Routing in Optical and Stochastic Networks

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Routing in Optical and Stochastic Networks

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magniﬁcus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 1 juni 2015 om 10:00 uur

door

Song YANG

Master of Computer Science Dalian University of Technology, China geboren te Anshan, Liaoning Province, China.

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This dissertation has been approved by the promotor: Prof. dr. ir. P.F.A. Van Mieghem copromotor: Dr. ir. F.A. Kuipers

Composition of the doctoral committee:

Rector Magniﬁcus, Voorzitter

Prof. dr. ir. P.F.A. Van Mieghem, Technische Universiteit Delft, promotor Dr. ir. F.A. Kuipers, Technische Universiteit Delft, copromotor Independent members:

Prof. dr. ir. G.J.T. Leus, Technische Universiteit Delft Prof. dr. ir. R.E. Kooij, Technische Universiteit Delft Prof. dr. ir. A. Pras, Universiteit Twente

Prof. dr. J.L. van den Berg, Universiteit Twente Prof. dr. C. Blondia, Universiteit Antwerpen

ISBN 978-94-6186-451-2

Keywords: Routing, Optical Networks, Energy-eﬃciency, Availability, Survivability, Stochastic Networks, Convex Optimization, Maximum Flow, Min-Cut.

Copyright c 2015 by S. Yang

This research was supported by the China Scholarship Council.

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the author or Delft University of Technology.

Printed in the Netherlands

Published by Uitgeverij BOXPress, ’s-Hertogenbosch An electronic version of this dissertation is available at http://repository.tudelft.nl/.

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Summary

In most types of networks (e.g., optical or transportation networks), finding one or more best paths from a source to a destination, is one of the biggest concerns of net-work users and providers. This process is known as routing. The routing problems differ accordingly depending on different application scenarios with their respective routing goals. For example, in the field of data communication networks, with many communi-cation applicommuni-cations emerging (such as Cloud Computing, Internet Protocol Television (IPTV), Video-On-Demand), finding optimum routing paths has been considered in the bandwidth-limited network or when it is expensive to increase the capacity of the ex-isting hardware. Moreover, for network providers, the Service Level Agreements (SLA) determine the level of service that is promised to the network user. The routing prob-lems should also incorporate these SLAs (e.g., connection availability). The energy consumption of the data communication networks is another example, due to its im-pact on the environment (e.g., green house effect). Energy-aware routings have been considered as one approach to efficiently deal with this issue.

Apart from that, the network itself is likely to behave in a stochastic manner. For instance, especially in large networks, it is diﬃcult to obtain an accurate view on the link characteristics like bandwidth utilization or latency, because their dynamics are usually of the same order as the time it would take to distribute information on the link state throughout the network. The routing problems become more diﬃcult to solve if the network dynamics have to be considered.

This thesis deals with routing problems in two kinds of networks, namely, (1) (de-terministic) optical networks and (2) stochastic networks. Optical networks have been widely deployed because of their high capacity, low bit rate, etc. Wavelength Division Multiplexing (WDM) technology enabled optical networks divides the capacity of a fiber into several non-overlapping wavelength channels that can transport data inde-pendently. These wavelength channels make up lightpaths, which are used to establish optical connections that may span several fiber links. Chapters 2 and 3 study the energy-aware path selection problem in IP-over-WDM optical networks and the energy-efficient network design problem, respectively. In Chapter 4, a more flexible optical network en-abled by OFDM technology is introduced, which is here called spectrum-sliced elastic optical path (SLICE) network. In this kind of network, the capacity of a fiber link is

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xii Summary

divided into a fixed number of (overlapping) low data rate transporting units, which are called subcarriers. Subsequently, Chapter 4 studies the routing problem in SLICE networks. Connection availability, which is defined as the probability that the corre-sponding connection is available, is a key element of many SLAs. Establishing a path over a connection should obey the availability agreements to avoid the loss of revenue. Chapter 5 studies the availability-based path selection problem. The energy-aware path selection problem in Chapter 2 is polynomial-time solvable, but all the other problems in Chapters 3, 4 and 5 are proved NP-hard. To solve them, exact algorithms or efficient heuristics are proposed.

In deterministic optical networks, the link weights (e.g., energy) are usually assumed to be known, while in the so-called stochastic networks they are uncertain. Although in some cases optical networks may behave in a stochastic manner, most of the routing problems in deterministic optical networks are already NP-hard as mentioned above, which suggests that similar routing problems in stochastic optical networks are even harder to solve. We therefore consider more general stochastic networks, which are not confined to any specified type of network. Chapter 6 studies the maximum-flow problem without and with a delay constraint under the assumption that the bandwidth and delay follow a general log-concave probability distribution. A convex optimization formulation is proposed to solve the maximum-flow problem in stochastic networks. When a delay constraint is imposed on each path, the problem becomes NP-hard. To solve it, an approximation algorithm and a heuristic are devised.

So far, all the studies do not thoroughly account for the correlations between link weights, but the link weights (such as failure probability) are correlated in some real-life networks. For example, in interdependent networks, where for instance the electricity network and Internet network are coupled and inter-connected, and one node or link failure in one network may cause failures of nodes or links in the other network. Chapter 7 addresses the shortest path problem and the min-cut problem (the dual problem of the maximum-ﬂow problem in conventional networks) in more general correlated networks. Correlated network can also be regarded as a special kind of stochastic network, since its link weight is uncertain instead of ﬁxed. Two correlated link weight models are studied, namely (1) deterministic correlated model and (2) (log-concave) stochastic correlated model. These two problems are proved NP-hard under the deterministic correlated model, and cannot be approximated to arbitrary degree, unless P=NP. But they are shown polynomial time solvable under a (constrained) nodal deterministic correlated model. Under the (log-concave) stochastic correlated model, two convex optimization formulations are proposed to solve the shortest path problem and the min-cut problem, respectively.

Finally, Chapter 8 concludes with the main ﬁndings and contributions of this thesis, and also points out possible future work.

