual number (a maximum of around fifty) of this type of article.

Also, the "Pharmacology and Pharmacy" section has the characteristic of having a very big percentage of journals specializing in the publication of reviews. These journals are those that very often present the best IF, which is logical since their review articles are more often cited than the original articles. However, ranking based on the h-index is not very sensitive to the percentage of reviews published.

The "Psychiatry" section has very few journals specializing in the publication of reviews, which makes this type of data not very important for comparing the IF and h-index.

Thus, from our two samples presenting different characteristics as regards the publication of reviews, we cannot deduce any influence from this parameter on the IF and h-index. Therefore, this parameter is not significant for comparing the two rankings.

The second part of this comparative study of the IF and h-index concerned the analysis of their correlation coefficient. Thus, we noted a high correlation for the "Psychiatry" sample. However, the low correlation, considered as insignificant between the IF and h-index of the journals in the "Pharmacology and Pharmacy" sample, led us to try to propose a categorization for it using the traditional method of reduced centred coordinates. The representation thus obtained, figure 2, shows four groups (quarters) of journals. Quarter 1, the most outstanding, includes the ten journals presenting both the highest IF 2006 and h-index 2006. The important fact that we can draw from this representation is that this group contains the journals that publish by far the highest percentage (71%) of articles in the form of reviews, which seems completely logical.

During the linearity test between the h-index and

1 1 1 .      n IF
we showed the great sensitivity of the model compared with  . In fact, as regards psychiatry, if we consider that the linear relationship remains acceptable up to a coefficient of determination of 93%, we can vary  between 1.6 and 4.2, which varies the slope of the straight line between 0.53 and 1.1! Furthermore, we can see that the variation of a according to  is approximately linear. We think that these many adjustments represent the different parameters of adjustment of distribution of articles in each journal. It is remarkable that we find a value close to that found by [START_REF] Schubert | A systematic analyse of Hirsch-type indices for journals[END_REF], who found 0.75 by taking the sizes of corpuses, all sciences combined, of 6000 or specific fields such as chemistry or biology. We also thought it interesting to take a single corpus by merging our two samples: we obtained the best adjustment ( 2 R = 0.94) for

2 . 2  
with a slope of 0.71. This result still seems to confirm Schubert's, meaning a coefficient a independent of the scientific field and an optimum value of  very nearly 2, meaning a Pareto coefficient law close to 3. Unlike Schubert's results, we do not have a zero intercept point, but a point close to 1. This can be explained by the fact that we took only the 50 journals with the highest IF and thus a minimum h-index of 5. In theory, the equation ( 8) is only valid for the extreme values.

More generally, and as shown in our study of two very small samples, the rating of journals starting from the h-index may represent an interesting and complementary alternative to the wellknown rating based on the IF. In fact, the h-index rating proposes a categorization of journals (several journals capable of having the same h-index) making it possible to create classes of journals with the same h-index: e.g. class 5, class 6, class 7 ... class 19, class 25. It is evident that, to have a meaning, this ranking must be made as for IF, within a collection of comparable journals of the same wellidentified scientific discipline. In order to put this new type of ranking into perspective, we could certainly propose to display beside the value of each class the maximum value found for the journal obtaining the best h-index for the discipline studied, using the following model: journal X  h-index=6/25 journal Y h-index=19/25 meaning that journal X is characterized by an h-index of 6 and journal Y by an h-index of 19, with the note that the journals rated first of the discipline studied have an h-index of 25.

This type of ranking by classes of journal is often appreciated and used by experts and scientific committees of evaluation as shown by [START_REF] Vanclay | Ranking forestry journals using the h-index[END_REF] in the study proposing a ranking of forestry journals based on an evaluation of the journals by experts and also on their h-index.

As shown by [START_REF] Braun | A Hirsch-type index for journals[END_REF], for a given journal, the h-index presents different and useful characteristics compared with the IF. Firstly, h-index is insensitive to an accidental excess of uncited papers and also to one or several outstandingly highly cited papers; secondly, it combines the effects of "quantity" (number of publications) and "quality" (citation rate) in a rather specific balanced way that should reduce the apparent "overrating" of some review journals.

