Luca Chiaraviglio
email: luca.chiaraviglio@inria.fr

Delia Ciullo
email: delia.ciullo@inria.fr

Marco Mellia
email: mellia@tlc.polito.it

Michela Meo
email: michela.meo@polito.it

Modeling Sleep Mode Gains in Energy-Aware Networks

Keywords: Energy-ecient networks, sleep modes, random graph models

Nowadays two main approaches are being pursued to reduce energy consumption of networks: the use of sleep modes in which devices enter a low-power state during inactivity periods, and the adoption of energy proportional mechanisms where the device architecture is designed to make energy consumption proportional to the actual load. Common to all the proposals is the evaluation of energy saving performance by means of simulation or experimental evidence, which typically consider a limited set of benchmarking scenarios. In this paper, we do not focus on a particular algorithm or procedure to oer energy saving capabilities in networks, but rather we formulate a theoretical model based on random graph theory that allows to estimate the potential gains achievable by adopting sleep modes in networks where energy proportional devices are deployed. Intuitively, when some devices enter sleep modes some energy is saved. However, this saving could vanish because of the additional load (and power consumption) induced onto the active devices. The impact of this eect changes based on the degree of load proportionality. As such, it is not simple to foresee which are the scenarios that make sleep mode or energy proportionality more convenient. Instead of conducting detailed simulations, we consider simple models of networks in which devices (i.e., nodes and links) consume energy proportionally to the handled trac, and in which a given fraction of nodes are put into sleep mode. Our model allows to predict how much energy can be saved in dierent scenarios. The results show that sleep modes can be successfully combined with load proportional solutions. However, if the static power consumption component is one order of magnitude less than the load proportional component, then sleep modes become not convenient anymore. Thanks to random graph theory, our model gauges the impact of dierent properties of the network topology. For instance, highly connected networks tend to make the use of sleep modes more convenient.

Modélisation des gains possibles avec les modes de sommeil dans les réseaux énergétiquement performants Résumé :

Pour réduire la consommation d'énergie des réseaux, il existe deux approches principales: l'utilisation d'un mode veille à basse consommation énergétique pendant les périodes d'inactivité et l'adoption de matériels dont la consommation d'énergie est proportionnelle à la charge. Dans ce rapport, nous formulons un modèle théorique, basé sur la théorie des graphes aléatoires, qui permet d'estimer les gains potentiels réalisables en adoptant des modes de mise en veille dans les réseaux où sont déployés des appareils consommant de l'énergie proportionnellement à leur charge. La mise en veille de certains appareils permet d'économiser sur l'énergie de ceux-ci, mais en contrepartie l'énergie consommée par les appareils actifs augmentera au vu de leur charge accrue. Le degré de proportionnalité entre la charge d'un appareil et sa consommation d'énergie joue un rôle important. Ainsi, il n'est pas simple de prévoir les cas dans lesquels une mise en veille ou une consommation proportionnelle à la charge serait plus avantageuse. Notre approche se base non pas sur des simulations détaillées mais sur des modèles simples de réseaux dans lesquels le matériel (n÷uds et liens) consomme de l'énergie proportionnellement au trac géré et dans lesquels une certaine fraction de n÷uds est mise en veille. Notre modèle permet d'estimer l'économie d'énergie dans des scénarios divers. Nos résultats montrent que la mise en veille et la proportionnalité de la consommation avec la charge peuvent être combinées avec succès. Toutefois, si la puissance de consommation statique est un ordre de grandeur inférieure à celle proportionnelle à la charge, la mise en veille perd ses avantages. Grâce à la théorie des graphes aléatoires, l'impact de diérentes propriétés de la topologie du réseau est étudié. A titre d'exemple, un réseau fortement connexe avantage la mise en veille.

Mots-clés : réseaux énergétiquement performants, modes de sommeil, modèles des graphes aléatoires 1 Introduction In networking, one of the main causes of energy waste is the fact that most of the devices do not consume energy proportionally to the work they sustain, but they consume much even when they are under-utilized [START_REF] Adelin | On the Impact of Monitoring Router Energy Consumption for Greening the Internet[END_REF]. On the contrary, network usage and trac follow the typical human being activity patterns, with signicant dierences between peak and o-peak values and typical daily periodicities. The network results thus highly under-utilized for long periods of time causing a large energy waste. Many solutions are being studied to reduce this waste, or, equivalently, to make the network consumption proportional to the trac load [START_REF] Bolla | The Potential Impact of Green Technologies in Next-Generation Wireline Networks -Is There Room for Energy Saving Optimization?[END_REF]. The proposed approaches can be divided into two main categories: i) Energy proportional approaches work on the individual devices and try to achieve energy consumption proportionality by adapting the speed (and capacity) of the devices to the actual load, over relatively short time-scales [START_REF] Barroso | The Case for Energy-Proportional Computing[END_REF]; ii) Sleep mode approaches involve the network as a whole and approximate load proportionality by carefully distributing the trac in the network so that some devices are highly utilized while others become idle and are put in sleep modes [START_REF] Chiaraviglio | Minimizing ISP Network Energy Cost: Formulation and Solutions[END_REF]. The two solutions can be merged so that energy proportional devices are present and sleep mode can be leveraged to possibly save additional energy.

In this context, several solutions have been proposed in the literature, including network management algorithms that optimize trac routing so as to maximize the energy saving oered by sleep mode enabled devices (see Sec. 7 for more details). However, to the best of our knowledge, all previous works adopt simulation or actual testbed experiments as main means to assess energy saving performance, and typically few benchmarking scenarios have been considered. Furthermore, either energy proportionality or sleep modes approach is assumed, with few works only considering the combination of the two. In this paper, we instead aim at: i) comparing and combining the two approaches, and ii) proposing an analytical methodology to estimate their benets.

Given a family of network topologies, and given a model of the energy consumed by a device as a function of its load, is it better to purely rely on device energy proportionality capability, or, on the contrary, is it better to couple it with sleep mode solutions? And, also, which is the minimum energy proportionality that would make sleep mode ineective? What is the impact of the network size, or topological properties on the benets of energy saving feature? The answer to these questions is the goal of this paper.

When a device is switched-o the trac passing through it has to be rerouted on dierent, typically longer, paths; thus, the benecial saving achieved by switching o the device is mitigated by the increase of the consumption of the devices that remain on, due to the higher load they have to sustain. To investigate this trade-o, we consider a general model to represent network topologies ranging from backbone networks to metropolitan networks, and a general model for device power consumption.

Several power consumption models for devices have been proposed [START_REF] Mahadevan | A power benchmarking framework for network devices[END_REF], [START_REF] Francini | Rate adaptation for energy eciency in packet networks[END_REF], [START_REF] Restrepo | Energy Prole Aware Routing[END_REF], [START_REF] Antonakopoulos | Power-aware Routing with Rate-adaptive Network Elements[END_REF]. Basically, all these models assume that the energy consumption of network devices, i.e., nodes and links, is composed by a constant amount and a variable part that is an increasing function of the trac that ows through the devices. We compute the total network energy cost as the sum of the xed and variable cost of network devices which depends on the trac they have to carry. In our previous work [START_REF] Chiaraviglio | Modeling Sleep Modes Gains with Random Graphs[END_REF] this problem has been faced considering that only links oer energy saving capabilities. In this work we go a step further by considering a more general model that includes also the cost of nodes. We start considering the variable cost of devices scales linearly with the load. Then, we extend the model to generic cost functions which include linear and super-linear costs. The network and its topological characteristics are represented by random graphs; leveraging then on random graph theory, the load on network devices is computed from Inria the knowledge of the shortest path between node pairs. Thus, the energy consumption of the whole network is easily derived.