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Chapter 1 Introduction

Routing [1] is the process of finding one or more best paths from a source to a destina-tion in the network. The routing problem is undoubtedly one of most widely researched topics due to its numerous industrial applications. Tons of routing problems have been proposed and solved depending on different types of networks with their respective routing goals. Among all of them, some representative routing problems are, for ex-ample, the shortest path problem [2], the min-sum link-disjoint paths problem1 [3], the maximum-flow problem2 [4], etc. The three aforementioned problems are polynomial-time solvable in conventional networks, but by adding or changing some problem condi-tions according to some specific applicacondi-tions, a lot of new problem variants appear, and in most of the cases, these problem variants are NP-hard. For example, the min-min link-disjoint paths problem, which seeks for a pair of link-disjoint paths such that the weight of the smallest path is minimized, is NP-hard according to [5]. This problem arises from the application that the weight of one path (primary path) needs to be minimized, which transmits data normally, while the weight of the other path (backup path) can be greater, since it is only activated in case of the failure of the primary path. Since routing problems in various kinds of networks can be rather a broad research field, in this thesis we draw our attention on two kinds of networks, namely (1) (de-terministic) optical networks, and (2) stochastic networks. Optical networks [6] which apply optical fiber as transfer medium can provide huge amounts of data compared to the traditional networks which use copper as medium. Apart from providing such huge bandwidth, optical fiber has the advantage of low cost, low bit error rates, low signal distortion, etc. Due to these reasons, optical networks have been widely deployed in most of the world’s basic physical network infrastructure. Typically, an optical telecom network can be further partitioned into 3 domain networks [7], namely, (1) core net-work, (2) metro network and (3) access network. Core network is the central part of the

1_{Find a pair of two link-disjoint paths such that the total weight is minimized.}

2_{Send as much ﬂow (information or goods) from a source to a destination, without exceeding the}

capacity of any of the used links.

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2 Chapter 1. Introduction

optical telecom network, which can span up to thousands of kilometers to connect dif-ferent cites/providers, which is the most energy and cost consuming domain of optical telecom network. On the other hand, metro network is usually deployed in a metropoli-tan region with at most hundreds of kilometers links, and enabled by ring topology (SONET/SDH ring). The access network usually connects subscriber/customer and their corresponding service providers, and spans only a few tens of kilometers. Since the core network is the dominant domain of optical telecom network, we focus on optical core network in this thesis.

In deterministic optical networks, the link weight such as bandwidth or delay is assumed to be known. However, in many real-life networks (and also sometimes in optical networks), the link weight usually varies and is uncertain. For example, due to the size and complexity of data communication networks, it is difficult and expensive to obtain an accurate view on the states of the links. Another example is that the delay and available bandwidth are affected by diurnal patterns, interference in wireless networks, or by failure and maintenance events. Although in some cases optical networks may behave in a stochastic manner, most of the routing problems in deterministic optical networks are already NP-hard (e.g., the routing and wavelength assignment problem [8], the impairment-aware path selection problem [9], etc.), which suggests that similar routing problems in stochastic optical networks are even harder to solve. We therefore consider more general stochastic networks, which are not confined to any specified type of network. We will also address relevant routing problems in stochastic networks. Throughout this thesis, we assume that the optical networks are deterministic if it is not specified.

Moreover, the link weights (such as failure probability) are correlated in some real-life networks. For example, in interdependent networks, where for instance the elec-tricity network and Internet network are coupled and inter-connected, one node or link failure in one network may cause failures of nodes or links in the other network. Another example is the overlay or multi-layer network, where the abstract links in the logical layer are mapped to diﬀerent physical links in physical layer. In this context, two or more abstract links which contain the same physical links may have correlated latencies [10], bandwidth usage [11] or geographically failures [12]. These kinds of correlated networks can also be regarded as stochastic networks, since their link weights are also stochastic/uncertain instead of ﬁxed/known.

1.1 Research Problems in Optical Networks

In Wavelength Division Multiplexing (WDM) technology enabled optical networks, the capacity of a ﬁber is divided into several non-overlapping wavelength channels that can transport data independently. These wavelength channels make up lightpaths, which are used to establish optical connections that may span several ﬁber links. With current

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1.1. Research Problems in Optical Networks 3

commercial technology, each lightpath can be independently operated at a data rate ranging up to 100 Gb/s [13].

In the absence of wavelength conversion [14], it is required that the lightpath occupy the same wavelength on all fiber links it traverses, which is referred to as the wavelength-continuity constraint. Alternatively, if the routing nodes are equipped with wavelength converters, it is possible to convert an input wavelength to a subset of the available wavelengths in the network. Wavelength converter can efficiently utilize the wavelength capacity efficiency, but it has not been widely deployed in the nowadays networks, due to its high price. Routing in optical WDM networks should therefore include both Routing and Wavelength Assignment (RWA).

In addition, the RWA problem should also incorporate some other issues (1) Traf-fic grooming, i.e., several independent traffic streams can be aggregated to share the capacity of a lightpath to efficiently utilize the available bandwidth (2) Survivability, i.e., two link-disjoint paths should be provided to accommodate each request in case of any link failure, and so on. The RWA problem together with these two issues in optical WDM networks have been extensively studied in the past [15]. Nevertheless, with the new necessities (e.g., energy concern, due to its impact on environment) emerging from society and commercial technology (e.g., Orthogonal Frequency-Division Multiplexing (OFDM) enabled optical networks) maturing from industry, we should revisit these related RWA problems. Hence, this thesis will first answer the following questions:

• How can we model the energy consumption in optical networks and how to solve

the RWA problem in optical networks such that the path’s energy consumption is minimized? Can traffic grooming aid to design an energy-aware path in optical networks? How do we design a most efficient network? Is a most energy-efficient network equal to a most cost-energy-efficient network?

• In current WDM-based optical networks, the gap between the available ﬁber link

capacity and the customer’s requested bandwidth cannot be filled up sufficiently, which results in a big bandwidth loss. Even if traffic grooming can efficiently utilize the available bandwidth, it only makes sense to groom the new request onto existing lightpaths that have the same source and/or destination, otherwise extra wavelength channels should be set up to connect the existing lightpaths. Hence, does a more elastic optical networks that allocate just enough capacity to accommodate each request exit? If so, how to solve the routing problem in this kind of networks?

• Can we quantitatively evaluate the survivability mechanism by taking the

reliabil-ity of individual links into account? Can we provide more than two link-disjoint paths to provide a user-requested or even more reliable survivability? What is the complexity of this problem and how do we solve this problem?

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4 Chapter 1. Introduction

1.2 Research Problems in Stochastic and Correlated

Networks

As was mentioned above, many real-life networks are likely to behave in a stochastic manner. In this thesis, we assume/deﬁne that in the so-called stochastic networks the link weight is stochastic. Apart from that, we also notice that the link weights are often correlated in e.g., inter-dependent networks, SRLG networks, etc. In fact, we can regard correlated network as a special kind of stochastic network, since the link weight in correlated network varies depending on some other links. In this context, the link weight in correlated networks is also stochastic instead of deterministic/ﬁxed. Nevertheless, we still distinguish stochastic networks and correlated networks in this thesis for the ease to clarify. Consequently, this thesis will answer the follow questions:

• How to model the stochastic link weights? What are the existing representative

stochastic link weight models? How to model the correlated link weights?

• Under the assumption that the bandwidth and delay follow a general log-concave

distribution, how to ﬁnd the maximum ﬂow in stochastic networks with and with-out delay constraint? What is the complexity of this problem?

• How to solve the shortest path problem and the min-cut problem in correlated

networks? Are they still polynomial-time solvable as in conventional networks?

