On the Effectiveness of Sleep Modes in Backbone Networks with Limited Configurations

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1 On the Effectiveness of Seep Modes in Backbone Networks with Limited Configurations Luca Chiaravigio, Antonio Cianfrani 2,3 ) Eectronics and Teecommunications Department, Poitecnico di Torino, Torino, Itay 2) DIET Department, University of Roma - La Sapienza, Roma, Itay 3) Consorzio Nazionae Interuniversitario per e Teecomunicazioni, Roma, Itay Emai: uca.chiaravigio@poito.it, antonio.cianfrani@diet.uniroma.it Abstract We study the probem of putting in seep mode devices of a backbone network, whie imiting the number of times each device changes its power state (fu power mode or seep mode). Our aim is to imit the number of network configurations, i.e., the change of the current set of network devices at fu power. We deveop a mode, based on random graph theory, to compute the energy saving given a traffic variation, QoS constraints, and the number of aowed network configurations. Resuts show that the energy savings with few configurations (two or three per day) are cose to the maximum one, in which a new configuration is appied for each traffic matrix. Thus, we can concude that a practica impementation of seep mode strategies for network operators is to define, on the basis of typica traffic trend, few configurations to be activated in specific time instants. I. INTRODUCTION The continuous growth of Internet traffic represents a chaenge for the network infrastructures of Internet Service Providers (ISPs). One of the consequences to be considered is the energy consumption increase of network devices [], since this effect coud represent a botteneck for ISP networks in the near future [2]. Starting from the semina work [3] different soutions have been proposed to increase the energy efficiency of wired networks (see [4] for an overview). The presented soutions rey on two main approaches [5]: dynamic power scaing and smart standby. In particuar, network devices expoiting dynamic power scaing adapt their working rate according to the actua utiization. On the contrary, with the smart stand-by devices are put in a ow-power state when not stricty needed. In this work we focus on the smart standby approach in backbone networks. Hereafter, we ca seep mode the period of time (which may ast for minutes or hours) during which a device is in a ow-power state. When some devices are in seep mode, the other devices that remain powered on have to meet Quaity of Service (QoS) constraints, which require the knowedge of the traffic exchanged in the network. We denote as configuration the set of devices in seep mode at a given time not vioating any QoS constraint. Since traffic varies over time, most of works in the iterature (see for exampe [6], The research eading to these resuts has received funding from the European Union Seventh Framework Programme (FP7/27-23) under grant agreement n (Network of Exceence TREND). [7], [8]) assume to compute and appy a new configuration for each traffic matrix, resuting in a arge number of appied configurations per day (one for each traffic matrix). However, appying a arge number of configurations has severa drawbacks in operationa networks. First, frequent transitions between fu power and seep mode resut in an increase of the Mean Time Between Faiures, since network devices are designed to be aways powered on. Second, routing protoco convergence at the IP ayer is negativey affected: this is due to the fact that topoogy changes have to be propagated in the network (even in the presence of a centra controer), and current routing protocos may show a sow convergence when many devices change continuousy their power state. Third, deciding when appying each configuration without vioating any QoS constraint is not trivia, since traffic frequenty changes. For a these reasons, the number of network configurations has to be imited. In this work, we present a mode based on random graph theory to quantify the energy saving of a network under QoS constraints and a maximum number of admissibe configurations. The mode ets us introduce an important insight: the energy saving with a imited number of configurations (two or three per day) is aready cose to the maximum one (a new configuration appied at each traffic variation). To the best of our knowedge, none of the reated works in the fied of energy-efficient backbone networks have conducted a simiar anaysis. The cosest papers to this work are [9], []. In [9] the probem of choosing a time interva to appy an energy saving network configuration is evauated. In particuar an heuristic is defined so that to determine a reduced network topoogy and the window size of its duration. The main difference with respect to our work is that in [9] a singe energy saving configuration is considered and a comparison with the maximum energy saving is not performed. In [] a mode to evauate seep mode gains with random graphs is proposed. The main differences of this work with respect to [] are the foowing. First, the scope of our previous work is to evauate the power saving of seep modes under different power modes (for current and future devices), whie in this work we evauate the energy saving given a power mode, a traffic profie and a maximum number of configurations.

