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Models and Heuristic Approaches for Network Power Management R.G. Garroppo, S. Giordano, G. Nencioni, and M. Pagano Dip. di Ingegneria dell’Informazione, Universit`a di Pisa, Italy email: {name.surname}@iet.unipi.it Maria Grazia Scutell`a Dip. di Informatica, Universit`a di Pisa, Italy email:
[email protected]
Abstract—The paper describes and compares different approaches that can be used to design Network Power Management methods, with the aim of reducing the power consumption of telecommunication networks. The approaches are based on the solution of optimization problems that have, in general, a Mixed Integer NonLinear Programming (MINLP) formulation. Given that the problems are NP-hard, exact methods for finding optimal solutions can be used only for scenarios of limited size. In this framework, the paper proposes a heuristic for finding a suboptimal solution of the Power Aware Routing and Network Design (PARND) problem, which is one of the more general design problems in Network Power Management. The simulation study highlights the capability of the proposed heuristic to obtain solutions near the optimum and to outperform the other approaches in terms of energy savings, while satisfying the constraints of the traffic demands.
I. I NTRODUCTION The Internet is rapidly becoming a major consumer of power, with significant economic and environmental impacts. Furthermore, energy consumption is one of the key issues for the future life. Considering these aspects is not surprising that communication operators and equipment vendors are trying to reduce the power consumption of networks, and a part of the research community is focused on the design of Green Networks. Recent works on Green Networks have defined the energy aware problems and (in some cases) solutions on two relevant aspects: the Network Device Design and the Network Power Management. The Network Device Design consists in energy efficient mechanisms implemented in network equipments. Examples of these works are Adaptive Link Rate (ALR) [1] and Low-Power Idle (LPI) [2]. The Network Power Management is expressed by methods aimed to achieve further energy savings by means of appropriate strategies that exploit the power behavior of green network devices [3] [4]. The paper focuses on Network Power Management, describing three different approaches to obtain energy savings: the Power Aware Routing (PAR), the Power Aware Network Design (PAND), and the Power Aware Routing and Network Design (PARND). The problems associated with these approaches lead to Mixed Integer NonLinear Programming (MINLP) formulations, which are NP-hard in the general case. Hence, exact methods for solving these problems are available only for small networks. To overcome this problem, the development
of simple heuristic could lead to find suboptimal solutions in an acceptable computational time also for medium-large networks. In this framework, the paper firstly presents suitable formulations for the described problems, then it proposes a heuristic for solving the PARND problem. The simulation analysis is aimed at comparing the power saving obtainable with the diverse approaches and at evaluating the ability of the proposed heuristic to provide a solution near to the optimum. The paper is structured as follows. Section II introduces the problem statement, while Section III describes the proposed heuristic for the PARND problem. Section IV reports the network scenarios considered in the simulations, whereas Section V presents the simulation settings and discusses the obtained results. The conclusions are drawn in Section VI. II. P ROBLEM S TATEMENT In this section, we present the formulation of the PAR, the PAND and the PARND problems, and for each one a brief summary of the solution methods proposed in the literature. Let us introduce the common parameters and notations of all the stated models. In particular, the starting point of the analysis is a network modeled as a directed graph G(V, E) where V is the set of nodes and E the set of links. We consider the following