1.3 Outline of the Thesis

The main focus of this thesis is to study various routing problems in both optical net-works and stochastic (and correlated) netnet-works. For these various problems, we study their complexities, establish proper models to better reflect them, and then propose exact, approximation, or heuristic algorithms to solve them. We also conduct extensive simulations to evaluate the proposed algorithms. This thesis mainly includes 8 chapters: Chapters 2-5 address the routing problems in (deterministic) optical networks, Chapter 6 studies routing problems in stochastic networks, and Chapter 7 focuses on the short-est path problem and the min-cut problem in correlated networks. More specifically, we begin with studying the energy-aware path selection problem and energy-efficient network design problem in Chapters 2 and 3, respectively. We proceed to study the routing problem in spectrum-sliced elastic optical path (SLICE) [16] optical networks in Chapter 43, and the availability-based path selection problem in Chapter 5. Af-ter that, we first summarize different representative stochastic link weight models in

3_{Enabled by OFDM technology, SLICE transports each traﬃc request by allocating just enough}

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1.3. Outline of the Thesis 5

Chapter 1 Introduction

Routing in Optical Networks

Chapter 2 Energy-Aware Routing Chapter 3 Energy-Efficient Network Design Chapter 4 Routing in SLICE Networks Chapter 5 Availability-Based Path Selection

Routing in (Uncorrected) Stochastic Networks

Chapter 6 Constrained Maximum

Flow

Routing in Correlated Networks

Chapter 7 Shortest Path and

Min-Cut

Chapter 8 Conclusions

Figure 1.1: Thesis structure.

stochastic networks in Appendix A. Considering that most of the current literature has been directed on single path ﬁnding problem in stochastic networks, we study the maximum-ﬂow problem (multipath routing problem) in stochastic networks in Chap-ter 6. In ChapChap-ter 7, we consider the shortest path routing problem and the min-cut problem in the so-called correlated networks. An overview of the work in this thesis can be found in Figure 1.1. The following is the outline of this thesis:

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aggrega-6 Chapter 1. Introduction

tion of multiple traffic streams on one channel or wavelength, has often been considered in the context of reducing blocking and improving capacity utilization. More recently, traffic grooming has also been advocated in the context of energy-aware routing. In this Chapter, we study the energy-aware path selection under the scheduled traffic model in IP-over-WDM optical networks. We show that there is indeed a strong relation between traffic grooming and energy efficiency, but also that it sometimes pays off not to groom. We propose an energy-aware routing algorithm that is based on traffic grooming, but which has the flexibility to deviate from it where needed. Our approach can be applied to networks with and without wavelength conversion.

Energy-Aware Network Design (Chapter 3): In this chapter, we study how to design an energy-efficient network. We have developed an Integer Linear Programming (ILP) formulation to optimize a network in terms of energy consumption, in the context of survivable impairment-aware traffic grooming of a given traffic matrix. Considering the time complexity of this problem, a fixed number of link-disjoint paths were pre-calculated.

Impairment-Rware Routing in SLICE Networks (Chapter 4): In this chapter, we study the impairment-aware dynamic routing and subcarrier allocation problem in translucent SLICE networks. We do not address the impairment factors that affect Quality of Transmission (QoT) and only consider that the signal’s transmission reach is related to the modulation used and that, when a signal traverses a longer distance than the selected modulation format’s acceptable transmission reach, it needs to be regenerated. We propose an impairment-aware routing algorithm that tries to balance traffic flows evenly across the network to reduce the blocking probability. We consider two cases, namely (1) a modulation will be selected that is used by the entire connection, and (2) the modulation can be changed during regeneration at regenerator nodes on the path.

Availability-Based Path Selection (Chapter 5): In this chapter, we study the problem of establishing a connection over at most k (partially) link-disjoint paths for which the availability is no less than δ (0 < δ ≤ 1). We consider networks with and without Shared-Risk Link Groups (SRLGs). We prove that this problem in general cannot be approximated in polynomial time, unless P=NP. We subsequently propose a polynomial-time heuristic and an exact Integer Nonlinear Program (INLP) formulation for availability-based path selection.

Constrained Maximum Flow in Stochastic Networks (Chapter 6): In this chapter, we study constrained maximum-ﬂow problems in stochastic networks, where the delay and bandwidth of links are assumed to follow a log-concave probability dis-tribution, which is the case for many distributions that could represent bandwidth and delay. We formulate the maximum-ﬂow problem in such stochastic networks as a convex optimization problem, with a polynomial (in the input) number of variables. When an additional delay constraint is imposed, we show that the problem becomes NP-hard and we propose an approximation algorithm based on convex optimization. Furthermore,

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1.3. Outline of the Thesis 7

we develop a fast heuristic that, with a tuning parameter, is able to balance accu-racy and speed. An overview of existing representative stochastic link weight models is presented in Appendix A in order to better understand stochastic networks.

Shortest Path and Min-Cut in Correlated Networks (Chapter 7): In this chapter, we deal with the shortest path problem and the min-cut problem in correlated networks. We ﬁrst propose two correlated link weight models, namely (1) deterministic correlated model and (2) (log-concave) stochastic correlated model. Subsequently, we prove that the shortest path problem and the min-cut problem are NP-hard under the deterministic correlated model, and even cannot be approximated to arbitrary degree in polynomial time, unless P=NP. In particular, these two problems are shown polynomial-time solvable under the (constrained) deterministic nodal correlated model. Under the (log-concave) stochastic correlated model, we propose convex optimization formulations to solve these two problems, respectively.

Conclusion (Chapter 8): Finally, we provide general conclusions and suggest pos-sible future work.

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Chapter 2 Energy-Aware Path Selection

2.1 Introduction

With the increasing size of data communication networks, the energy consumption of network equipment is increasing exponentially, which will result in very heavy envi-ronmental pollution (e.g., green house eﬀect). Reducing energy consumption in data communication networks is therefore a crucial issue for society. Since optical networks constitute most of the current physical network infrastructure, reducing their caused en-ergy consumption is indispensable. Typically, the most enen-ergy consuming components in it are router ports, line cards, ampliﬁers, transponders, Optical Cross Connect(OXC), etc. As was mentioned earlier, since optical core network is the most dominant part of optical telecom network, we introduce three common energy-aware techniques in it according to [7]:

1. Energy-aware traffic engineering [17]: Under the assumption that the net-work is given/fixed, find routing paths such that the energy consumption is min-imized.

2. Energy-efficient network design [18]: It refers to the case where the resources of a network (e.g., transponders and amplifiers) are unknown and the aim is to reach the most energy efficient design such that the given (static) traffic matrix can be accommodated.