2 Second, whie in [] the configuration is assumed to be given by an orace, here we compute the best configuration which fufis different QoS constraints. Third, in our previous work we did not consider the variation of traffic and the maximum number of configurations, whie here we expicity take them into account. The paper is organized as foows. In Section II a detaied description of the network mode is proposed; In Section III we evauate the energy saving when a imited number of network configurations is avaiabe whie in Section IV the numerica resuts are shown; finay in Section V concusions are drawn. II. NETWORK MODEL The aim of the network mode is to reate the number of inks to put in standby to the network topoogy, the traffic matrix and the QoS constraints. The mode is based on the foowing assumptions: i) we focus on inks ony, ii) we consider average ink power consumption and average number of inks per node, iii) we assume a random poicy to seect which devices to be powered off for a given configuration, iv) we consider traffic uniformy exchanged among a nodes, v) we assume that during peak hour a inks have to be powered on. A these assumptions represent a conservative scenario, and the actua savings of operationa networks may be even arger than the ones shown in this paper: for instance a smart seection of standby inks by means of a power aware routing strategy coud highy increase energy saving. However, the indications provided by our anaysis(i.e. the fact that few configurations are abe to save an amount of power comparabe to the maximum one) are sti vaid and genera. In the foowing, we detai how the mode is buit. In particuar, we start from a graph withn nodes andlundirected inks. The average node degree is K = 2L N. This means that each node is connected to other K nodes on average. Moreover, the average power consumption of a ink is P L. The power consumption of the network when a inks are powered on is: P TOT = N K 2 P L () We then consider a fraction of p [,] inks put in seep mode. When a ink is put in seep mode, we assume that its power consumption is negigibe. The tota power of the network then becomes: P TOT = N K 2 P L (2) where K is the average node degree when seep mode is appied. We assume the set of inks powered off is randomy chosen. Under such hypothesis, the graph composed by the inks powered on maintains the same properties of the initia graph, and the new degree can be computed as K = K( p) []. Our mode can be extended to the case in which nodes are put into seep mode. However, we eave this feature as future work. A. QoS Constraints We then introduce the QoS constraints which imit the fraction of inks p that are put in seep mode. ) Network Connectivity: The minimum network connectivity is guaranteed when p is ower than the fragmentation threshod []: p < E[(K ) 2 ] K where E[(K ) 2 ] is the second moment of the degree distribution when seep mode is appied. If p is arger than the fragmentation threshod, then connectivity is not guaranteed among a node pairs, and custers of nodes not connected together appear. However, recent measurements studies have proven that normay K ranges between 4 and 8 for teecommunication networks [2]. Thus, each node is connected from 4 to 8 other nodes in the network on average. In our study, we assume that a node is connected to other two nodes on average when seep mode is appied, i.e., K 2. Intuitivey, this correspond to the condition in which each node acts as transport device by moving traffic from a source node and then sending it to a destination node. Thus, we impose the minimum degree constraint as: p 2 K Lemma If the degree constraint (4) hods, then aso the fragmentation constraint (3) is satisfied. Proof: We prove this condition by contradiction, assuming that E[(K ) 2 ] K > 2 K From [] it hods that: E[(K ) 2 ]=E[K 2 ]( p) 2 + p( p)k. Moreover, we express E[K 2 ] = K 2 + σ 2 K, where σ2 K is the degree variance. By substituting E[(K ) 2 ] we find that condition (5) is never satisfied. 2) Average ink utiization: Let us define the average ink oad ρ (,) when a devices are powered on. The ink oad is normay kept beow a maximum utiization threshod δ to avoid congestion, i.e. ρ δ. Then, we express the average ink oad as: ρ = D max N K 2 C (6) where D max is the maximum amount of traffic in the network, is the average shortest path ength, and C is the average ink capacity. By imposing the oad threshod, we compute D max as: D max = δn K 2 C (7) In our mode we consider D max as the peak traffic matrix. Note that this is a conservative assumption, since we consider a peak traffic matrix that saturates a network inks i.e., the utiization of every ink is equa to δ. We then consider the oad when devices are in seep mode. The seep mode can be enabed ony when traffic decreases, so (3) (4) (5)