elements characterizing the power consumption of a router [5]: • line (slot) cards with port cards; • chassis; • route processor. Therefore,the following parameters are assumed to be given in order to characterize the power consumption of a network element: • PLC is the power consumption of the considered line card; • PvC is the power consumption concerning the chassis of the node v; RP • Pv,T (v) is the power consumption concerning the route processing of the node v at the traffic throughput T (v); • PvMAX is the maximum power consumption of the node v. In order to define the power consumption concerning the L , we take into account that for connecting router u link uv, Puv and router v (both uv and vu links), two line cards are needed, one for each router. Considering that uv and vu are strictly connected (see Equations 9), the power consumption of the
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line cards in the two routers can be assumed to represent the predominant component of the power consumption of the links L equals to the uv and vu respectively. Hence, we assume Puv LC P of the line card used in the router for the link uv. Two set of binary variables are defined: • xv ∈ {0, 1} is set to 1 when the node v is switched on, and to 0 otherwise; • yuv ∈ {0, 1} is set to 1 when the link uv is switched on, and to 0 otherwise; About the traffic demand and capacity of nodes and links, we define: • D is the set of node pairs of the traffic matrix, • dsd is the traffic demand between the source node, s, and the destination, d, sd is the amount of d flowing through the link uv, • f uv sd • CvN is the node capacity of v, L is the link capacity of uv. • and Cuv The traffic throughput of node z can be calculated as follows: sd T (z) = ∑ ∑ fuv + ∑ dsd . (1) uv∈E sd∈D v=z
on Dijkstra’s algorithm, and have deeply investigated the impact on the performance of the traffic load, and the network topology. In general, PAR is a nonlinear multicommodity flow problem that can be efficiently solved by means of Ipopt [8], a software package for large-scale nonlinear optimization that implements an interior-point line-search filter method, considering the actual power consumption of nodes and finding the optimal solution. B. Power Aware Network Design (PAND) The target of the PAND problem is to switch off links and nodes in order to minimize the overall power consumption of the network by taking into account the traffic matrix, D, and L and PMAX . the power consumption of links and nodes, i.e. Puv v The PAND problem can be formulated as follows. Minimize
∑
RP ∑ Pv,T (v)
(2)
subject to
∑
uv∈E u=z
−
∑
sd fuv
=
uv∈E v=z
∑
dsd if z = s ∀sd ∈ D ∀z ∈ V −dsd if z = d 0 otherwise (3)
sd L fuv ≤ Cuv
∀uv ∈ E
(4)
sd∈D
T (v) ≤ CvN
sd L fuv ≤ Cuv · yuv
∀uv ∈ E
(7)
sd∈D
v∈N
sd fuv
(6)
subject to the flow conservation constraints (Eq. 3) and
sd∈D s=z
The solution of the PAR problem represents the traffic routing strategy that permits to minimize the overall power consumption of the network by taking into account the traffic matrix, D, and the power consumption of the nodes concerning RP . This problem does not consider the route processor, i.e. Pv,T (v) the possibility of switching off nodes or links; hence the constant components of the network power consumption are insignificant. The objective is to reduce the power consumption with a power-aware routing solution, by minimizing the power component depending on the load of the nodes. The PAR problem can be formulated as follows.
L Puv · yuv
uv∈E
v∈N
A. Power Aware Routing (PAR)
Minimize
∑ PvMAX · xv + ∑
∀v ∈ V.
(5)
Equations 3 are the classical flow conservation constraints, instead Equations 4 and 5 are the node and the link capacity constraints, respectively. The authors of [3] have proposed this problem and solved it by using a linear programming solver (e.g CPLEX [6]), by considering coarse linear approximations of power consumption of the nodes, which is generally a nonlinear function. The authors of [7] have proposed a heuristic solution based
T (v) ≤ CvN · xv yuv = yvu
∑
∀v ∈ V
(8)
∀uv ∈ E
yuv ≤ |V | · xz
(9)
∀z ∈ V.