3. Selectively switching off nodes or links: Switching off network elements can significantly save energy [19]. The nodes or links can be switched off when they are (1) unused/idle, or (2) the traffic goes below a given threshold, or (3) it is possible to reroute the traffic to backup paths. It is argued that it is less realistic to switch off nodes, since turning off a core node would cause a large portion of traffic to be rerouted, which may induce extra delay, large congestion controlling messages, etc. Therefore, most of current research focuses on switching off links

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10 Chapter 2. Energy-Aware Path Selection

(and their associated amplifiers) [20]. With this, it is still necessary to make sure the network is robust after some links are switched off. For example, in case of any single-link failure, at least two link-disjoint paths for each node pair should exist in the network after some link are switched off.

In this chapter, we study the energy-aware path selection problem for dynamic scheduled traﬃc in IP-over-WDM networks corresponding to technique (1), and in Chapter 3, we will address the energy-eﬃcient network design problem with respective to technique (2). We do not address technique (3) since it is out of the scope of this thesis.

In wavelength division multiplexing (WDM) technology, the capacity of a fiber is divided into several non-overlapping wavelength channels that can transport data in-dependently. These wavelength channels make up lightpaths that are used to establish optical connections that may span several fiber links. With current commercial tech-nology, each lightpath can be independently operated at data rates of several Gb/s. However, traffic between a pair of nodes may not be able to fill up the available band-width of a lightpath. In order to efficiently utilize the available bandband-width, several independent traffic streams can be aggregated to share the capacity of a lightpath. This is known as traffic grooming (e.g., see [21]). While traffic grooming has obvi-ous potential to increase throughput, the grooming of traffic may also lead to energy efficiency, although this is not always the case.

The outline of this chapter is as follows. Section 2.2 presents related work. Section 2.3 describes our network model and analyzes the energy consumption with traﬃc grooming and when setting up a new lightpath. An energy-aware routing algorithm for dynamic traﬃc is proposed in Section 2.4. Section 2.5 provides our simulation results and we conclude in Section 2.6.

2.2 Related Work

Cavdar [22] addresses dynamic energy-aware traffic provisioning at the WDM layer by allocating weights to links and selecting a path with minimum weight. However, the author assumes that the capacity of each traffic demand is equal to the maximum capacity of each wavelength, which means that traffic grooming cannot be applied.

Xia et al. [23] discuss the energy and traffic flow details of every operation in an IP-over-WDM network. They subsequently propose an energy-aware routing algorithm that uses an auxiliary graph to represent the consumption of each operation, both at the IP and WDM layers. The proposed routing method can deal with both static and dynamic traffic, although for dynamic traffic an infinite holding time is assumed. Chen and Jaekel [24] do take holding time into account and use an ILP to show that the holding time affects the energy consumption in traffic grooming. However, they do not propose a scalable energy-efficient traffic grooming algorithm for scheduled traffic.

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2.3. Network Model and Energy Analysis 11

Zhang et al. [7] incorporate holding time in energy-aware traffic grooming and solve both the static and dynamic case. For static traffic, they propose an ILP. For dynamic traffic, they find the shortest paths in an auxiliary graph with specific weights. Their algorithm is compared to two routing algorithms from [25], which are “minimum light-paths” that tries to minimize the number of newly established lightpaths and “minimum hops” that tries to minimize the number of lightpath hops. Simulation results show that the algorithm of Zhang et al. performs best under low traffic, but performs worst under high traffic.

The algorithm proposed in this chapter uses a more reﬁned energy model than in [7]. For instance, we distinct between energy consumed in router ports at the IP layer and in the components at the optical layer that consume a ﬁxed energy for each full wavelength connection.

2.3 Network Model and Energy Analysis

2.3.1 Network Architecture

We adopt the transparent IP-over-WDM with optical bypass and grooming model of Musumeci et al. [26]. It is a two-layered network architecture, with routers in the IP layer and optical cross connects (OXCs) in the WDM layer. The following components consume energy, where the values are taken from [26]:

1. Transponders (Etr = 34.5 W per 10 Gb/s wavelength): Transponders have two

functionalities: (i) O/E and E/O conversion between OXC and IP and (ii) trans-mitting and receiving signals.

2. OXCs (Eos = 1.5 W per 10 Gb/s wavelength): Optical cross connects optically

switch traffic, either when traffic is bypassed at a node or when adding/dropping traffic.

3. Router ports (Ees = 14.5 W per Gb/s): The ports mainly deal with electronic

processing.

We are only interested in the energy consumption involved in accommodating a new request. We therefore do not consider energy that is consumed continuously, irrespective of whether there is traffic or not. For instance, optical amplifiers consume a constant value of energy regardless of the presence of traffic. When our objective would be to switch off components, then those static energy costs would have to be taken into account.

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2.3.2 Energy Analysis of Traﬃc Flow in Diﬀerent Cases

We will discuss the energy model that we adopt in this chapter and analyze the energy consumption in the cases of both traﬃc grooming and setting up a new lightpath.

Often, e.g. in [27], [23], [28], it is assumed that energy consumption E has the following linear relationship with traﬃc:

E = E₀+ Et· τ (2.1)

where E₀ is the overhead which represents the idle energy consumption, Et symbolizes

the traffic-dependent energy factor, and τ represents the amount of traffic in Gb. We consider different energy calculation methods for different operations. Some operations are only wavelength (but not traffic) dependent, in which case Et = 0. For instance,

once transponders and OXCs are switched on they will consume a ﬁxed amount of energy corresponding to a full wavelength capacity (and not to the fraction of traﬃc transported).

E =

34.5 W/λ ∀ transponders

1.5 W/λ ∀ OXCs (2.2)

For electronic switching (processing) at the IP layer, the energy consumption is traﬃc dependent.

E = 14.5· τ ∀ router ports (2.3)

Although in reality E₀ = 0, we have discarded its contribution (i.e., set E₀ = 0), since - like optical amplifiers - the router ports (currently) cannot be automatically switched on/off, and hence their energy consumption is always fixed.

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2.3. Network Model and Energy Analysis 13

Let us consider the example of setting up a new lightpath in the network of Figure 2.1, where already two lightpaths (6−2−3) and (3−5) exist. Lightpath 1 still remains for 5 hours and lightpath 2 remains for 6 hours. The new request r(6, 5, τ, 3 : 00, 4) asks for a lightpath from node 6 to node 5, needing τ Gb/s, starting at time 3 : 00 and lasting for 4 hours. We discuss the energy consumption of two routing strategies to route this request: (i) making use of existing lightpaths, i.e. grooming the new request into the existing lightpaths, and (ii) setting up a new lightpath, for instance along the shortest-hop path from source to destination.

OS ES E/O TX AMP RX TX RX OS

O/E E/O O/E

AMP AMP

OS OS

Node 6 Node 2 Node 3 Node 5

Router OXC

ES ES

Figure 2.2: Traﬃc ﬂow in the grooming case.