The h-index could be very interesting and a complementary tool of IF if it would not be calculated for a "life-time contribution" as suggested by [START_REF] Hirsch | An index to quantify an individual's scientific research output[END_REF] for individual scientists, but for a definite period as we did in this study with the same parameters as IF 2006. In our study, IF and h-index were exactly and strictly comparable and thus complementary in the rating of journals of the same discipline. [START_REF] Braun | A Hirsch-type index for journals[END_REF], who worked on 2001 as source year (one publication year), had to eliminate the first and second journals of the 2001 IF list. Since these journals published 24 and 23 papers, respectively, in 2001, they had no chance to compete with the chart toppers (obviously the h-index cannot be larger than the number of papers it is based on). If we take a two publication year period as in our study, all journals will have published enough articles (probably at least fifty) and this will avoid having to possibly eliminate some journals having very high IF because they published a very low number of articles.

As suggested by [START_REF] Rousseau R | A case study: Evolution of JASIS' h-index[END_REF], one might also consider calculating a relative h-index by dividing it by the yearly number of articles of the journal, which could be another research lead for the assessment of the different ranking methods of scientific journals.

Some questions are open about the link to Pareto's law. For example, how can we interpret the linear variation of a according to  .

  the journal "Antimicrobial Agents and Chemotherapy" ranked thirtieth with IF 2006 (IF = 4.143), was ranked third with h-index 2006 (h-index 2006 = 19), equal to the journal "Annual Review of Pharmacology and Toxicology" (itself ranked first in the IF 2006 ranking, IF 2006 = 22.808). In table 2B, the journal "Psychopharmacology" ranked twenty-fourth with IF 2006 (IF 2006 = 3.625), was ranked seventh with h-index 2006 (h-index 2006 = 15), equal to the journals "British Journal of Psychiatry" (whose IF ranking -seventh -did not change) and "Schizophrenia Research" (itself ranked fourteenth in the IF 2006 ranking, IF 2006 = 4.264).

Figure

  Figure 1A: Comparison of IF 2006 and h-index 2006 for Pharmacology and Pharmacy journals (hindex ranking)

  by two different numerical values (reduced central values) into four groups around an average value. average and standard deviation In order to make this ranking, we calculate the Z standard scores (reduced central values) of

	Let i IF journal's impact factor,	i		,.. 1	N	, IF and		IF	average and standard deviation
	Let	i H journal's h-index	i ,	i		,.. 1	N	; H and

H



Table 1 :

 1 Calculation of the h-index 2006

	Annual Review of Pharmacology and Toxicology
	Articles 2004* Citations 2006 Articles 2005* Citations 2006
	1	17	1	16
	2	33	2	7
	3	21	3	109
	4	11	4	36
	5	14	5	18
	6	6	6	5
	7	18	7	23
	8	23	8	6
	9	49	9	0
	10	18	10	11
	11	67	11	14
	12	26	12	12
	13	6	13	19
	14	28	14	6
	15	25	15	12
	16	33	16	21
	17	49	17	8
	18	27	18	5
	19	18	19	10
	20	33	20	15
	21	17	21	54
	22	54	22	0
	23	17	23	3
			24	4
			25	18
			26	33
			27	12
			28	17
			29	19
	13 papers cited at least 19 times 8 papers cited at least 19 times
	Total: 21 papers 2004-2005 cited at least 19 times
	13 papers cited at least 20 times 6 papers cited at least 20 times
	Total: 19 papers 2004-2005 cited at least 20 times
		h-index = 19
	*Order of the articles given by the Web of Science

Table 2A :

 2A Impact Factor 2006 ranked list of the 50 first Pharmacology and Pharmacy journals