Since we are not interested in proposing a novel algorithm to select which link and node can be put into sleep mode, we consider a generic policy, according to which some elements are turned o. This results in a change in the topological characteristics of the network, which is modeled as a new graph, whose energy consumption is evaluated using graph theory again. To the best of our knowledge, the only previous work that is similar to ours is [START_REF] Antonakopoulos | Power-aware Routing with Rate-adaptive Network Elements[END_REF]. Yet, only simple simulations have been used, so that the set and generality of presented results is limited. In our work, we present modeling results that corroborate the intuition of [START_REF] Antonakopoulos | Power-aware Routing with Rate-adaptive Network Elements[END_REF] and derive more general insights.

We present an extensive sensitivity analysis to show the impact of model parameters. We include both small-world and power-law graph models that are claimed to reect actual network topology properties [START_REF] Pastor-Satorras | Evolution and structure of the internet: A statistical physics approach[END_REF]. Our results show that:

• when the variable part of the cost model is small with respect to the constant part, as is typical of today devices, sleep modes are convenient;

• for future devices, whose consumption will probably be more load proportional, sleep modes might not be convenient anymore provided the static cost would be one order of magnitude smaller than the variable part;

• network topology characteristics have limited impact on energy saving. Yet, well-connected topologies show larger benets in terms of energy saving margins when sleep-mode policies are in place.

This suggests that, given the today technological constraints that make the constant energy consumption of devices quite large, sleep mode enabled networks will allow to save more energy than purely energy proportional approaches for long time. Finally, we emphasize that, despite being simple, our model gives general insights of sleep modes eectiveness in actual telecommunication networks.

The rest of the paper is organized as follows. The system model and methodology is detailed in Section 2. The adopted network models are reported in Section 3. We present the evaluation of sleep modes in Section 4. The comparison of dierent switching o policies is reported in Section 5. A discussion about model assumptions is reported in Section 6. Section 7 reviews related work. Finally, Section 8 concludes the paper.

System Model and Methodology

In this section we provide a general overview of the methodology we use to evaluate sleep mode gains.

The network is composed by access and transport devices. Access devices are the possible sources and destinations of trac, therefore they can never be powered o. On the contrary, some of transport devices can be turned o if their trac can be supported by other devices that remain on.

We adopt the following assumptions: i) trac is uniformly exchanged among all access nodes; ii) trac is routed on the shortest paths among nodes ; iii) the set of devices to be switched o is given a-priori, e.g., it has been previously chosen based on collection of trac measurements and predictions so as to guaranteeing the minimum required Quality of Service (QoS); iv) node/link power consumption is composed by a xed amount of power, and a variable part that is an An energy-aware routing protocol could be adopted here rather than a shortest path routing. For example, trac could be routed over the paths consuming the lowest power. However, this issue is outside the scope of this paper.

RR n°8187

increasing function of the current trac owing on the link; v) the same power consumption model is applied to all the links/nodes in the network.

Modeling the power consumption of today's networks is still an open issue, since it mainly depends on the technology of network devices. Current network devices consume a large amount of static power and a limited amount of power that depends on the current load [START_REF] Adelin | On the Impact of Monitoring Router Energy Consumption for Greening the Internet[END_REF]. However, future devices will instead be more energy-proportional [START_REF] Bolla | The Potential Impact of Green Technologies in Next-Generation Wireline Networks -Is There Room for Energy Saving Optimization?[END_REF], with a predominant term of power scaling with load. To capture the eectiveness of sleep modes in both the aforementioned scenarios, we rst assume a linear power consumption model, in accordance with [START_REF] Antonakopoulos | Power-aware Routing with Rate-adaptive Network Elements[END_REF]. Then, as a second step, we extend our model to super-linear power functions, focusing on the case in which power scales quadratically with load. The latter is representative of devices exploiting Dynamic Voltage Scaling (DVS) techniques [START_REF] Restrepo | Energy Prole Aware Routing[END_REF]. This allows us to investigate the impact of sleep modes when extremely optimized devices are deployed.

Basic formulation and metrics

Let the transport network topology be described by an undirected graph G(N , L), with N the set of nodes, with cardinality N = |N |, and L the set of links, with cardinality L = |L|. The average node degree is K = 2L N . The link rate is denoted by R, while T is the total trac owing in the transport network from access nodes at a given time.

The jth link power consumption, A(j), is modeled by a xed amount F and a variable term that is a function f (•) of the link load, ρ(j):

A(j) = F + f (ρ(j)) (1)
For the sake of simplicity, we assume the same xed power F and function of the load f (•) for all links. Similarly, the ith node power consumption A N (i) is modeled by a xed amount F N and a variable term that is a function g(•) of the node load ρ N (i):

A N (i) = F N + g(ρ N (i)) (2)
Again, the same xed power F N and function g(•) is assumed for all the nodes ! . The total network consumption, C, can be computed as

C = j∈L (F + f (ρ(j))) + i∈N (F N + g(ρ N (i))) = LF + j∈L f (ρ(j)) + N F N + i∈N g(ρ N (i)) (3)
Let us consider the case where the variable part of the cost model increases linearly with the load. For links we have:

f (ρ(j)) = αρ(j) (4) thus, j∈L f (ρ(j)) = j∈L αρ(j) = LαE[ρ] = Lαρ (5)
We assume values for R and T such that the network is not overloaded, i.e., the link load ρ is smaller than one.

! Note that this model can be easily extended to integrate dierent classes of links (nodes), each of them with a dierent value of the parameters F (F N ). where ρ is the mean link load and α is a scaling coecient. Similarly, for nodes we assume:

g(ρ N (i)) = βρ N (i) (6)
By assuming that the node variable power consumption depends on the total amount of trac passing through the node and that the link load is independent on the node degree, we dene the average load on a node ρN as the sum of the average load carried by the K links connected on average to a node: ρN = K ρ.

Under these assumptions, the total network consumption is

C = N K 2 (F + αρ) + N F N + β K ρ (7)
As an index of the relative importance of variable and xed costs, we dene the constant cost equivalent load,

ν = F α ; ν N = F N β (8)
where ν (ν N ) is the amount of load that, added to a link (node), makes its energy consumption increase of a quantity F (F N ). Or, in other terms, whenever the load increases by an amount ν (ν N ), the part of the energy consumption due to links (nodes) increases by F (F N ). The parameters ν, ν N and their ratio play a crucial role in the evaluation of sleep mode schemes, as we show in Section 4. Fig. 1 shows a graphical representation of a link power consumption A.

The average link load can be computed as:

ρ = T d LR ( 9
)
where d is the average shortest path length and LR is the total capacity oered by the network. We call C the all-on network consumption and we take this value as a reference for the nominal consumption of the network. Table 1 summarizes the notation introduced so far.

Node Switch-o Policy

We now compute the network consumption when some nodes enter sleep mode. We assume that the scheme according to which the nodes enter sleep mode has been preliminarily planned so that, when the nodes are removed (powered o), the resulting network still meets the QoS Let p ∈ (0, 1) be the fraction of nodes that are switched o. We model the network resulting from the sleep mode scheme through the new graph in which we randomly remove a fraction p of the nodes; we take p smaller than the critical probability after which the network becomes disconnected. In this regime, random node elimination makes the new graph maintain the same structure of the original graph; refer to [START_REF] Albert | Statistical mechanics of complex networks[END_REF] for details. The number of nodes after the random node removal becomes N ′ = N (1p), and the new average degree becomes K′ = K(1p).

From [START_REF] Restrepo | Energy Prole Aware Routing[END_REF], the average network consumption in sleep mode C ′ is now:

C ′ = N K 2 (1 -p) 2 (F + αρ ′ ) + N (1 -p) F N + β K(1 -p)ρ ′ (10) with ρ′ = T d′ N ′ K′ 2 R = T d′ N K 2 (1 -p) 2 R ( 11
)
where d′ is the average shortest path length in the new graph.

Our aim now is to compare the energy consumption of the all-on network consumption, C, to the one of the network in sleep modes, C ′ . To this purpose, we dene the ratio E = C ′ /C as the energy reduction ratio. Intuitively, the use of sleep modes for network devices saves energy when E < 1.