For traffic grooming, according to [23], the procedure is shown in Figure 2.2. The traffic is electronically switched (ES) at node 6 and the signal is converted to an optical signal (EO). The signal is then optically switched (OS) and transmitted (TX). After being amplified (AMP) in the fiber, it is optically switched (OS) by node 2 and amplified to be received by node 3 (RX). As argued before, amplifiers’ energy costs are fixed and hence not taken into account. After OS, the signal is converted to an electronic signal (OE), and multiplexed to the next connection by electronic switching (ES). There the signal is converted again to an optical signal (EO), and transmitted (TX) to node 5. Node 5 follows the same receiving procedure. Summing all these energy consumptions of grooming this traffic to lightpath 1 and lightpath 2 will lead to:

Egroom = (Ees+ Eeo+ Eos+ Etx) + Eos+

(Erx+ Eos+ Eoe+ Ees+ Eeo+ Eos+ Etx)

+(Erx+ Eos+ Eoe+ Ees) (2.4)

Transponders are responsible for either O/E/O conversions or transmitting/receiving signals and their energy consumption is denoted by Etr = Eeo+ Etx= Erx+ Eoe.

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(overhead) of optical switching and transponders is already paid for by the existing lightpath (at least for the time it still remains, after which the cost corresponds to that of setting up a new lightpath for the remaining time), which leads to

Egroom = 3Ees (2.5) OS ES E/O TX AMP RX O/E ES OS Node 6 Node 5 Router OXC

Figure 2.3: Traﬃc ﬂow in the new lightpath case.

The case for directly setting up a new lightpath is shown in Figure 2.3 and can be expressed as:

Enew = 2(Ees+ Etr+ Eos) (2.6)

For traﬃc grooming, the IP layer electronic processing will be equal to

Ees = Et· τ · h (2.7)

where h represents the request’s holding time. Eq. (2.7) indicates that electronic processing is always traﬃc dependent and has no energy savings compared to setting up a new lightpath. After the end of the previously allocated lightpaths, the energy costs of traﬃc grooming equal those of setting up a new lightpath (for the remaining time h), where Ees= Et· τ · h and Eos and Etr behave as E0· h.

Let us look at some cases in which traﬃc grooming is not the best solution. In our example Egroom− Enew = Ees− 2(Etr+ Eos) = Ees− 72. Unlike Etr and Eos, Ees

depends on the requested bandwidth τ . Hence, if Ees= 14.5τ > 72, i.e. τ > _14.572 Gb/s,

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2.4. Energy-Aware Auxiliary Graph 15

example would be where a request would need to be groomed onto many lightpaths, in which case the amount of electronic switching could become too costly. Finally, in some cases it may not be possible to groom traﬃc all the way from source to destination, which in some cases may render a new lightpath a more eﬃcient solution.

2.4 Energy-Aware Auxiliary Graph

In this section we propose an energy-aware algorithm for dynamic scheduled traffic. First, we introduce an auxiliary graph to represent the original topology and then each link in the auxiliary graph will be assigned energy weights, as specified in Section 2.3. Finally, we apply Dijkstra’s shortest path algorithm on each layer of the auxiliary graph to find the minimum energy consumption path to route the request. We present three types of auxiliary graphs, namely for when the network can offer (1) no wavelength conversion, (2) selective wavelength conversion, and (3) full wavelength conversion.

2.4.1 Auxiliary Graph without Wavelength Conversion

We use Wa to denote the number of wavelengths that each ﬁber link contains. We ﬁrst

assume that there is no wavelength conversion, such that the network can be regarded as Wa separate sub-layered graphs, one for each wavelength. Later, we also include

wavelength conversion. Each network node is split into a physical IP node and an optical WDM node. These two types of nodes are connected via three types of links:

• Physical node: the IP source and destination nodes of a traﬃc request.

• Optical node: source and destination for a new established lightpath in the optical

layer.

• Conversion link (conv_link): connects the physical and optical nodes, i.e. O/E

(conv_link_rx) or E/O (conv_link_tx) conversion.

• Lightpath link (light_link): connects two physical nodes if they are the start and

end of an existing lightpath.

• Optical link (opt_link): connects two optical nodes and could be used to establish

a new lightpath.

For example, in the 3-node network of Figure 2.4(a) with two existing lightpaths (using the same wavelength wx), the corresponding auxiliary graph on the wx layer will

be represented as in Figure 2.4(b).

In this way Wa auxiliary graphs will be created to represent each wavelength of the

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(a) Original graph (b) Corresponding auxiliary graph

Figure 2.4: Auxiliary graph for energy-aware path selection with no conversion.

2.4.2 Weights Allocation

We will discuss how to allocate weights to the 3 diﬀerent types of links. Optical and Conversion Links

This case corresponds to the weight allocation when setting up a new lightpath. Note that the link between a physical node and an optical node represents the energy con-sumption at the IP layer. Hence, the energy concon-sumption of the conversion link can be calculated as

Econv_{_link_rx} = Etr + Eos (2.8)

Econv_link_tx = Ees+ Etr (2.9)

where Etr and Ees represent the energy consumption in the case of setting up a new

lightpath. The conversion link which originates at an optical node will be allocated according to Eq. (2.8), while the conversion link originating at a physical node will be allocated according to Eq. (2.9). The conversion link originating at the optical node does not contain Ees when its corresponding physical node is not the destination node,

because this is included in Econv_{_link_tx}when the traﬃc leaves the physical node. Since

traﬃc does not leave the destination node, it should be added there (as indicated in Section 2.3). However, from an algorithmic perspective, the Ees contributions at the

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2.4. Energy-Aware Auxiliary Graph 17

destination nodes are the same for traﬃc grooming and setting up a new lightpath, and they therefore do not aﬀect the solution. To make the equations simpler, they have therefore been disregarded in the auxiliary graph. The same applies to the lightpath link weight. The weights of the optical link will represent the energy of optical switching, namely Eos.

Lightpath Link

The energy weight per time unit for a lightpath link will be set according to two cases, depending on whether the request’s ending time is earlier than that (end) of an existing lightpath. We use Egnew to represent setting up a new lightpath for the remaining time.

Elight_link =

Ees arrival + h≤ end

Ees+ Egnew arrival + h > end (2.10)

where

Egnew = 2Etr+ Ees+ nEos during time (arrival + h, end) (2.11)

and arrival denotes the arrival (starting) time of the request, and n denotes the number of nodes (including source and destination) that a lightpath traverses.