(10)
uv∈E v=z
Equations 7 and 8 are the node and the link capacity constraints extended to switch off nodes and links, respectively. Equations 9 impose that, when link uv is switched off, the link vu can not be switched on, and vice versa. Finally, Equations 10 affirm that a node can be switched off only when all its links are switched off. The authors of [4] have proposed and compared diverse heuristics to solve this problem. The PAND is a problem very hard to solve exactly, therefore we have found an optimal solution using an integer linear programming solver (CPLEX) only for a small network scenario. C. Power Aware Routing and Network Design (PARND) This problem minimizes the overall power consumption of the network by considering jointly both the possibility of switching off links and nodes of the network, and a power-aware routing strategy. The PARND problem can be formulated as follows. Minimize
C RP ∑ Pv,T (v) + ∑ Pv · xv + ∑
v∈N
v∈N
L Puv · yuv
(11)
uv∈E
subject to the same constraints of the PAND problem (i.e. Eq. 3, 7, 8, 9, and 10). The authors of [9] have formulated a similar problem but they assume that nodes and even links have a linear energy
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behavior. They have solved their problem using CPLEX. The authors of [10] have considered only the link energy consumption, which is linearly proportional to the traffic that flows through. They have solved this problem by means of a heuristic. Differently from [9] and [10], we consider an energy model of a router based on [5]. In particular, the considered energy model is given by the sum of two constant addends associated with the power consumption of the chassis and the line cards, and one variable (nonlinear) element associated with the route processor. III. T HE
PROPOSED SOLUTION FOR THE
PARND
PROBLEM
For solving PARND we propose a new heuristic (see Algorithm 1), which is based on a PAND solution proposed in [4]. Algorithm 1 - Heuristic for the PARND problem 1: SN=sort nodes (V ); 2: min pow cons = ∞; 3: for all v ∈ SN do 4: disable node v; 5: solve PAR problem ⇒ (status, power consumption); AND 6: if (status == Solve Succeeded power consumption < min pow cons) then 7: min pow cons = power consumption; 8: else 9: enable node v; 10: end if 11: end for 12: SL=sort links (E); 13: min pow cons = ∞; 14: for all uv ∈ SL do 15: disable link uv; 16: solve PAR problem ⇒ (status, power consumption); 17: if (status == Solve Succeeded AND power consumption < min pow cons) then 18: min pow cons = power consumption; 19: else 20: enable link uv; 21: end if 22: end for 23: solve PAR problem. The heuristic begins by sorting the nodes by means of a least-flow (LF) policy, namely the nodes are sorted basing on their traffic throughput after solving the PAR problem on the network with all elements powered on (i.e., xv = 1 ∀v ∈ V and yuv = 1 ∀uv ∈ E). Afterwards, each node, and consequently all its adjacent links, is disabled according to the considered ordering and the PAR problem is solved on the restricted network. Note that only nodes without traffic demand towards/from any other node can be disabled. If the resolution successfully ends and the overall power consumption is reduced, then the network power consumption is updated and the iterations continue, otherwise the node is enabled again. When all nodes are considered, the PAR problem is solved for the new network topology where the previous disabled
nodes and their adjacent links are switched off. Based on this solution, the links are sorted by taking into account the link flows (LF policy). Subsequently, a procedure similar to that used to switch off the nodes is applied to the sorted list of the links. The solution of PAR applied to the resulting network topology finally provides the power-aware routing strategy and the overall power consumption. It is relevant to observe that in the case the nodes cannot be switched off, the heuristic can be limited to the iterative procedure associated with the links. IV. N ETWORK S CENARIOS In the following section we present the two core network scenarios that we have considered to analyze the performance of the different approaches. In both network scenarios, each node represents a core router; we assume the use of the Juniper T1600 core router, having a total throughput capacity of 1600Gb/s and a power consumption of 8352W [11]; thus all nodes of the networks have the same energy behavior. For each link we assume to use multiple SONET/SDH OC768c/STM256 PICs, which have a payload bandwidth of 38.486Gb/s and a power consumption of 65.7W [12]. We assume that all links have the same capacity. To determine the number of PICs for each link, we have computed the maximum load of a link using the shortest-path routing (SPR), aimed at minimizing the number of hops between the source and the destination of each traffic demand. Thus, the link capacity is the minimum multiple, α, of the PIC capacity higher than or equal to the maximum load. The results reported in [13] show that the power consumption of a chassis is equal to about 200W for all classes of routers; thus, for each node v we have set PvC = 200W. In summary, we have considered the following parameters: • PvMAX = 8352W and CvN = 1600Gb/s ∀v ∈ V ; L = α · 65.7W and CL = α · 38.486Gb/s ∀uv ∈ E; • Puv uv C • Pv = 200W ∀v ∈ V ; Concerning the component of the route processor for the energy characterization of devices, we focused our attention on a cubic curve (see Figure 1) since it represents the state-of-the-art of circuit-level energy-efficiency mechanisms [3]. In particular, the cubic curve is the energy behavior of network equipments that use energy savings techniques such as dynamic voltage and dynamic frequency scaling (DVSDFS), which permit energy consumption to scale with resource requirements. Current routers do not implement such techniques and are very energy inefficient, but we assume that in the near future they could be used. Based on the previous cited parameters of actual routers, we have computed the power consumption concerning the route processor as follows. RP Pv,T (v) =
PvMAX − PvC · T (v)3 ∀v ∈ V (CvN )3
(12)
A. Network Scenario ”EUropean core topology (EU)” The European core topology is taken from [14], and given by the Nobel 2 project. The network, shown in Figure 2, is
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Fig. 3. Network topology from the dataset AS 3967 of the RocketFuel study
To obtain the traffic matrix, we set the mean value of traffic demand from s to d as in the EU scenario, i.e. dsd = 2.5Gb/s. Then, each element of the matrix dsd is extracted from a uniform distribution: dsd = U 0.5 · dsd , 1.5 · dsd ∀s, d ∈ V . Taking into account this traffic demand, after applying the SPR and calculating the maximum load link, the link capacity L = is 115.458Gb/s (i.e. α = 3) and the power consumption is Puv 197.1W ∀uv ∈ E. V. S IMULATION R ESULTS
Fig. 2.