By computing E and comparing C and C ′ in ( 7) and [START_REF] Pastor-Satorras | Evolution and structure of the internet: A statistical physics approach[END_REF], it is possible to evaluate when sleep modes are convenient:

C > C ′ i N K 2 (F + αρ) + N (F N + β K ρ) > N K 2 (1 -p) 2 (F + αρ ′ ) + N (1 -p) F N + β K(1 -p)ρ ′ (12)
Then, we have

C > C ′ i F + 2 K(2-p) FN α+2β > T (2p-p 2 )N K 2 R ( d′ -d) (13)
This equation denes the region in which the sleep mode approach is convenient. Inria Note that, if the network technology is such that the link cost is negligible with respect to the node cost, i.e., F << F N and α << β, we can disregard the link cost and we have

C > C ′ i ν N > 2T pN R ( d′ -d) (14)
Similarly, if the node cost is negligible with respect to the link cost, i.e., F N << F and β << α, we obtain

C > C ′ i ν > T (2p -p 2 )N K 2 R ( d′ -d) (15)
Lemma 1: If devices do not implement load proportionality (α = β = 0), sleep mode is always convenient, i.e., C ′ < C and E < 1.

Proof. With α = β = 0, [START_REF] Haddadi | Network topologies: Inference, modelling and generation[END_REF] [START_REF] Haddadi | Network topologies: Inference, modelling and generation[END_REF] becomes: d > d′ , i.e., it is veried if the average shortest path after some node switch o, d′ is smaller than the initial average shortest path d, which is never true " . Consequently, E ≥ 1.

simplies to N K 2 F + N F N > N K 2 (1 -p) 2 F + N (1 -p)F N , which is veried for any p ∈ (0, 1). Lemma 2: If the xed power consumption is zero (F = F N = 0), sleep mode is never convenient, i.e., E ≥ 1. Proof. If F = F N = 0,

Generalization of the cost function

We now extend our analysis to a generic cost model, in which the variable part of the cost increases according to a polynomial function of the load. For simplicity, we rst consider the cost due to links and then compute the total cost due to nodes too. For each link j, we can write the cost function as:

A(j) = F + ∞ z=1 α z (ρ(j)) z (16)
The total cost C becomes:

C = j∈L F + ∞ z=1 αz(ρ(j)) z + i∈N FN + ∞ z=1 βz(ρN (i)) z = LF + j∈L ∞ z=1 αz(ρ(j)) z + N FN + i∈N ∞ z=1 βz(ρN (i)) z = LF + L ∞ z=1 αzE[ρ z ] + N FN + N ∞ z=1 βzE[ρ z N ]
% where the last term E[ρ z N ] can be computed as:

E[ρ z N ] = ∞ k=1 E[ρ z N |K = k]P(K = k) = ∞ k=1 E k q=1 ρ(q) z P(K = k), ( 18
)
where K is the random variable representing the degree distribution. The last result follows from the node load computation: we assume that the load of a node is given by the sum of the load carried by the links connected to it.

Observe that, given all the moments of the link load distribution and the degree distribution up to the z -th moment, [START_REF] Newman | Mean-eld solution of the small-world network model[END_REF] allows the computation of the total cost for any type of polynomial cost function. Moreover, we can extend the previous analysis to any other cost functions, e.g., logarithmic or any sub-linear functions, by using the Maclaurin series to approximate the cost function, so that cost can still be expressed by [START_REF] Blondel | Distance distribution in random graphs and application to network exploration[END_REF].

Quadratic Cost Model As a special case of ( 16), we assume that the variable part of the cost model increases quadratically with the load. Thus, the jth link cost can be modeled as: F + α(ρ(j)) 2 and (3) becomes:

C = LF + α j∈L (ρ(j)) 2 + N F N + β i∈N (ρ N (i)) 2 = L(F + αE[ρ 2 ]) + N (F N + βE[ρ 2 N ]) (19)
Considering the node load distribution, we have:

E[ρ 2 N ] = σ 2 ρN + E[ρ N ] 2 = σ 2 ρN + (ρ K) 2 , where σ 2
ρN is the variance of the node load distribution. Note that, being the node load given by the sum of the load of the node's links, the node load distribution is a compound distribution that depends on the link load and the degree distributions. Assuming that link load ρ is independent on the node degree K, and that the load on the links are i.i.d. random variables, we obtain:

σ 2 ρN = Kσ 2 ρ + ρ2 σ 2 K . Thus, E[ρ 2 N ] = (ρ K) 2 + Kσ 2 ρ + ρ2 σ 2 K = ρ2 E[K 2 ] + σ 2 ρ K (20)
Considering the link load distribution we have:

E[ρ 2 ] = E[ρ] 2 + σ 2 ρ = ρ2 + σ 2
ρ , where σ 2 ρ is the variance of the link load distribution. From (9), we can observe that ρ is a linear function of the average shortest path d. Thus,

ρ = E[ρ] = E[(T /LR)d] = (T /LR)E[d],
and the link load variance is:

σ 2 ρ = T LR 2 σ 2 d , (21)
where σ 2 d is the variance of the shortest path distribution. Therefore, by knowing the rst and the second moments of the shortest path and the degree distribution, we can compute the total cost given by [START_REF] Haddadi | On the importance of local connectivity for Internet topology models[END_REF]. Note that to compute the network cost after the random node removal, we need to evaluate the second moment of the degree distribution in the new graph E[(K ′ ) 2 ]. From [START_REF] Albert | Statistical mechanics of complex networks[END_REF] we know that:

E[(K ′ ) 2 ] = E[K 2 ](1 -p) 2 + p(1 -p) K, where E[K 2 ]
is the second moment of the degree distribution in the original graph. We will show the analysis referred to the quadratic cost model in Section 4.4.

Network Models

Many network topologies and the Internet, in particular, satisfy the following properties: (i) small-world property, according to which the average number of hops between each node pair is quite limited, (ii) local clustering, according to which the topology has highly connected zones, (iii) heavy-tailed distributions of the node degree, meaning that, in general, most of the nodes have few links while a few nodes have a large number of links. In the literature, several random graph models have been proposed to generate topologies matching the properties of the real ones. However, deciding which model better ts the real topology is an open problem [START_REF] Haddadi | Network topologies: Inference, modelling and generation[END_REF]. Therefore, Inria instead of focusing on a single model, we consider a few among the most popular models, showing that common properties regarding energy consumption can be inferred in all cases. In particular, we consider three well-known graph models: Erdös and Rényi, Power Law and Watts-Strogatz models.

In the Erdös and Rényi (ER) model [START_REF] Erdös | On random graphs[END_REF] nodes are connected by links according to a given probability, and the resulting degree distribution follows a Poisson distribution. The properties of this model are well-known in the literature and have been extensively studied. In particular, the ER model exhibits the small-world property, according to which the diameter of the graph scales as log(N ). However, the local clustering and heavy-tailed properties are not met.

In the Power Law (PL) model [START_REF] Albert | Statistical mechanics of complex networks[END_REF] the distribution P K (k) of the node degree K follows a power-law distribution, i.e., P K (k) ∼ k -γ . The intuition is that some nodes behave like hubs, and have many more connections than others.

The Watts-Strogatz (WS) model [START_REF] Newman | Renormalization group analysis of the small-world network model[END_REF] is built starting from a regular lattice in which each node is linked to a xed number of neighbors. Then, additional edges are inserted between randomly chosen pairs of nodes # . The resulting graph is an interpolation between ordered lattices and purely random graphs. This model matches both small-world and the local clustering properties, but the degree distribution is not heavy-tailed.

In the next sections, we discuss more in detail the properties of the considered models focusing, in particular, on the rst and second moments of the shortest path and the degree distributions that we need in order to compute the rst and second moments of the load distribution and, hence, the energy consumption.