3 TABLE I NETWORK AVERAGE SHORTEST PATH LENGTH Mode og(n) og(n) ER og(k) og(k( p)) ( ) og( K PL + N) og N K( p) og[(e[k 2 + [( ] K)/K] E[K og 2 ) ] ] K ( p) K we assume an amount of traffic D < D max. The new average oad ρ can be expressed as: ρ = D N K 2 ( p)c (8) where is the average shortest path ength when seep mode is appied. Ceary, the new oad must be ower than the maximum utiization threshod, i.e. ρ δ. Thus, we can bound the maximum p as: p D N K 2 Cδ (9) To compute ρ and ρ, we need to define the shortest path ength and, which depend on the graph mode considered. In the iterature, different graph modes have been proposed (see [3] for an overview). However, deciding which mode fits better current Internet topoogies is an open issue. Therefore, in this work we adopt two different graph modes: the Erdös and Rényi (ER) mode and the Power Law (PL) mode. In the ER mode [4] nodes are connected by inks according to a given probabiity, and the resuting degree distribution foows a Poisson distribution. On the contrary, in the Power Law (PL) mode [] the distribution P K (k) of the node degree K foows a power-aw distribution, i.e., P K (k) k γ. The intuition is that some nodes behave ike hubs, and have many more connections than others. Tab. I reports the shortest path ength and. We refer the reader to [] for detais on how these expressions are obtained. 3) Increase of the Shortest Paths: Finay, an Internet Service Provider might be interested in imiting the shortest path ength when seep mode is appied. We define the increase in the shortest path ength as, and we introduce a threshod φ (,), i.e. φ. For an ER mode, the maximum p for the shortest path constraint is expressed as: eogk φ+ p () K Simiary, it is possibe to define an equivaent constraint aso for the PL mode: og p e ( (E[K 2 ) ] K) K +og( K)[ (φ+)] N 2 (φ+) () with defined in Tab. I for the PL mode. 4) Putting things together: Given the network mode (ER or PL), the QoS threshods (δ and φ), and the traffic matrix D, we are abe to compute the maximum p which jointy satisfies connectivity (4), maximum ink utiization (9) and increase of the shortest path () or (). In this way, we can derive the tota power consumption P TOT and estimate the power saving. In the next section, we wi show how to compute the energy savings from the variation of D over time and a imited number of configurations. III. TRAFFIC VARIATION AND NUMBER OF CONFIGURATIONS In this section we consider the traffic variation to evauate the reationship among the energy saving and the number of network configurations. A network configurationiis a network state with a non empty set of inks in seep mode and satisfying QoS constraints. The overa set of network configurations to be appied during a day is denoted with i, where i is the index of a network configuration, and it is bounded by the maximum number of configurations G. For exampe G = means that one configuration is appied over the day (e.g. during night when traffic is ow) to save energy. In particuar when G = the network has two states: the peak hours state, when a inks are powered on, and the off-peak state, when configuration i = is appied and a subset of network inks is in seep mode. Each configuration is abe to satisfy a maximum amount of traffic and so the set of configurations i depends on the traffic behavior. In this work we consider a traffic profie D(t), symmetric around T/2, 2 being T the traffic period; we focus our attention on the decreasing traffic phase i.e., [,T/2], but a simiar anaysis can be derived for the traffic increase phase. Let us define as [τ i,τ i+ ] the time interva during which the configuration i is appied. Since the traffic is decreasing, the number of inks that configuration i is abe to put in seep mode depends on the traffic at time τ i i.e., the maximum traffic in the considered time interva. As a consequence we define as p(τ i ) the fraction of inks put in seep mode when configuration i is appied; p(τ i ) satisfies the conditions (4), (9) and () for ER mode or () for PL mode considering traffic D(τ i ). The energy consumption of the network E TOT when G configurations are appied can be expressed as: E TOT(G) = N K 2 P L G (τ i+ τ i )( p(τ i )) (2) i= with τ =, p(τ ) = and τ G+ = T/2. Let us define the energy saving S when G configurations are appied as: S(G) = E TOT (G) E TOT (3) where E TOT is the energy consumption of the network with a inks powered on, which can be expressed as: E TOT = N K 2 P L T 2. Given the number of configuration G, our aim is to maximize the saving (3). Thus, we need to find the time instants (τ, τ2,.. τg ) that maximize S(G). Fig. reports an exampe of inear traffic variation and corresponding power consumption variation for G =,2,3. The intuition is that, 2 We eave as future work the extension for the asymmetric case.