European core topology
composed of 28 nodes and 82 directional links (or, equivalently, 41 bi-directional links). The traffic matrix is obtained by the data file ”Nobel-2 directed graph” downloaded from [14]; the file contains the undirected traffic demand between each couple of nodes of the considered network scenario. To obtain the directed traffic demands, we have randomly split the demand between each direction. The total amount of traffic demand is 1898 Gb/s, distributed among the 756 pairs (i.e. all couples of nodes sd ∈ D). The mean traffic demand of an active pair is of about 2.5 Gb/s. Considering this traffic demand, we have computed the link capacity as previously described and we have obtained a link capacity equals to 307.888Gb/s (i.e. α = 8) and a power L = 525.6W ∀uv ∈ E. consumption Puv B. Network Scenario ”RocketFuel (RF)” The Rocketfuel network topology is obtained from the dataset AS 3967 in the Rocketfuel study [15]. Figure 3 shows the network topology obtained from the original data set, merging in one router all routers located at the same city and in one link all parallel links.
The simulation study is devoted to compare the performance in terms of network power consumption provided by the different strategies presented in the previous sections. In particular, the results indicated as SPR, PAR, PAND and PARND refer to the optimal solution of the shortest path routing, the PAR, the PAND and the PARND problems. On the contrary, HPAND and HPARND refer to the results obtained by using the C++ code implementing the LF heuristic of [4] for the PAND problem, and the proposed heuristic for the PARND problem. In particular, the code implementing HPARND exploits the Ipopt for solving the PAR problems during the iterations of Algorithm 1. For analyzing different network load conditions, we have multiplied the traffic matrix by a scalar β. It is relevant to note that the overall power consumed by the L and the PC of network is calculated by summing up the Puv v RP the active links and nodes, as well as the Pv,T (v) calculated for each active node by taking into account its throughput T (v). A. Results for the EU The first parameter considered in the comparison is the overall power consumed by the network for transporting the traffic demands. The results are summarized in Figure 4 for three values of β and for the SPR, the PAR, the HPAND and the HPARND strategies. The results concerning the PAND and the PARND have not been added to the Figure, due to the difficulties in finding the optimal solution for these problems. In particular, CPLEX (Optimization Studio Accademic Research Edition 12.2) did not provide a solution of the PAND problem after 72 hours, when running on a PC with 12 Intel Xeon L5440 @2.27GHz CPUs, 24GB RAM, and an Intel S5520HC Motherboard. Furthermore, it is relevant to note that, for β = 1.1, SPR and HPAND (using SPR in its iterations) did not find a solution satisfying the constraints on the link capacity. Indeed, in both cases we observed that the link transporting the traffic from Hamburg to Berlin has a
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Fig. 4.