The Erdös-Rényi model

We rst compute the network consumption C in the case of linear cost models, i.e., the variable part of the cost model increases linearly with the load. To compute C we need the expression of the average shortest path that, for an Erdös-Rényi graph, is given by [START_REF] Pastor-Satorras | Evolution and structure of the internet: A statistical physics approach[END_REF]:

d ≃ log(N ) log( K) (22)
Using ( 7), ( 9) and ( 22), we can compute the all-on consumption of the network as:

C = N K 2 F + αT R log(N ) log( K) + N F N + 2 βT R log(N ) log( K) (23)
After randomly removing a fraction p of the nodes, the average shortest path becomes: d′ = log(N (1-p)) log( K(1-p)) and, from [START_REF] Pastor-Satorras | Evolution and structure of the internet: A statistical physics approach[END_REF], the network consumption becomes:

C ′ = N K 2 (1 -p) 2 F + αT R log(N (1-p)) log( K(1-p)) + N (1 -p)F N + 2 βT R log(N (1-p)) log( K(1-p)) (24)
To compute the network consumption when the quadratic cost model is used, we need to know the second moments of both the degree and shortest path distributions, as shown in the previous section. The degree distribution in an ER graph follows a Poisson distribution, thus:

K = σ 2 K and E[K 2 ] = K2 + K.
For what concerns the variance of the shortest path distribution # In the original WS model presented in [START_REF] Watts | Collective dynamics of small-world networks[END_REF] shortcuts are rewired from the lattice. However, the resulting graph is aected by a not negligible probability to be disconnected. Therefore, we adopt the modication of the WS model proposed by [START_REF] Newman | Renormalization group analysis of the small-world network model[END_REF], in which shortcuts are additionally inserted as new links. In this way, the resulting graph is always connected. σ 2 d , it can be shown that it tends to zero when the number of nodes N is high (see [START_REF] Blondel | Distance distribution in random graphs and application to network exploration[END_REF] for details). Thus, we assume that σ 2 d = 0 and we obtain

E[d 2 ] = d2 + σ 2 d = d2 ≈ log(N ) log( K) 2
. Finally, we can compute the network consumption using [START_REF] Haddadi | On the importance of local connectivity for Internet topology models[END_REF], [START_REF]Orbis topology[END_REF] and [START_REF]Geant topology[END_REF].

Power-law model

We consider a graph in which the distribution of the degree K follows a power law, i.e., P K (k) ∼ k -γ . In this case, the average shortest path can be computed as in [START_REF] Pastor-Satorras | Evolution and structure of the internet: A statistical physics approach[END_REF]:

d ≈ 1 + log(N/ K) log[(E[K 2 ] -K)/ K] (25)
In particular, we consider a Pareto distribution for the degree since it is one of the most widely used and studied power laws in the literature. The Pareto distribution is described by the parameters (a, k m ), where k m is the minimum possible value of K, and a is a positive parameter. For this distribution we have:

K = ak m /(a -1) and E[K 2 ] = ak 2 m /(a -2).
Using the linear cost model, the all on network consumption C can be computed similarly to what done for the ER model using ( 7), ( 9) and [START_REF] Bolla | Enabling Backbone Networks to Sleep[END_REF]. As reported in [START_REF] Albert | Statistical mechanics of complex networks[END_REF], a power law graph remains power law even after a random removal of nodes. Thus, the average shortest path of the network with sleep modes is:

d′ ≈ 1 + log(N/ K) log E[K 2 ]- K K (1 -p) (26)
from which the network consumption with sleep mode C ′ can be computed.

To compute C in the case of quadratic cost model, we need also to estimate the second moment of the shortest path distribution. For the PL graph we have: [START_REF] Dorogovtsev | Pseudofractal Scale-free Web[END_REF]). Thus, the second moment of the shortest path distribution is: The Watts-Strogatz (WS) model interpolates between ordered lattices and purely random graphs [START_REF] Newman | Mean-eld solution of the small-world network model[END_REF]. Starting from a lattice of N vertices in which each node is symmetrically connected to its K L nearest neighbors, randomness is introduced by independently adding x shortcuts between randomly chosen pairs of nodes [START_REF] Newman | Renormalization group analysis of the small-world network model[END_REF]. The mean node degree, K, is given by K = K L + 2x/N . We rst focus on the computation of the network consumption when a linear cost model is used. In the literature, there are some models for the estimation of the average shortest path length when a WS model with rewiring is considered. For example, for small values of x, i.e., x << N/K L , approximation of [START_REF] Newman | Mean-eld solution of the small-world network model[END_REF] holds,

σ 2 d ≃ 2 9 log(N ) log(3) (see
E[d 2 ] ≈   1 + log(N/ K) log E[K 2 ]- K K (1 -p)   2 + 2 9 log(N ) log (3)
d ≃ N K L /2 log(2x) 4x (28)
In this case, d is similar to the shortest path of a lattice, scaled by a factor that takes into account the number of random links. For large values of x, i.e., x >> N/K L , the WS graph with rewiring Inria is similar to a purely random graph [START_REF] Newman | Renormalization group analysis of the small-world network model[END_REF] and the average shortest path can be approximated as,

d ≃ log(N ) log( K) (29)
However, both approximations do not hold in our case, since we are using the WS model without rewiring, i.e. we add the random links to the graph, and the number of random links is quite large. We therefore propose a model that ts the scenarios we are interested in. By considering a large number of simulation results for cases of interest to this study, we have derived the following expression to compute d:

d ≈ N K L /2 log(2x) 4x + log(N ) log( K) (30)
Intuitively, we have found that in our scenarios d scales like a random graph plus the corrective term of [START_REF] Gupta | Using Low-Power Modes for Energy Conservation in Ethernet LANs[END_REF].

During the sleep mode regime, the average shortest path becomes:

d′ ≈ N K L /2 log(2x(1 -p) 2 ) 4x(1 -p) 2 + log(N (1 -p)) log( K(1 -p)) (31)
Similarly to the previous graph models, using ( 12), ( 30) and ( 31), we derive the cost C and C ′ for the WS graph in this regime.

To evaluate the total cost with a quadratic cost model we also need the second moment of the shortest path and the degree distributions of a WS graph. We propose the following new model for E[d 2 ] that is derived by interpolation from simulation results: For what concerns the degree distribution, we use E[K 2 ] ≈ K2 + K L , assuming a behavior similar to a ER graph.

E[d 2 ] ≈ log(N ) φ τ log(K L ) + log 2x

Model Validation

We validated the proposed WS model by comparison against simulation results in a large number of scenarios. We consider networks with a given value of N and of the constant part of the neighbor degree, K L , but dierent values of x. By varying x we make the random component of the degree vary. Top plot of Fig. 2 reports the average shortest path d for the case N = 10, 000, K L = 4, and x ∈ [5, 000 -20, 000]. For each value of x, the results are averaged over 20 independent runs in which dierent random seeds are used for adding the shortcuts. The gure reports d computed from: [START_REF] Gupta | Using Low-Power Modes for Energy Conservation in Ethernet LANs[END_REF] that corresponds to the model of a scaled lattice (SL label in the gure), ( 29) that is the ER model, our proposed model [START_REF] Gupta | Greening of the Internet[END_REF], and simulation results. Clearly, the SL model does not match the measured d for the considered scenarios; the ER model underestimates d, since it does not consider the presence of many links to neighbors. Our model presents a good matching, since it is tted for these scenarios. We have also validated the model in other scenarios, including scenarios with sleep modes, obtaining similar results.

Bottom plot of Fig. 2 shows the energy reduction ratio, E, computed by simulation and with the proposed model, for the case of a network with sleep modes, N = 10, 000 nodes, K L = 4, x = 10, 000 and ν = ν N ; the fraction of nodes that are switched o varies between 0 and 0.4. Bars indicate minimum and maximum values of the energy reduction ratio. Moreover, we have also computed the condence intervals of E, nding that they never exceed 1% of the estimated value at 95% condence level. Again, observe how accurate the proposed model is.