4 D max D(t) P TOT (t) Traffic P TOT (t) G= T/2 t τ T/2 t G=2 P TOT (t) G=3.8.6 PL K=6 ER K=6 ER K=5 ER K=4 ER K=3 p.4 τ τ 2 T/2 t τ τ τ 2 3 T/2 t.2 Fig.. An exampe of traffic variation and power consumption variation for G =, G = 2 and G = 3. in order to maximize the saving, the vaues of (τ, τ2,.. τg ) are chosen in order to minimize the shaded area of the figure, which corresponds to (2). For exampe, G = corresponds to the case in which we seect a singe set of powered off inks that maximizes the energy saving S(), which can be expressed as: ( ) S() = τ T 2 p(τ ) (4) Since the traffic is decreasing, the saving is maximized for the vaue τ soving the equation ds() dτ =. Simiary, we can extend this process to generic G configurations. Finay, it is possibe to define an upper bound on savings, by computing a new configuration for each variation of traffic: S(U) = 2 T T 2 p(τ)dτ (5) By comparing S(G) with S(U) we are abe to evauate the effectiveness of seep modes when G is imited. IV. RESULTS We first investigate how much p is affected by the traffic and the connectivity constraint. Uness otherwise specified, we adopt the foowing set of parameters: N =, δ = 5% [5], C = Gbps [5], P L = 5 W [6]. We initiay vary D continuousy between D max and zero. For each D, we compute the maximum p not vioating the constraints. Fig. 2 reports the variation of p versus the normaized traffic D/D max. The figure reports the resuts for the PL and ER graph modes. For the PL mode, we adopt a Pareto distribution (see [] for detais). With high traffic, p is ow, since many devices has to be powered on to satisfy the traffic demands. Note that at the peak D/D max = p is zero since the network is exacty dimensioned to carry the peak amount of traffic, and it is not possibe to put in seep mode any device. However, as traffic decreases, p steadiy increases. Interestingy, for ow traffic, p is constant, due to the fact that the connectivity constraint has been reached. Focussing then on the mode, both PL and ER presents a simiar trend, showing that the network mode performs simiary for both these graph famiies. Finay, the figure reports aso the variation of the degree K. Interestingy, p decreases with K. This is due to the fact that, since K is ower, the network is composed by p D/D max Fig. 2. Variation of p vs traffic for ER and PL modes. φ=% φ=3% φ=5% φ=7% φ=9% D/D max Fig. 3. Variation of p vs traffic for different vaues. ess inks. Consequenty, the connectivity constraint becomes tighter than the maximum oad constraint, and so it is possibe to put in seep mode ess devices. On the contrary, for highy connected networks (i.e, high vaues of K), the connectivity constraint is reached for extremey ow vaues of traffic, i.e. D/D max =. for K = 6. Note that the degree of current networks ranges between 4 and 8, thus we are more ikey in the situation in which p mainy depends on traffic rather than the connectivity constraint. We then consider aso the constraint on the increase of the shortest path ength φ. Fig. 3 reports the variation of p versus D/D max for the ER graph mode with K = 6. When φ = %pis beow 2%, meaning that the fraction of inks that are put in seep mode is rather imited. However, as φ increases, p steadiy increases, being the maximum p equa to 58% for φ = 9%. Thus, we can concude that the setting of φ strongy infuences the quantity of inks that can be put in seep mode. In particuar, an operator shoud carefuy choose the vaue of φ which trades between QoS and power saving, e.g. φ = 5% is a good tradeoff in our mode. In the foowing, we vary D over time and we consider the impact of appying a imited number of configurations. Rather than focussing on a singe profie, we consider a famiy of