2.4.3 Auxiliary Graph with Wavelength Conversion

Although wavelength conversion consumes additional energy, it may reduce the blocking probability. Wavelength conversion can be easily incorporated into our auxiliary graph by connecting the Waindependent sub-layers. To do so, we make use of an extra virtual

layer on which the path search will start and connect all layers to that virtual layer. Besides using the virtual layer as a starting point, it has the additional advantage that the existing lightpaths with weight Elight_link that were previously (in Section 2.4.1)

present at each sub-layer need now only be present at the virtual layer, which reduces the total number of lightpath links. The physical nodes are connecting to their physical counterpart in the virtual layer by a virtual link (virt_link) of cost Evirt_link = 0. If,

at some optical node, one can convert from that wavelength to another, then we add a wavelength link (wave_link) between those two optical nodes at the two corresponding sub-layers. The cost of that link equals the energy cost of wavelength conversion, which in our case is set to Ewave_{_link} = Etr since wavelength conversion relies on an O/E/O

operation. For example, in Figure 2.4(a) there are already two existing lightpaths present in the network. To obtain the auxiliary graph, we ﬁrst follow the procedure described in Section 2.4.1, thereby excluding the lightpath links. Subsequently we add the virtual layer, place the existing lightpath links, connect the physical nodes with their corresponding physical node in the virtual layer, and also add the wavelength

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links to obtain Figure 2.5. In the example, nodes A and B are assumed to be able to convert to the other wavelength, and hence a link is drawn between the respective optical nodes. PA B C A PC PB PA PB PC PA B C A PC PB Virtual Topology Wavelength Layer 2 Wavelength Layer 1 Wavelength Link Virtual Link Lightpath Link

Figure 2.5: Auxiliary graph for energy-aware path selection with wavelength conversion.

The above-described way of connecting the wavelength conversion links is most versatile, since it can also capture the cases where wavelength conversion is only possible from a wavelength to a restricted range of other wavelengths (e.g., see [29]). If this restriction is not there and a node with wavelength conversion capabilities can convert to any other wavelength, we can use the virtual layer to reduce the number of wavelength conversion links. This time we would need to also represent the optical nodes in the virtual layer. If a node (say B) has wavelength conversion ability, then for all sub-layers a link connecting node B to its virtual companion is added. Instead of Wa(Wa−1)

2 links

per node with wavelength conversion capabilities, Wa links now suﬃce (each with half

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2.5. Simulations 19

2.4.4 Algorithm and Complexity

For each new request we should update the auxiliary graph to reﬂect the proper weights and available capacity. If there is no wavelength conversion, after allocating weights to the links on each sub-layer of the auxiliary graph, running Dijkstra’s shortest path algorithm on each sub-layer will allow us to choose the most energy-eﬃcient route. In this case, the complexity of the algorithm per request is dominated by running Dijkstra’s algorithm Wa times, which leads to an overall complexity of O(WaN log(N ) + WaN2)

where N denotes the number of nodes in the original topology, and Wa represents the

number of wavelengths in a ﬁber. N2 instead of L (the number of links in the original topology) is used, because in the worst case N (N − 1) lightpaths (reﬂected in links) could be present.

In the general case with wavelength conversion, Dijkstra’s algorithm is ran only once - starting from the source node at the virtual layer - but now on a larger graph. This leads to a complexity of O(WaN log(WaN ) + WaL + N2+ Wa2) for each request, where

the term W_a2 could be dropped if wavelength conversion goes through the virtual layer.

2.5 Simulations

Figure 2.6: NSFNet including link distances in km.

We compare our energy-aware routing algorithm, in terms of energy consumption and blocking, to the method of directly setting up new lightpaths (Direct New Light-path) based on the shortest path and to a traﬃc grooming algorithm. The traﬃc groom-ing algorithm tries to select paths on which the request can be completely groomed. If multiple such paths exist, the one using the least amount of lightpaths is chosen. If no such path exists, then a new (shortest hopcount) lightpath will be set up. If this

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20 Chapter 2. Energy-Aware Path Selection 800 1000 950 1900 2600 ₁₂₀₀ 700 1300 1300 900 ₆₀₀ 1000 1000 1000 1000 1000 1000 1200 1400 1100 250 850 800 1200 1150 900 950 1000 850 650 900 1100 1200 800 1100 1000 800 850 300 600 1000 900 900

Figure 2.7: USANet including link distances in km.

option also fails, we select a path that partly grooms and partly uses a new lightpath. Else, the request is blocked. We simulate these three algorithms under two cases: one where wavelength conversion is permitted and each node has full wavelength conver-sion abilities (represented by WC), and the other one where wavelength converconver-sion is not permitted (represented by NWC). We simulate on two realistic carrier backbone networks, namely the NSFNet of 14 nodes and 20 links and the USANet of 24 nodes and 43 links (see Figures 2.6 and 2.7).

We vary the amount of traﬃc requests from 1000 to 10000, where s and d are ran-domly generated and the holding time h also ranran-domly varies. The requested capacity

τ will be generated according to the distribution OC − 1, OC − 3, OC − 12, OC − 48

and OC − 192 as 20 : 10 : 10 : 4 : 1. The number of wavelengths per link is chosen as 40 and 200.

Figure 2.8 gives the energy consumption of the three algorithms in the NSFNet and the USANet. We set the number of wavelengths to 200, so that no blocking happens. From this figure we can see that the energy consumption grows almost linearly with the amount of traffic, but the slope is smallest for energy-aware routing. To not clutter the figure, we omitted the results for wavelength conversion, since, with ample wavelengths, wavelength conversion was not needed and thus never used (giving the same results as for without wavelength conversion).

Figures 2.9(a) and 2.9(b) show the blocking probability in the two networks when the number of wavelengths is set to 40. Due to the use of traffic grooming, energy-aware routing obtains a lower blocking probability than shortest path routing. However, there is even a slight improvement over the traffic grooming algorithm. This is because in some cases the traffic grooming algorithm tries to groom over many lightpaths, while the energy-aware algorithm would opt for a short new lightpath. Clearly, the use

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2.6. Conclusion 21 10000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.5 1 1.5 2 2.5x 10 4 Traffic Demand Energy Consumption (kWh) Energy−Aware (NSFNet) Direct New Lightpath (NSFNet) Energy−Aware (USANet) Traffic Grooming (NSFNet) Direct New Lightpath (USANet) Traffic Grooming (USANet)

Figure 2.8: Energy consumption in two networks.

of wavelength conversion reduces the blocking probability even further. Due to the different blocking probabilities, comparing based on total energy consumption would give a wrong reflection. Hence, in Figures 2.10(a) and 2.10(b), we present the average energy consumption per accepted request. Reflected in such a metric, it is possible to see the extra energy cost in wavelength conversion. We believe that in practice, these extra energy costs pale in comparison to the extra revenues gained by being able to allocate more requests.

2.6 Conclusion

In this chapter, we have proposed an energy-aware routing algorithm for dynamic sched-uled traﬃc. By applying an energy model to compute the energy consumption at the IP and optical layer, the algorithm can attain practical energy consumption weights. Our model also allows to take the energy costs of full or sparse wavelength conversion into account. Simulation results show that the proposed algorithm can achieve a lower energy consumption and blocking probability compared to directly setting up new light-paths or traﬃc grooming in an energy-oblivious way. Because wavelength conversion costs energy, it is only used when it is really need to prevent blocking a request and otherwise it prefers wavelength continuous routes.