We then validate the WS model also for the case of quadratic cost function. Top plot of Fig. 3 reports E[d 2 ] for N = 10, 000, K L = 4, and x ∈ [5, 000 -20, 000], considering the ER model, the simulations, and the proposed model of [START_REF] Soteriou | Dynamic power management for power optimization of interconnection networks using on/o links[END_REF]. The lattice model (not reported in the gure) highly overestimates E[d 2 ], with values in the order of 10 6 . Our model presents a very good matching with the simulation results.

Bottom plot of Fig. 3 reports the energy reduction ratio E for N = 10, 000 nodes, K = 6, x = 10, 000 and p ∈ [0, 0.4]. In this case the condence intervals of E never exceed 5% of the estimated value at 95% condence level. Our proposed model matches the simulated E considering dierent values of ν. In the following we therefore adopt our models for computing C and C ′ .

Performance Evaluation

In this section, we compare the eectiveness of the approaches based on sleep modes under the dierent network models proposed in the previous section.

For the numerical results, unless otherwise specied, we set N = 10, 000 and the link rate R = 100 Gbps. The link and node linear power model of ( 4) and ( 6) is used. The average degree K is set to 6. This reects the results of recent measurement studies about actual network topologies like the Internet (see [START_REF] Haddadi | On the importance of local connectivity for Internet topology models[END_REF] for an overview), according to which K ranges between 4 and 8. We assume that the total trac scales with the number of nodes N and the degree K. Notice also that we impose an amount of trac that does not overload the network when the maximum amount of devices is switched o, i.e., ρ′ < 1 for p = 0.4.

For the PL model, we choose a Pareto distribution of the node degree, setting a = 3 and k m = 4, so that K = 6; for the WS model, we set K L = 4 and x = 10, 000 and the average node degree is K = K L + 2x/N = 6.

Impact of Sleep Modes

We rst evaluate the impact of sleep modes on energy consumption for dierent graph models. Fig. 4 reports the energy reduction ratio versus p for two distinct cases: high equivalent load (ν = ν N = 10) and low equivalent load (ν = ν N = 0.01). When the constant part of power is predominant (ν = 10) the introduction of sleep modes saves energy in the network, i.e., E < 1, for all the graph models. Moreover, the energy reduction ratio decreases as p increases, suggesting that the saving increases with the number of devices that are switched o. Notice also that the curves of WS, PL and ER are overlapping since the constant part of power is the same for the three graph models, while the impact of the variable power is negligible. On the contrary, when highly energy-proportional networks are considered (ν = 0.01), the network consumes a higher amount of energy when devices are switched o and E > 1. Thus, in this scenario, sleep modes are not convenient. In particular, E is higher for WS than for ER and PL models, due to the fact that, in the WS graph model, node removal has an important impact on the shortest paths.

We then analyze the impact of sleep modes by separately analyzing the impact of links and nodes power consumption. In particular, from ( 14) and ( 15) we compute the minimum values of ν and ν N for which E becomes smaller than 1; these values represent breakeven points for which sleep mode saves energy. The top (bottom) plot of Fig. 5 reports the breakeven curve for links (nodes). If the value of ν falls above the breakeven curve, sleep mode is convenient; otherwise, the network consumes a higher amount of energy when devices are switched o. In all cases, as p increases the minimum value of ν increases too, meaning that for large values of p a sleep mode approach is convenient only when the constant part of the energy cost is high with respect to the variable part. In particular, from the top plot of Fig. 5, it can be observed that the breakeven curve of the ER model ranges between 0.06 and 0.14. The WS breakeven curve is above the other two curves, due to the fact that the average shortest path length increases faster for WS (and load increase is thus more signicant) than in the other models; or, in other terms, even with devices that present quite signicant load proportionality, sleep modes are convenient. From bottom plot of Fig. 5, it can be observed that the breakeven curves of nodes are one order of magnitude larger than the corresponding ones of links. This is due to the fact that the variable term of nodes power grows much faster than the links one, thus higher xed costs are required to balance the increase in the node load.

Impact of Technology Constraints

We now evaluate the impact of the equivalent load. Fig. 6 reports the values of E versus ν for the three proposed models; the dierent curves correspond to dierent values of p where for simplicity we set ν = ν N . In all cases, the breakeven point for which E = 1 occurs when the xed cost F (F N ) is one order of magnitude lower than α (β) (ν ≈ 0.3). Two dierent regimes are possible: i) sleep mode is not convenient (E > 1), and ii) sleep mode is convenient (E < 1).

In the rst regime, the higher the probability to switch o devices is, the higher the additional energy is, being WS the worst case. In the second regime, instead, sleep mode leads to high energy saving for all models, and the saving strongly increases with p. Notice that with today technology, we are in the right part of the gures (sleep mode is always convenient, with saving up to 30-40%), while in the future, the values of ν will probably decrease, meaning that sleep mode will become less convenient.

In the following, we consider the impact of varying the relative cost of nodes and links. In particular, we introduce a scaling factor r for the cost of nodes, so that the total network cost C becomes:

C = j∈L A(j) + 1 r i∈N A N (i)
In this way, we nd the tradeos that emerge as the costs of nodes and links are weighted dierently. For example, the cost of links is normally marginal for wireless technologies: the largest amount of power is needed to keep powered on Base Stations in UMTS or Access Points in WiFi networks [26]. On the contrary, for backbone networks the cost of links is not negligible, due to the presence of multiple ampliers for optical networks or DSL lines for home access networks [27]. Left plot of Fig. 7 reports the breakeven curves for which E = 1 varying both ν and ν N , considering dierent values of the parameter r. In this case, we are considering an ER graph model and p = 0.3. Points below a breakeven curve correspond to conditions in which sleep modes are not convenient. For low values of ν and ν N sleep modes are not convenient, i.e., E > 1 (bottom left corner of the plot). Conversely, for high xed costs (top right corner of the plot) sleep modes are convenient, E < 1. As r increases, the network becomes almost insensitive to nodes equivalent load. For example, for r = 100, sleep modes are convenient if ν > 0.12 or ν N > 40. On the contrary, for r = 10 -2 sleep modes are convenient if ν > 20 or ν N > 1.

To give more insights, central and right plots of Fig. 7 detail the energy reduction ratio for r = 10 2 and r = 10 -2 , respectively. The vertical colorbars report the values of the energy reduction ratio. As expected, for r = 10 2 the energy reduction ratio mainly depends on the value of ν. Notice that for ν >> 1 (i.e., F >> α), the energy reduction ratio of the network approaches the limit (1p) 2 = 0.49 obtained when α = 0. Interestingly, for r = 10 -2 (right plot) the energy reduction ratio still depends on the cost of the links, so that the network saves up to 25% of energy when ν = 100, independently of ν N . This is due to the fact that, for a given p, the fraction of links powered o is always larger than the fraction of nodes.

Impact of Network Properties

We now consider the impact of the network properties on the possible energy saving. In particular, we start by setting K = 6, while we vary N in [10 2 , 10 5 ]. For the WS model we set x = N . Fig. 8 reports the energy reduction ratio E for the considered models, for ν = ν N = 0.5 and p = 0.3. The highest saving is obtained by the PL model. In all the cases, E increases (sleep mode eectiveness reduces) with the number of nodes; this is due to the fact that we increase N while letting the node degree unchanged, so that the set of alternative paths does not scale with N and the impact of switching o some devices on the path length increases. This is particularly visible under the WS model. Now we consider the impact of the average node degree K and select values of K that mimic those used in [START_REF] Haddadi | On the importance of local connectivity for Internet topology models[END_REF] to represent the average node degree of large topologies. We set ν = ν N = 0.5 and p = 0.3. Moreover, for the WS model we x x = 10, 000 and we vary K L ∈ [START_REF] Bolla | The Potential Impact of Green Technologies in Next-Generation Wireline Networks -Is There Room for Energy Saving Optimization?[END_REF][START_REF] Antonakopoulos | Power-aware Routing with Rate-adaptive Network Elements[END_REF]. Fig. 9 shows E versus K. For all the models, the energy reduction ratio is decreasing as K is increasing. Indeed, for large values of K, the number of possible paths between any two nodes is large and, once some nodes are powered o, it is easy for the network to nd alternative paths; or, in other terms, the increase of the average shortest path due to sleep modes is limited if K is large. For low values of K, sleep modes are not convenient. For example, for the WS model E is around 1.27 when K = 4, meaning that the sleep mode wastes an additional 27% of power with respect to an always on solution. Conversely, when K increases, the additional cost due to the increase of the path length is smaller, so that sleep modes are more ecient. Finally, notice that the energy reduction ratio of the ER and PL models is consistently lower than the WS one. This is due to the better path length properties the former two graphs have.