Power consumption vs. the compared strategies - EU Scenario
load higher than the maximum capacity, fixed to 307.888Gb/s. On the contrary, HPARND permits to satisfy the constraints on the links capacity also for β = 1.1. The performance of HPAND and HPARND are quite similar for β = 0.9 and β = 1.0; in both cases the two heuristics lead to the same number of switched on links, equal to 56 and 58, respectively. Hence the small differences between the values of the power consumed using these heuristics is due to the routing strategy. Exploiting a power-aware routing strategy, HPARND permits to save further energy with respect to HPAND. Furthermore, for β = 1.1 HPARND permits to satisfy all traffic demands and simultaneously to switch off 24 unidirectional links. In this condition, the maximum link load is equal to about 253 Gb/s, which represents about the 82% of the link capacity. Figure 5 shows the load of the links obtained with SPR, HPAND and HPARND. Comparing the curves, we can note that HPARND leads to transfer traffic demands from links that would be underutilized with the SPR strategy towards other selected links. This optimization of the routes leads to the increase of the average load of the links, but permits also to switching off some links that in the case of the SPR strategy would be lightly loaded. B. Results for the RF This small network scenario allowed us to solve PAND and PARND also via CPLEX. The results in terms of the power consumption are summarized in Figure 6 for three values of β. Before the analysis of the results, we report that the simple SPR algorithm does not allow to satisfy the constraints on the maximum link load for β = 1.1. In this scenario, the load of the links transporting the traffic from Herndon to New York and in the opposite direction, is higher than the link capacity. Analyzing the Figure 6 we can deduce that the proposed HPARND permits to achieve results near the optimum, calculated by solving PARND with CPLEX. Furthermore, the distance between the overall power consumption obtained with HPARND and the optimum one increases as the traffic demands increase; indeed, whereas for β = 0.9 HPARND and PARND produce similar values of power consumption, for β = 1.1 HPARND gives a power consumption that is about 2% higher than the optimum one. Moreover, for the same value of β HPAND leads to an overall power consumption that is about
Fig. 6.
Power consumption vs. the compared strategies - RF Scenario
13% higher than the optimum one, calculated by solving the PAND problem with CPLEX. This different behavior of the two heuristics with respect to their corresponding optimum solutions is further emphasized by the analysis of the number of active links needed to satisfy the traffic demands. In particular, whereas HPAND requires 40 active links to satisfy the traffic demands for β = 0.9, only 36 links are sufficient for PAND, HPARND and PARND. In the case β = 1.1, 48 links are needed for HPAND, whereas only 38 are required by the PAND and the PARND strategies. Furthermore, in this case HPARND requires 40 links; this result points out that when the traffic demand increases the worsening of the HPARND performance with respect to the PARND solution is less apparent than that observed by comparing the HPAND and the PAND solutions. Figures 7 and 8 show the load of the links obtained respectively with the HPAND and the PAND strategy for β = 1.1. By comparing the curves, we can note that HPAND does not converge to a solution that provides an adequate concentration of routes permitting to switching off some links. This drawback is due to that, if the algorithm make the off target choice by switching off a suboptimal node/link, it will never backtrack to correct the mistake. The results of different simulation runs have highlighted that the local minimum obtained with this procedure is always slightly far from the optimum. Furthermore, as already observed, the performance worsens when the traffic demands increase. On the contrary, this drawback is limited with the proposed HPARND, due to the load balancing action of the PAR strategy used during the iterations. VI. C ONCLUSIONS The simulation study presented in this paper highlights that very high power savings are provided by the joint utilization of power-aware routing strategy and of a network design aimed at minimizing the power consumption. As shown in the RF scenario, the optimal solution of the PAND problem provides interesting results, but the heuristic from the literature, i.e. HPAND, does not permit to achieve results near the optimum, especially when the network load increases. In this situation, the solution provided by the HPAND algorithm permits to obtain a power saving to the 9% with respect to the RF
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ACKNOWLEDGMENTS This work has been partially sponsored by the EFFICIENT project funded by MIUR. We thank Prof. Jordi Castro for his valuable suggestions.
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network adopting the simplest strategy, i.e. SPR, whereas the proposed HPARND doubles this value. Furthermore, the power aware routing strategy, used by HPARND, produces a load balancing that permits to satisfy the constraints on the link capacity even in conditions of network load, where the solutions based on the SPR routing fail. Finally, the simulation results have pointed out that the PAR strategy does not provide high power savings, since these are mainly provided by the switching off of links or nodes. Hence, the utilization of a power aware routing strategy provides interesting advantages with respect to the simple SPR solution only when it is jointly used with a network design strategy minimizing the power consumption.
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