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22 Chapter 2. Energy-Aware Path Selection 10000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Traffic Demand ocking ro a i ity Energy−Aware (NWC) Direct New Lightpath (NWC) Energy−Aware (WC) Direct New Lightpath (WC) Traffic Grooming (NWC) Traffic Grooming (WC) (a) NSFNet 10000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 Traffic Demand ocking ro a i ity Energy−Aware (NWC) Direct New Lightpath (NWC) Energy−Aware (WC) Direct New Lightpath (WC) Traffic Grooming (NWC) Traffic Grooming (WC)

(b) USANet

Figure 2.9: Blocking probability in two networks.

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Traffic Demand Energy Consumption (kWh) Energy−Aware (NWC) Energy−Aware (WC) Direct New Lightpath(NWC) Direct New Lightpath (WC) Traffic Grooming (NWC) Traffic Grooming (WC) (a) NSFNet 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 Traffic Demand Energy Consumption (kWh) Energy−Aware (WC) Traffic Grooming (NWC) Direct New Lightpath (NWC) Direct New Lightpath (WC) Energy−Aware (NWC) Traffic Grooming (WC)

(b) USANet

Figure 2.10: Average energy consumption per accepted request in two networks when links have 40 wavelengths.

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Chapter 3 Energy-Eﬃcient Network Design

3.1 Introduction

In Chapter 2, we studied the energy-aware path selection problem. In this chapter, we study the energy-efficient (optical) network design problem. Besides energy efficiency, another important concern is that networks should be robust. However, robustness and energy efficiency seem conflicting goals, since reaching robustness usually requires installing redundant components/equipment, which consume energy. We will study how to design an energy-efficient network for a (predicated) traffic matrix, while taking into account robustness constraints.

In our work [30] for SURFnet we have developed, for several given traﬃc scenarios, an Integer Linear Program (ILP) formulation to compute a network conﬁguration that uses the fewest possible transceivers. The constraints of the problem were:

• Each request is accommodated by two link-disjoint paths (survivable

provision-ing): a primary path and a backup path.

• Considering that the quality of an optical signal degrades with distance and some

other important factors (e.g., cross talk, ampliﬁer spontaneous emission, etc.), regenerators should be deployed to restore the signal when necessary.

• Traﬃc grooming is applied when the bandwidth of a request is less than the

capacity of a lightpath.

The model proposed in [30] is a two-step algorithm:

1. First, an ILP is set up to minimize the number of transceivers for a set of pre-calculated link-disjoint paths, without considering the impairment constraint.

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24 Chapter 3. Energy-Eﬃcient Network Design

2. Subsequently, the requests that violate the impairment constraints are selected one by one, and for each selected request, the approach ﬁrst tries to groom it over the existing lightpaths to establish two link-disjoint paths. If grooming fails, then regenerators are placed on the original paths for this request.

The regenerators are assumed to be implemented by back-to-back transceivers and it is also assumed that the add/drop multiplexers can be implemented by the transceivers. This simpliﬁcation allows to reﬂect the total cost of transponders, add/drop multiplex-ers, and regenerators by only the number of transceivers.

In this chapter, we study the energy-efficient network design problem that follows the same assumptions and constraints as presented in [30], and also based on an additional energy consumption model. According to [31], the traffic grooming problem is proved NP-hard, even in ring topologies. Hence, the considered network design problem in this chapter, which contains the traffic grooming problem, is also NP-hard. We subsequently develop an ILP-based heuristic to solve the energy-efficient network design problem, which is also applied on SURFnet network. The outline of this chapter is organized as follows: In Section 3.2, we overview related work. Section 3.3 introduces the adopted network architecture and describes the energy consumption model in it. Section 3.4 proposes the ILP model and Section 3.5 provides our simulation results. Finally, we conclude in Section 3.6.

3.2 Related Work

Energy-efficient or cost-efficient network design refers to the case where the resources of a network (e.g., transponders and amplifiers) are unknown and the aim is to reach the most energy or cost efficient design.

Shen and Tucker [32] first express energy consumption based on different components in IP-over-WDM networks, like Erbium Doped Fiber Amplifiers (EDFA), transponders, router ports, etc. Following that, under the constraints of flow conservation, wavelength capacity and so on, it can obtain the optimum (smallest) number of components ac-cording to a specific topology. The authors also propose two heuristics. While this paper suggests that energy efficient design is often also cost effective, Palkopoulou et

al. [33] claim that the most cost eﬃcient architecture is not always the least energy

consuming one (an opinion also reﬂected in [34]).

Shen et al. [34] take a more practical approach, thereby considering the Cisco GSR 12008 router (where the line card can be switched off when its interface is idle). Secondly, it is more specific in the ILP which not only considers about the design problem but also deals with the traffic dependent part. Unfortunately, the extra ILP features make it more time consuming (and simulations were therefore only done on a 6 node network).

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3.3. Network and Energy Model 25

Musumeci et al. [26] discuss about four IP-over-WDM architectures, which are IP with no Bypass (IP-NB), IP with Bypass and Grooming (Transparent-BG and Opaque IP-BG) and Time Driven Switching (TDS). In IN-NB, all traffic flows are electronically processed in each node that they traverse therefore traffic grooming cannot save energy. In transparent-BG, the IP flows can optically bypass the intermediate nodes and so there is no O-E-O conversion happens, besides the signal only needs to be electronically processed at end nodes when traffic grooming happens. Although in opaque IP-BG the signal will not need to be electronically processed at intermediate nodes the O-E-O conversion happens at each intermediate nodes that the signal traverses in order to regenerate the signal. Since Semiconductor Optical Amplifier (SOA)-based optical switches which support fast switching is deployed in TDS, switching and grooming operations are performed in the optical domain and transponders are used only in source/destination nodes. By establishing ILP for the four architectures, the authors show that TDS consumes the least energy and IP-NB consumes the largest energy under the same traffic amount.

Chowdhury et al. [35] assume transponders have mixed-line-rate (MLR) in trans-parent optical network and use ILP to model the energy consumption of the network. The simulation results show that MLR consume less energy compared to single-line-rate (SLR) network when SLR set the line rate 10 Gbps, 40 Gbps and 100 Gbps.

Dong et al. [36] consider about connecting data centers and IP-over-WDM optical networks and establish ILP model to solve the problem of the location of data centers in IP-over-WDM networks in order to minimize the overall energy consumption. Besides, the authors propose to take renewable energy into account. The problem of whether to locate data centers next to renewable energy or to transmit renewable energy to data centers has been studied by simulation. The results show that optimizing the data center locations with renewable energy under the multi-hop bypass heuristic can save up to 71% of the power consumption.