Impact of Power Consumption Model

We consider here the impact of the power consumption model. In particular, we compare the linear power consumption model of ( 7) and the quadratic model of [START_REF] Haddadi | On the importance of local connectivity for Internet topology models[END_REF]. Fig. 10 details the breakeven curves for which E = 1, comparing the two models for dierent values of r under the ER network model. Interestingly, the area for which E > 1 steadily increases with the quadratic model, since the weight of the variable term is higher than the linear case.

We then consider the impact of the quadratic cost function when the fraction of switched o nodes varies. Fig. 11 shows the energy reduction ratio versus the probability to remove nodes p, for the ER and PL graph models (refer to Fig. 3 for the WS model). In this case, we consider ν = ν N = 0.005 and ν = ν N = 50. The gures report also the average results obtained by simulation. Bars indicate minimum and maximum values. Interestingly, when ν = 0.005 the variable term of power is predominant and E > 3 for p = 0.4 considering the ER model. This means that a strongly energy proportional network with sleep modes can consume up to three times an all-on network. Notice also that, with the same parameters, the WS model performs consistently worse, while the PL performs quite better. On the contrary, as high xed costs are considered, all the models save energy, being E < 1 for all values of p. Finally, all the models present a good matching with the simulations, indicating that the introduced approximations do not impact on the variation of E.

Comparison of Switching Policies

Finally, we investigate the case in which the nodes to be switched o are chosen according to another strategy than a simple random removal policy. Note that, the choice of the node removal strategy is out of the main scope of this paper. Nevertheless, in the following we compare the simple random removal strategy with the least-ow policy of [START_REF] Chiaraviglio | Minimizing ISP Network Energy Cost: Formulation and Solutions[END_REF] to assess the eectiveness of sleep modes under a smart switch-o policy. Specically, the least-ow strategy consists in switching o the devices with the least amount of trac owing through them, where the total amount of ow crossing a node is computed as the sum of the trac owing through its incident links. This strategy could be particularly eective in reducing the rerouted trac in the network when a node is powered o, in order to maintain an adequate QoS for users. Note that our models apply only to random strategy, thus, the results of the least-ow policy are derived by simulation.

Fig. 12 shows the energy reduction ratio versus p for the random (R) and least-ow (LF) policies considering the ER graph model and r = 1/10. Several considerations hold in this case. First, with little proportional networks (left plot) the energy reduction ratio of the least-ow policy is higher than a pure random policy. This is due to the fact that most of least loaded nodes have low degree, thus the number of actually switched o links is lower than in the random case, in which a fraction 1 -(1p) 2 of links is switched o. Second, as predominant variable terms of power are considered (central plot), the energy reduction ratio of the least-ow policy is consistently lower than the random policy. This is due to the fact that the average path length after the nodes removal grows slower than in the random policy, since the least loaded devices carry less trac. This is conrmed by the right plot of the same gure that reports the average path length in the considered cases.

Discussion

In this section we briey discuss the main assumptions of our model and how much the results are impacted by these assumptions.

Network Topology. In the literature, the eectiveness of sleep modes has been evaluated on benchmarking topologies (see for example [START_REF] Chiaraviglio | Minimizing ISP Network Energy Cost: Formulation and Solutions[END_REF] and [START_REF] Idzikowski | Saving energy in IP-over-WDM networks by switching o line cards in low-demand scenarios[END_REF]). However, the obtained results are tailored to specic reference topologies, and are dicult to be generalized. In this work, we follow a dierent approach: we exploit graph theory to model network topologies, and we derive general insights, showing what happens when network parameters, such as the number of nodes and the average node degree, are varied. In this way, we are able to overcome the limitations derived from the analysis over a specic topology. However, we recognize that actual topologies can be more complex than a single graph model. To partially cope with this issue, rather then showing results for a single model, in this work we have considered dierent graph models, showing that common properties can be inferred in all cases.

Set of Devices in Sleep Mode Another crucial aspect is how to choose the set of devices to be put in sleep mode. In the literature, several algorithms have been proposed to select the set of devices to be powered o (see for example [START_REF] Chiaraviglio | Minimizing ISP Network Energy Cost: Formulation and Solutions[END_REF][START_REF] Idzikowski | Saving energy in IP-over-WDM networks by switching o line cards in low-demand scenarios[END_REF][START_REF] Coiro | Reducing power consumption in wavelength routed networks by selective switch o of optical links[END_REF][START_REF] Fisher | Greening Backbone Networks: Reducing Energy Consumption by Shutting O Cables in Bundled Links[END_REF]). In this work, we assume that the Inria set of devices in sleep mode is given and is such that the minimum required QoS is guaranteed to all the users. For analytical tractability, we adopt the conservative assumption that the devices in sleep mode are randomly chosen. We recognize that the actual savings may be even larger if smarter algorithms are considered. However the most interesting nding of the paper, i.e., the fact that sleep mode is eective even when load proportional devices are deployed, is still valid and general. To further support this insight, in this paper we compare the random policy with the least-ow policy of [START_REF] Chiaraviglio | Minimizing ISP Network Energy Cost: Formulation and Solutions[END_REF], which is more eective in choosing the set of devices in sleep mode. We show that the general message still holds even when the this advanced policy is adopted. Trac Variation. In a telecommunication network trac varies over time, suggesting that sleep mode can be dynamically applied in order to follow the trac trend [START_REF] Gupta | Using Low-Power Modes for Energy Conservation in Ethernet LANs[END_REF], [START_REF] Alonso | Opportunistic power saving algorithms for Ethernet devices[END_REF]. In this work, we consider a snapshot of the network at a given time, and evaluate the eectiveness of sleep mode for the selected snapshot. Our model gives indications for the considered scenario. Finally, note that transient behavior of devices and of the whole network has to be considered. This includes the latency introduced to recover from a sleep mode state, and how to react to sudden load uctuations. These aspects are out of the scope of this paper.

Trac Distribution. Our framework takes as input the total amount of trac exchanged in the network, which we assume to be uniformly distributed among all nodes. If a real trac matrix shows unbalanced trac demands, e.g., a peering point collecting most of the trac in the network [START_REF] Chiaraviglio | Minimizing ISP Network Energy Cost: Formulation and Solutions[END_REF], a uniform matrix is a conservative scenario, since many devices are required to be at full power. This intuition has been conrmed in previous work too [START_REF] Chiaraviglio | Minimizing ISP Network Energy Cost: Formulation and Solutions[END_REF].

Power and Capacity Distribution. We assume that all the devices (nodes or links) consume the same power. This assumption can be considered representative of current core networks, in which all devices have a similar capacity [START_REF] Chiaraviglio | Minimizing ISP Network Energy Cost: Formulation and Solutions[END_REF]. Extensions to classes of devices are possible, and we leave it for future work. Considering power consumption model of devices, several measurements corroborate the simple linear relationship with oered trac. More advanced scaling mechanisms (such as DVS) are also modeled by quadratic functions. Extensions to higher degree polynomial functions are possible, but not straightforward.