5 f(t) (-H)L+H H H L Saving S [%] S(U) S(3) S(2) S() LT/2 Fig. 4. T/2 T Traffic Profie Mode. t (a) Without maximum increase of the shortest path ength.2 symmetric profies of period T, defined as foows: (L )( H) 2t L H T + t < T L H 2 H D(t) = (L H)( H) L ( 2t T )+H T L H 2 H t T 2 (6) In particuar, parameter L (, ) varies the width of the off-peak zone, whie H (, ) varies the difference between peak traffic and off peak traffic. In this way, we are abe to capture different traffic behaviors and to generaize as much as possibe our resuts. Fig. 4 reports a graphica representation of traffic profies. We consider initiay the variation of L, whie we keep H =. This case is representative of traffic profies with a peak during the day and a deep off-peak during the night. Fig. 5(a) reports the saving for different configurations and an ER graph mode with K = 6. Moreover, the figure reports aso the upper bound S(U). Savings are ow when L, since the width of the off-peak zone is reduced. Conversey, the saving steadiy increases whenldecreases. The saving with one configuration is beow S(U) (as expected). The gap between S(U) and S() is maximum for L.5, which corresponds roughy to the case in which the traffic profie is a straight ine (as reported in Fig. ). In this case, a singe switch off is far for the upper bound. However, the savings steadiy increase with G. For exampe, the saving S(3) is ony % ower than the upper bound. Thus, a imited number of configurations are abe to save an amount of energy comparabe with the upper bound. We then repeat the variation by imposing a maximum increase of the shortest path ength φ = 5%. Fig. 5(b) reports the savings obtained with different configurations. Differenty from the previous case, the savings are ower, since this constraint strongy infuences the fraction of inks that can be put in seep mode. However, the gap between the upper bound and the fixed configurations is even reduced, being aso the saving S(2) cose to the upper bound. This is due to the fact that, as the constraint φ becomes tight, p is ess dependent on the variation of traffic. Thus, few configurations are enough to achieve a saving cose to the maximum one. Finay, we investigate the impact of H, which contros the difference between peak and off peak traffic. We consider again an ER mode with K = 6. Fig. 6 reports the variation Saving S [%] S(U) S(3) S(2) S() (b) With maximum increase of the shortest path ength (φ = 5%) Fig. 5. Saving vs parameter L for ER mode with K = 6. of H for G =,2,3 and the upper bound U. With H the saving decreases, since the difference between off-peak and peak traffic is ow. On the contrary, when H the saving increases. Note that nowadays, we are in the bottom of the figures, i.e. the difference between the peak and the offpeak traffic is high). For L <. and H <. the savings S(), S(2) and S(3) are cose to the upper bound S(U), i.e., typicay arger than 5%. Considering then the fu range of vaues, S() and S(2) save ess energy than S(U). However, savings(3) is cose to the upper bounds(u). This fact further corroborates our intuition that few configurations are abe to obtain energy savings comparabe to the maximum one. V. CONCLUSIONS We have evauated the effectiveness of seep modes in backbone networks with a imited number of configurations. Resuts show that the saving obtained with few configurations (at most three) is ony % far from the maximum saving. We think that this is an important indication for operators, since few configurations require a simpified seep mode management as we as a margina impact on the QoS perceived by users. As next steps, we wi consider the case in which aso nodes can be put into seep mode. Moreover, we wi extend our findings on a rea topoogy with measured power consumption figures and measured traffic profies..2

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