	IF 2006	Abbreviated	IF	h-index	Nb articles	%
	ranking	Journal Title	2006	2006	2004-2005	Reviews
	1	ANNU REV PHARMACOL	22.808	19	52	100
	2	NAT REV DRUG DISCOV	20.970	30	135	63
	3	PHARMACOL REV	16.854	17	48	100
	4	TRENDS PHARMACOL SCI	10.400	21	190	64
	5	PHARMACOL THERAPEUT	8.657	15	137	97
	6	CLIN PHARMACOL THER	8.066	16	212	2
	7	ADV DRUG DELIVER REV	7.977	18	217	100
	8	PHARMACOGENETICS	7.221	13	95	1
	9	MED RES REV	7.218	11	55	69
	10	DRUG DISCOV TODAY	7.152	15	243	98
	11	CURR OPIN PHARMACOL	6.916	17	179	76
	12*	NEUROPSYCHOPHARMACOL*	5.889	16	468	6
	13	CURR DRUG METAB	5.762	11	80	100
	14	DRUG METAB REV	5.754	9	61	69
	15	REV PHYSIOL BIOCH P	5.625	5	32	100
	16	PHARMACOGENET GENOM	5.391	11	87	6
	17	CURR PHARM DESIGN	5.270	16	588	100
	18	DRUG RESIST UPDATE	5.268	9	56	99
	19	CURR MED CHEM	5.207	15	382	93
	20*	INT J NEUROPSYCHOPH*	5.184	9	114	10
	21	CRIT REV THER DRUG	5.000	5	21	100
	22	ANTIVIR THER	4.982	11	228	9
	23	TOXICOL APPL PHARM	4.722	16	472	15
	24*	J CLIN PSYCHOPHARM*	4.561	10	171	4
	25	DRUGS	4.472	14	337	58
	26	MOL PHARMACOL	4.469	16	765	2
	27	CURR OPIN DRUG DISC	4.319	11	138	100
	28	CURR DRUG TARGETS	4.274	10	146	100
	29*	CNS DRUGS*	4.210	11	157	50
	30	ANTIMICROB AGENTS CH	4.153	19	1621	1
	31	CLIN PHARMACOKINET	4.115	12	156	99
	32	CONTROL CLIN TRIALS	4.025	6	40	8
	33	J CONTROL RELEASE	4.012	14	678	3
	34	PHARMACOGENOMICS J	3.957	9	92	8
	35	J PHARMACOL EXP THER	3.956	17	1240	0
	36	J ANTIMICROB CHEMOTH	3.891	17	780	6
	37	NEUROPHARMACOLOGY	3.860	15	513	3
	38	BRIT J PHARMACOL	3.825	15	767	5
	39*	EUR NEUROPSYCHOPHARM*	3.794	10	160	6
	40	DRUG SAFETY	3.673	9	165	48
	41	DRUG METAB DISPOS	3.638	13	461	1
	42*	PSYCHOPHARMACOLOGY*	3.625	15	801	5
	43	PHARMACOGENOMICS	3.603	11	136	51
	44	BIOCHEM PHARMACOL	3.581	15	873	1
	45	CNS DRUG REV	3.474	6	38	89
	46	ALIMENT PHARM THERAP	3.287	14	773	23
	47*	J PSYCHOPHARMACOL*	3.255	8	149	14
	48	INT J IMMUNOPATH PH	3.213	9	127	8
	49	EUR J PHARM BIOPHARM	3.185	11	270	10
	50	EXPERT OPIN INV DRUG	3.174	12	242	72
		*journals belonging to both JCR sections : «Pharmacology and Pharmacy» and «Psychiatry»

Table 2B :