Related Work

The study of power-saving network devices has been introduced in recent years, starting from the pioneering work of [START_REF] Gupta | Greening of the Internet[END_REF]. Rate adaptation and sleep mode techniques have recently attracted the attention of the research community (see [START_REF] Bolla | Energy Eciency in the Future Internet: A Survey of Existing Approaches and Trends in Energy-Aware Fixed Network Infrastructures[END_REF][START_REF] Bianzino | A Survey of Green Networking Research[END_REF] for an overview). In [START_REF] Nedevschi | Reducing Network Energy Consumption via Sleeping and Rate-Adaptation[END_REF] authors focus on two power management schemes for power saving, based either on sleeping or rate adaption. Dierently from our work, they consider to put into standby mode network elements during idle times, i.e., among subsequent packet arrivals. In our work, instead, we assume that devices are put in sleep mode over coarser time scales, i.e., of the order of minutes and hours. Moreover, they consider the impact of the two techniques in isolation, while in our work we assess the impact of sleep mode over a variety of scenarios, including energy-proportional networks.

Moreover, in the last years, Dynamic Voltage Scaling techniques that dynamically adjust the frequency and voltage of links have been shown to reduce power consumption [START_REF] Shang | Power-ecient interconnection networks: Dynamic Voltage Scaling with links[END_REF]. However, these techniques require sophisticated hardware mechanisms to ensure correct link operation during scaling, and the potential power savings can be smaller than the ones achievable with sleep modes since links are still powered on even when they are in an idle state. Furthermore, the authors of [START_REF] Soteriou | Dynamic power management for power optimization of interconnection networks using on/o links[END_REF] investigated the savings achievable dynamically turning links on/o in response to trac variation. However, their work is targeted to a rather dierent eld, i.e., the interconnection network fabrics. In particular, the topology of an interconnection network is represented by a grid, while an ISP network has a less regular structure. Moreover, the authors of [START_REF] Soteriou | Dynamic power management for power optimization of interconnection networks using on/o links[END_REF] consider only the case in which links only are put in sleep mode and their analysis is limited to a simple power model on/o, whithout investigating the case in which energy-proportional devices are used.

In [START_REF] Lange | Energy Consumption of Telecommunication Networks and Related Improvement Options[END_REF] authors estimate the total power consumption of a telecommunication network, given the power requirements of network devices and the trac from users. In particular, they consider all the network portions of a telecom operator, foreseeing that the highest energy consumption growth rates will be in the data centers and IP backbone networks. Basing their assumptions on technology forecasts, they claim that future networks with enabled sleep modes and load adaptation will save a consistent amount of energy, i.e., typically larger than 56%. While this intuition is inline with our results, our model is able to derive more general insights, being able to highlight the breakeven points that emerge adopting dierent power models and technology assumptions.

Power consumption of current network devices is mostly independent of the current load, considering backbone [START_REF] Adelin | On the Impact of Monitoring Router Energy Consumption for Greening the Internet[END_REF][START_REF] Mahadevan | A power benchmarking framework for network devices[END_REF] and access networks [26]. Moreover, the power consumption of links is not negligible, and can be of the same order of magnitude of nodes [27]: this is especially true for long-haul optical links, in which several ampliers are needed to regenerate the signal.

Step functions are adopted to model power consumption of such devices. Several works have therefore targeted the minimization of network devices powered on while satisfying QoS [START_REF] Bolla | Enabling Backbone Networks to Sleep[END_REF][START_REF] Bolla | Energy Eciency in the Future Internet: A Survey of Existing Approaches and Trends in Energy-Aware Fixed Network Infrastructures[END_REF][START_REF] Bianzino | A Survey of Green Networking Research[END_REF]. Our work conrms that, in this scenario, networks with sleep nodes are the best solution in terms of power saving.

Researchers from universities and industries are now studying solutions to adapt power with current load. In particular, in [START_REF] Restrepo | Energy Prole Aware Routing[END_REF] the authors exploit the idea of exchanging energy proles among devices to reduce the overall power consumption during routing and trac-engineering operations. They consider dierent models to represent device power consumption, including linear and super-linear models. In [START_REF] Antonakopoulos | Power-aware Routing with Rate-adaptive Network Elements[END_REF] authors evaluate the impact of sleep modes over dierent network topologies. They consider dierent power models, evaluating the total power consumption when some elements are put into sleep modes. All these works are mainly focussed on specic case-studies, while our framework is able to produce more general insights over a variety of network graphs, showing that the gain adopting sleep modes can be consistent even when energy-proportional devices are deployed.

Overall, the consciousness of power-aware telecommunication networks is growing. Several projects are investigating the impact of power adaptive technologies for network devices [START_REF]ECONET Project[END_REF]38,[START_REF]D-Link Green Products[END_REF][START_REF]EARTH Project[END_REF]. Indeed, this work is undertaken under the project TREND (Towards Real Energyecient Network Design) [START_REF]TREND Project[END_REF], a Network of Excellence funded by the European Commission through the FP7 Program.

Conclusions

In this paper, we have proposed an analytical framework for the evaluation of the potential energy saving that can be achieved by applying sleep modes to the devices of a complex network, like the Internet. We have modeled the network device (nodes and links) power consumption by means of a simple function composed of a constant cost and a variable cost proportional to the device load. Leveraging on random graph theory, we have then computed the overall power consumption of networks with dierent topological properties. Random graph theory results are used to evaluate the total power consumption of a network with all the devices powered on, or a fraction of devices only. The comparison of these gures has enlightened when the sleep mode adoption is convenient.

Our results suggest that with today technology, with device consumption that varies very Inria little with the load, the use of sleep modes is very eective in reducing the network energy consumption. In the future, with devices whose consumption will likely be more load proportional, the eectiveness of sleep mode approaches will reduce. However, we have found that sleep modes reduce power consumption even when device power scales quadratically with load. Our results indicate also that highly connected networks, with large node degree and high randomness, tend to make the use of sleep modes more convenient.

Figure 1 :

 1 Figure 1: Linear power consumption model for a link.

% 3 . 3

 33 The Watts-Strogatz model

  φ = 2x/N KL , τ = 0.08 and χ = 15.

Figure 2 :

 2 Figure 2: Validation of the proposed model to compute d for WS graphs: (top) average shortest path for dierent models and by simulation, considering N = 10, 000, K L = 4, x ∈ [5, 000 -20, 000], (bottom) energy reduction ratio versus p for the proposed model and the simulation, considering N = 10, 000, K L = 4, x = 10, 000.

Inria

Figure 3 :

 3 Figure 3: Validation of the proposed model for the quadratic load function for WS graphs: (top) E[d 2 ] for dierent models and by simulation, considering N = 10, 000, K L = 4, x ∈ [5, 000 -20, 000], (bottom) energy reduction ratio versus p of dierent models considering N = 10, 000, K L = 4, x = 10, 000.

Figure 5 :

 5 Figure 5: Equivalent link load ν (top) and equivalent node load ν N (bottom) versus the switch-o probability p for the ER, PL and WS model.

Inria

Figure 6 :

 6 Figure 6: Energy reduction ratio E versus equivalent load ν for dierent values of p, from left to right: ER, PL, WS model.

Figure 7 :

 7 Figure 7: ER model: (left) breakeven curves corresponding to E = 1 versus ν and ν N , (central) energy reduction ratio for r = 10 2 , (right) energy reduction ratio for r = 10 -2 .

Inria

Figure 8 :

 8 Figure 8: Energy reduction ratio E versus the number of nodes N for ν = 0.5 and p = 0.3.

Figure 9 :

 9 Figure 9: Energy reduction ratio E versus average node degree K for ν = 0.5 and p = 0.3.

  quadratic) r = 1 (linear) r = 1/100 (quadratic) r = 1/100 (linear) r = 100 (quadratic) r = 100 (linear)

Figure 10 :

 10 Figure 10: ER quadratic-linear comparison for p = 0.3.

Figure 11 :

 11 Figure 11: Energy reduction ratio versus p with the quadratic cost model: ER model (top) and PL model (bottom).

Figure 12 :

 12 Figure 12: Least-Flow (LF) versus Random (R) policy comparison for the ER graph model: energy reduction ratio versus p for ν = 10 (left) and ν = 0.01 (center); average path distance (right).