Iqbal et al. [37] consider an Ethernet-over-SONET/SDH-over-WDM network, which is usually used to represent a metro optical network. To configure an optical network for serving a given traffic matrix, they propose an ILP to minimize the energy consumption or the cost expenditure with and without considering traffic survivability. Similarly to [33], their simulations reveal that the most energy-efficient network is not always equal to the most cost-effective one.

3.3 Network and Energy Model

Since a precise energy model for SURFnet is not known, we have used the transparent IP-over-WDM with optical bypass and grooming model of Musumeci et al. [26]. The main objective is to demonstrate the feasibility of our approach. If a diﬀerent model emerges in future studies, then we can adapt our approach to that particular model.

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We assume a two-layered network architecture, with routers at the IP layer and optical cross connects (OXC) at the WDM layer. The energy-consumption values for the various relevant components are taken from [26] and [38] and are listed below:

1. Transponders (34.5 W per 10 Gb/s wavelength): Transponders have two function-alities: (1) O/E and E/O conversion between OXC and IP and (2) transmitting and receiving signals.

2. OXC (1.5 W per 10 Gb/s wavelength): Optical cross connects optically switch traﬃc.

3. Router ports (14.5 W per 10 Gb/s wavelength): The ports mainly deal with electronic switching.

4. Regenerators (50 W per wavelength): Regenerators are assumed to be imple-mented by back-to-back transceivers, and their function is to restore the signal to the maximum strength in terms of the Figure-of-Merit (FoM) value [30].

5. (R)OADM (7.5 W per 10 Gb/s wavelength)1: (Reconfigurable) Optical Add/Drop Multiplexers. When the nodes perform traffic grooming/degrooming, they add/drop traffic requests. To be consistent with [30], (R)OADMs are assumed to be imple-mented by back-to-back transceivers. In this sense, a regenerator is similar to a (R)OADM, but since a (R)OADM does not have O-E-O conversion, it consumes less energy than a regenerator.

Optical amplifiers (0.07 W per 80 km) are deployed on each fiber in a link to amplify the signal, e.g. every 80 km. However, since they consume a constant value of energy regardless of the presence of traffic and since they are always turned on, they are not taken into consideration in our model.

3.4 ILP Model

In this section, we will set up an ILP model with the goal of minimizing the energy consumption of the whole network provided that it can accommodate each request with primary and backup paths. The links along the primary and backup paths are not allowed to overlap. According to [39], protection can be at lightpath (PAL) level or at connection (PCL) level. PAL denotes that both the primary and backup paths of a traffic request should use the same (logical) lightpath, while PCL denotes that the primary and backup paths can use different lightpaths to accommodate the same traffic request. Both PAL and PCL can provide end-to-end protection since the physical links

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3.4. ILP Model 27

along the primary and backup paths are link disjoint. Our model adopts PAL (as it leads to a faster time complexity).

The basic idea of our model is to establish flow conservation constraints for the IP layer (Eqs. (3.3)-(3.7)) to accommodate all the traffic requests. Since we allow for grooming at the optical layer, we also add flow conservation constraints for the optical layer (Eq. (3.8)). To reduce the time complexity, we pre-calculate a fixed number

K of link-disjoint path pairs between each node pair. Also, the required amount of

regenerators and the energy consumption of optical switching can be calculated in advance for each path pair, which largely reduces the time complexity. We will proceed to explain the notation used in our model.

3.4.1 ILP notation

R: the set of |R| traﬃc requests, where the i-th request is represented as ri(si, ti, τi),

with source si, destination ti and requested bandwidth τi.

G(N , L): physical topology, where N is the set of N nodes and L is the set of L

links.

C: the capacity of a wavelength.

Wa: the set of Wa wavelengths in each ﬁber.

K: the number of pre-calculated pairs of disjoint paths between each node pair. Etr: energy consumption of transponders per wavelength.

Eos: energy consumption of optical switching per wavelength.

Ees: energy consumption of IP-layer electronic switching per Gbit/s. Ereg: energy consumption of regeneration per wavelength.

Eadd/drop: energy consumption of adding/dropping traﬃc per wavelength.

oxck

i,j: amount of optical switching between nodes i and j when using the k-th pair

of pre-calculated disjoint paths.

regk

i,j: the required number of regenerators between nodes i and j when using the k-th pair of pre-calculated disjoint paths.

Lu,v_i,j,k: its value is 1 if the k-th pair of pre-calculated paths between nodes i and j traverses link (u, v); 0 otherwise.

3.4.2 ILP variables

The following variables are used:

P_i,jr,ω : boolean variable equal to 1 if request r is accommodated by lightpath (i, j) on wavelength ω; 0 otherwise.

Vi,j : number of lightpaths between nodes i and j in the virtual (i.e., lightpath)

topology.

Vω

i,j : number of lightpaths between nodes i and j in the virtual topology on

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ηi,j,ω

u,v : boolean variable equal to 1 if a lightpath between nodes i and j traverses link

(u, v) on wavelength ω; 0 otherwise.

Rr,ω_i,j,k: boolean variable equal to 1 if the k-th pair of pre-calculated disjoint paths between nodes i and j is selected for request r on wavelength ω; 0 otherwise.

H_i,jk,ω: boolean variable equal to 1 if the k-th pair of pre-calculated disjoint paths between nodes i and j on wavelength ω is selected; 0 otherwise.

Aω

i: boolean variable equal to 1 if an add/drop multiplexer is used at node i on

wavelength ω; 0 otherwise.

3.4.3 ILP formulation

Objectives:

Minimize energy consumption:

4 i,j_∈N:i=j Vij · Etr +2 r_i∈R (i,j)∈N ×N :i=j&i=su W_a ω=1 τi· Pi,jri,ω · Ees + Wa ω=1 K k₌₁ (i,j)∈N ×N :i=j

H_i,jk,ω · oxck_i,j · Eos

+ W_a ω₌₁ K k=1 (i,j)∈N ×N :i=j

H_i,jk,ω · reg_i,jk · Ereg

+ i∈N Wa ω=1 Aω_i · Eadd/drop (3.1)

Minimize number of transceivers:

2 W_a ω=1 ⎛ ⎝ (i,j)∈N ×N :i=j Vij + i∈N Aω_i + i,j_∈N:i=j k_∈K

H_i,jk,ω · regk_i,j

⎞

⎠ _(3.2)

Constraints:

Flow conversation at the IP layer

m∈N P_m,hri,ω = n∈N P_h,nri,ω, ∀ri ∈ R, ω ∈ Wa, h∈ N | h = si & h= ti (3.3) m∈N Pri,ω m,s_i = 0, ∀ri ∈ R, ω ∈ Wa (3.4)

Routing in Optical and Stochastic Networks