 2B Impact Factor 2006 ranked list of the 50 first Psychiatry journals

	IF 2006	Abbreviated	IF	h-index	Nb articles	%
	ranking	Journal Title	2006	2006	2004-2005	Reviews
	1	ARCH GEN PSYCHIAT	13.936	27	233	3
	2	MOL PSYCHIATR	11.804	21	204	20
	3	AM J PSYCHIAT	8.250	25	592	49
	4	BIOL PSYCHIAT	7.154	22	635	5
	5*	NEUROPSYCHOPHARMACOL*	5.889	16	468	6
	6	J CLIN PSYCHIAT	5.533	16	552	19
	7	BRIT J PSYCHIAT	5.436	15	326	25
	8*	INT J NEUROPSYCHOPH*	5.184	9	114	10
	9	J AM ACAD CHILD PSY	4.767	13	287	41
	10*	J CLIN PSYCHOPHARM*	4.561	10	171	4
	11	AM J MED GENET B	4.463	14	298	1
	12	SCHIZOPHRENIA BULL	4.352	9	122	11
	13	PSYCHOTHER PSYCHOSOM	4.333	9	93	4
	14	SCHIZOPHR RES	4.264	15	503	3
	15*	CNS DRUGS*	4.210	11	157	50
	16	J PSYCHIATR NEUROSCI	4.100	10	70	26
	17	ADDICTION	4.088	11	319	11
	18	ACTA PSYCHIAT SCAND	3.857	11	258	13
	18	PSYCHOSOM MED	3.857	11	287	6
	20	PSYCHOL MED	3.816	11	299	5
	21*	EUR NEUROPSYCHOPHARM*	3.794	10	160	6
	22	J PSYCHIATR RES	3.700	9	140	2
	23	J NEUROL NEUROSUR PS	3.630	13	700	3
	24*	PSYCHOPHARMACOLOGY*	3.625	15	801	5
	25	BIPOLAR DISORD	3.494	11	156	23
	26*	J PSYCHOPHARMACOL*	3.255	8	149	14
	27	DRUG ALCOHOL DEPEN	3.213	10	300	8
	28	J AFFECT DISORDERS	3.138	12	456	4
	29	INT CLIN PSYCHOPHARM	3.080	8	113	5
	30	EUR ARCH PSY CLIN N	3.042	8	120	5
	31	AM J GERIAT PSYCHIAT	2.894	8	198	5
	32	PHARMACOPSYCHIATRY	2.849	6	119	7
	33	PSYCHIAT RES-NEUROIM	2.755	8	151	2
	34	MENT RETARD DEV D R	2.671	8	82	100
	35	PROG NEURO-PSYCHOPH	2.584	11	320	94
	36	DEPRESS ANXIETY	2.549	8	113	5
	37	CAN J PSYCHIAT	2.531	6	192	23
	38	DEMENT GERIATR COGN	2.511	9	223	3
	39	GEN HOSP PSYCHIAT	2.500	7	118	0
	40	J CHILD ADOL PSYCHOP	2.486	7	148	8
	41	PSYCHIAT SERV	2.430	9	316	0
	42	HUM PSYCHOPHARM CLIN	2.386	7	127	19
	43	J INT NEUROPSYCH SOC	2.367	9	188	7
	43	NEUROPSYCHOBIOLOGY	2.367	7	158	2
	45	J PSYCHOSOM RES	2.322	8	273	0
	46	PSYCHIAT RES	2.310	9	303	1
	47	COMPR PSYCHIAT	2.181	7	138	2
	48	WORLD J BIOL PSYCHIA	2.094	5	53	30
	49	CNS SPECTRUMS	2.051	7	175	53
	50	EPILEPSY BEHAV	2.026	8	345	14
		*journals belonging to both JCR sections : «Pharmacology and Pharmacy» and «Psychiatry»

Table 3 :

 3 Pharmacy and Pharmacology journals

	Quarter 1 : 10 journals	Quarter 3 : 23 journals
	ANNU REV PHARMACOL	ANTIVIR THER
	ADV DRUG DELIVER REV	CLIN PHARMACOKINET
	CLIN PHARMACOL THER	CNS DRUG REV
	CURR OPIN PHARMACOL	CNS DRUGS
	DRUG DISCOV TODAY	CRIT REV THER DRUG
	NAT REV DRUG DISCOV	CURR DRUG METAB
	NEUROPSYCHOPHARMACOL	CURR DRUG TARGETS
	PHARMACOL REV	CURR OPIN DRUG DISC
	PHARMACOL THERAPEUT	DRUG METAB DISPOS
	TRENDS PHARMACOL SCI	DRUG METAB REV
	Quarter 2 : 15 journals	DRUG RESIST UPDATE
	ALIMENT PHARM THERAP	DRUG SAFETY
	ANTIMICROB AGENTS CH	EUR J PHARM BIOPHARM
	BIOCHEM PHARMACOL	EUR NEUROPSYCHOPHARM
	BRIT J PHARMACOL	EXPERT OPIN INV DRUG
	CONTROL CLIN TRIALS	INT J IMMUNOPATH PH
	CURR MED CHEM	INT J NEUROPSYCHOPH
	CURR PHARM DESIGN	J CLIN PSYCHOPHARM
	DRUGS	J PSYCHOPHARMACOL
	J ANTIMICROB CHEMOTH	PHARMACOGENET GENOM
	J CONTROL RELEASE	PHARMACOGENOMICS
	J PHARMACOL EXP THER	PHARMACOGENOMICS J
	MOL PHARMACOL	REV PHYSIOL BIOCH P
	NEUROPHARMACOLOGY	Quarter 4 : 2 journals
	PSYCHOPHARMACOLOGY	MED RES REV
	TOXICOL APPL PHARM	PHARMACOGENETICS