1 2 . 3 20 5

 12320 Basic formulation and metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Node Switch-o Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Generalization of the cost function . . . . . . . . . . . . . . . . . . . . . . . . . Erdös-Rényi model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Power-law model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 The Watts-Strogatz model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3.1 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Performance Evaluation 17 4.1 Impact of Sleep Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Impact of Technology Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Impact of Network Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4 Impact of Power Consumption Model . . . . . . . . . . . . . . . . . . . . . . . . Comparison 2004 route des Lucioles -BP 93 06902 Sophia Antipolis Cedex Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr

Table 1

 1

	Symbol	: Notation Denition
	N	number of nodes
	L K	number of links average node degree
	T	trac
	R	link rate
	F (F N )	link (node) xed cost
	A (A N )	link (node) total cost
	ρ (ρ N ) d	average link (node) load average shortest path length
	α (β)	link (node) scaling coecient
	C	network consumption
	ν(ν N )	link (node) constant cost equivalent load
	constraints, e.g., the graph is connected and maximum link load is below the admissible value.
	Clearly, when a node is powered o, all links connected to it are switched o too.

RR n°8187

" Note that, in our context, d′ ≥ d is always veried since we are considering only connected graphs, and we assume we cannot remove source/destination nodes.RR n°8187

Acknowledgment

This work received funding from the European Commission Seventh Framework Programme under grant agreement 257740 (Network of Excellence TREND).

IEEE Network, V. [START_REF] Bolla | Enabling Backbone Networks to Sleep[END_REF]


Esteban Vanegas Márqueza

Katia Mocellin

Louise Toualbi

Yann De Carlan

Roland E Logé

E Vanegas Márquez

Finite element simulation of cold pilgering of ODS tubes

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Oxide dispersion strengthened (ODS) ferritic/martensitic alloys are promising cladding materials for sodium-cooled fast nuclear reactors intended within Generation IV. This is due to their excellent creep properties and superior irradiation resistance [1], compared to other conventional heat resistant steels such as 9Cr ferritic/martensitic steels and austenitic steels. Excellent creep properties are obtained thanks to the nanometric oxide particles dispersed in the matrix. These nanometric oxides are supposed to be very stable under neutron irradiation and at high temperatures. ODS materials are obtained by powder metallurgy. Once the powder has been obtained, consolidation of the ODS materials is achieved either by hot extrusion, or by hot isostatic pressing. Subsequently, the cladding is cold formed by a sequence of cold pilgering rolling passes with intermediate heat treatments, for stress relief purposes [START_REF] Toualbi | Proceedings of DIANA I workshop ''Dispersion Strengthened steels for advanced nuclear applications[END_REF]. The HPTR cold pilgering process is a widespread seamless tube forming operation in which the tube wall thickness and the inner diameter are reduced simultaneously. This technique provides a high forming rate, narrow tolerances and good turnouts. The process uses a reciprocating, or back-and-forth, motion. After each back and forth movement (stroke) of the dies, the raw tube is translated by a small distance and rotated around its axis. A volume element takes several dozens of strokes before deformation is completed. This complex mechanical history may lead to the nucleation of defects in low-cycle fatigue regime [START_REF] Montmitonnet | [END_REF]. The goal is to built a robust numerical model to describe the mechanical conditions that prevail during a pilgering pass. In this work, the influence of the ferritic ODS constitutive law, on the prediction of defect, is assessed. Consequently deformation path should be identified using a numerical simulation because of the complexity of the kinematics. Moreover the constitutive law must be accurate enough to describe the pilgering mechanical loadings. Comparing with the bibliography where only monotonic laws are used to simulate in 3D the pilgering process. The full stroke numerical simulation of the cold pilgering process leads to the determination of the strain path undegone by a material point. This task in new comparing to litterature.

Simulation set up

Optimizing numerically the whole process with a long tube is simply not feasible at the time being, because of the huge computation time involved. Consequently, a short tube approach is used. A pileging pass where deformation is close to 25% is simulated. The tube is turned 39° and moved forward (feed) 1.7 mm after each stroke. A material point takes 120 strokes to pass through the working zone. Initial position of the simulated systems is displayed in Figure 1. Lagrangian numerical sensors have been placed at the mid-length of the tube at different angles.

Monotonic constitutive model

A standard constitutive model adapted to described monotonic loadings, tensile is used. The elastic-viscoplastic Hansel-Spittel law is chosen, it is given by : ( ) ( )

T m m m T m m m m T m e e T Ae 8 3 7 5 4 2 9 1 1 0 ε ε ε ε ε σ ε ε & & + =
where ε is the equivalent deformation (total strain), ε & the equivalent deformation rate (total strain rate), T is the temperature and A , A isothermal tensile test is considered , no strain rate effects is assumed: parameters 3 m and 8 m are therefore set to zero.

Cyclic elastic-plastic constitutive model

The cyclic deformation induced by cold pilgering is fairly well approximated by a sequence of compression tests along two perpendicular directions [4]. The constitutive model considered in this study is therefore based on Chaboche for the cyclic behavior, it is denote as cyclic C/C. The latter utilizes multi-components forms of kinematic and isotropic hardening variables with nonlinear evolutionary rules to describe the hysteresis loops and the transient behavior of the material. The formulation chosen here is the following [5]. The elastic domain is defined by:

0 2 ≤ - -       - = ∑ ∑ y i i i i R X J f σ σ
In the previous equation y σ is the initial yield surface size. X is the kinematic internal stress tensor and R is the isotropic hardening. The formulation of the kinematic hardening tensor is based on the non-linear Chaboche kinematic model.

Damage

The criterion introduced by Latham and Cockroft and used in this work establishes the damage function from the principal stresses. This damage criterion was already used for VMR cold pilgering (e.g. [6]) and seems to be a good indicator to predict defects frequency. Compressive stresses have a negligible effect on damage compared with tensile ones; so in its original version this criterion does not take them into account. The classical damage function used in [6] is the sum of the product of the maximum tensile stress at each stroke. This function represents an indicator of the material state.

Considering the HPTR cold pilgering process, and the experimental evidence of longs cracks along the rolling axis, it is proposed to consider each strain increments in all directions (

i ε ∆ , rz z r zz rr i , , , , , θ θ θθ =
). Similarly, the expression of the maximal stress is modified in order to consider only positive stress components. Finally, authors have shown that damage occurs for mechanical states where the triaxiality is higher than -1/3. A modified cumulative damage function (Lat&Co) is then proposed: σ the equivalent stress. Thus, it is possible to obtain a cumulative damage field in each mechanical solicitation direction. Longitudinal damage function is maximal when comparing with other directions (not showed here). Figure 2 shows the sensor (located at mi-length and mid-thickness). Longitudinal damage function evolution for each constitutive law. Longitudinal damage is highest for the cyclic C/C behavior law. Even with the huge stress values predicted by monotonic laws; the longitudinal damage value is lower than the cyclic law. Hence, when the cyclic constitutive law is taken into account, the material follows more damaging situations.

+ - ≥ + ∆ * = ∑ i y triaxialit Stroke i i Co Lat ε σ

Conclusions

Accurate modeling of the material constitutive behavior is one of the most important features needed in order to simulate the process properly. Full numerical simulations considering the monotonic and cyclic constitutive behavior of ODS steels were carried out. A systematic analysis of all strokes has helped defining which stroke could contribute mainly to the oligocyclic fatigue of the material. Future work will include a numerical simulation of ODS tubes cold pilgering considering anisotropic-cyclic constitutive behavior. The influence of this feature on the mechanical history undergone by a material point during the process will be assessed.

Figure 1 .

 1 Figure 1. FEM simulation set-up.

m

  are the regression coefficients.

Figure 2 .

 2 Figure 2. Longitudinal damage function history in sensor S2 (in MPa).