Preserv of Optic Net Design

7/28/2019 Preserv of Optic Net Design

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A comparison of OTN and MPLS networks under

traffic uncertaintyPietro Belotti, Kireeti Kompella, and Lloyd Noronha

Abstract—We study a special case of the problem of installingcapacity on backbone networks in the presence of an uncertaintraffic matrix. The uncertainty model is, in general, defined bya set of linear constraints on the traffic demands. Owing to theavailability of statistics on the nominal (average) traffic matrix,in our experiments each traffic demand is characterized by an

average and a peak volume traffic. To allow for traffic variability,we assume that, throughout network operation, any subset of atmost K demands, whose elements can change over time, areallowed to take on a peak value.

The problem consists in allocating sufficient capacity on thenetwork so as to route all traffic matrices included in the

uncertainty set. Network links can be equipped with differenttechnologies: optical transport network (OTN) switches and multiprotocol label switching (MPLS) routers. These can be combinedin several ways to create: an OTN-only network, i.e., with OTNswitches only; an MPLS-only network; a combined network withan OTN and an MPLS layer; and a bypass network, where trafficenters and leaves the core of the network using MPLS routers, butall the interim transit nodes are OTN switches. While the cost of all these types of network is influenced by uncertainty, a networkusing MPLS routers can take advantage of statistical multiplexingand reserve capacity by considering both the average and peakvalues, thus limiting the total capacity to be installed.

When only one type of router is considered, the design problemcan be trivially solved without resorting to optimization methods.A more general problem, where more than one technology canbe used to route the matrix, is modeled by Robust Optimization,a paradigm yielding a solution that guarantees to work inany scenario within the uncertainty set. Our computationalexperiments prove that, for a realistic cost differential betweenMPLS and OTN ports, the former yields the minimum-costnetwork by taking advantage of statistical multiplexing whilesupporting all possible demand combinations.

Index Terms—Statistical multiplexing, robust optimization,traffic uncertainty, MPLS, OTN.

I. MOTIVATION

Provisioning backbone telecommunication networks is a key

task in the design of the communication infrastructure. Net-

work operators have to cope with an ever-changing technologymarket, which makes it difficult to choose the most appropriate

and cost-effective technology. One of the most important

factors is uncertainty in the traffic demand, which introduces

the tradeoff between an inexpensive circuit-switched network

(at the risk of having to drop demands due to insufficient

capacity) and a more conservative and robust network that

manages well any statistical variation, albeit more costly.

P. Belotti is with the Deptartment of Mathematical Sciences, ClemsonUniversity, Clemson, SC 29630. Email: [email protected]

Kireeti Kompella and Lloyd Noronha are with Juniper Networks, Sunny-vale, CA 94089. Email: {kireeti,noronha}@juniper.net

In this article, we compare two routing technologies, MPLS

and OTN, in terms of the total network cost, using a realistic

backbone network. Due to the high variability of the traffic

matrix (TM) – or, equivalently, the fact that it is not known a

priori – we target all of our experiments at a set of TMs.

The variability and unpredictability of TMs, which may

occur even on an hourly basis, is best tackled by routing

technologies that can take advantage of statistical multiplexing,

such as MPLS, which however have a higher per-port cost.

We show that several scenarios of traffic variability favor

MPLS-based networks, which are more efficient at accom-modating volatile demands. Our claims are supported by an

optimization model that determines the minimum-cost network

as a combination of MPLS routers and OTN switches that

satisfies a set of TMs. We outline below the context of our

work and the main challenges we have addressed.

A. Optimization in network design

Network design problems can be broadly defined as follows:

the network is represented by a graph G = (V, E ), where V is

the set of nodes and E is the set of links. A set Q of demands,

each represented by a triplet (sq, tq, vq), for q ∈ Q, is given.

Each triplet defines a source node sq, a destination node tq,and a traffic volume vq , expressed in Mbps. The problem is to

determine the capacity to be installed on each link or node of

G so as to route all traffic demands while minimizing the total

installation cost, which is a function of the capacity installed

both on links and on nodes.

Various problems in the design of telecommunication net-

works can be formulated as optimization problems [1, 2, 3, 4].

Optimization models describe concisely and completely all

possible configurations of a network, and, when solved, pro-

vide the one with the best performance—intended here as a

generic term that can mean throughput, total cost, number of

demands routed, congestion, etc.

Optimization models consist of a set of variables, a set of constraints, and an objective function. Variables can be contin-

uous when modeling flow or portions of demand, integer (e.g.

when modeling number of ports, fibers, etc.), or binary, suited

for yes/no decisions. The value of these variables at the end

of the optimization process defines the optimal configuration

of the network, if any exists. The constraints identify a set of

configurations of the system, and have to be specified in such

a way that all and only possible configurations of the network

are considered. The objective function defines the criterion to

be used for choosing the best among the (often infinitely many)

configurations. State-of-the-art solvers for several classes of



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optimization problems are available; depending on the size

(number of variables/constraints) and type of problem (e.g.,

integer variables or nonlinear constraints), the solution times

can vary significantly, hence special care is required to limit

the model’s complexity in the modeling phase.

In the remainder of the paper, we consider a network

optimization problem under the following assumptions:

1) All routing paths must be set up according to the OSPF(Open Shortest Path First) protocol: given a function

w : E → Z+ defining the weight of each link, the

routing path of a demand q from source sq to destination

tq must be one of the shortest paths between sq and

tq according to the metric defined by the weights wij ,

which are known in advance. Further constraints on the

routing proportions, such as those required by Equal

Cost Multi-Path (ECMP), can also be considered.

2) The traffic matrix is unknown a priori and changes over

time, but varies within a bounded, non-empty set that is

known in advance.

3) The capacity to be installed corresponds to the number

of ports of different capacity (1.25, 2.5, and 10 Gbps,for instance) at each node.

The first assumption greatly simplifies the network design

problem, and makes our models viable for relatively large

networks. In fact, if the weights wij, for ij ∈ E , are given as

input, then the routing paths of all demands can be determined

by shortest path computations, which take negligible time even

in large networks. Thus, in most cases, the capacity allocation

can be done automatically and the set of feasible solutions can

be significantly reduced. The last two assumptions, however,

require a more detailed description: assumption 2 is explained

in the remainder of this section, while the technologies we

have considered are detailed in Section II.

B. Network optimization under traffic uncertainty

Real-world network design problems are subject to uncer-

tainty in several contexts: cost, capacity, and reliability of link

and node equipment, and, most importantly, traffic matrix,

which is the focus of this paper. Although the set of source-

destination pairs may be known, the volume of traffic of each

demand is seldom known with accuracy. This happens for

several reasons: inaccurate measurements, dynamic behavior

of traffic, network failures inducing a shift in the demand,

to name only a few. The sources of traffic uncertainty are

multiple and need to be identified and accurately described:for instance, it helps to know whether there are lower/upper

bounds on the volume of a demand from a source s to a

destination t of the network, bounds on the total traffic volume

or, again, statistical information on the TM.

From a modeling standpoint, the more accurate the approx-

imate traffic matrix, the better: a network that accommodates

a loose uncertainty set can be overly conservative, hence

very expensive. From a more computational point of view,

solving a network provisioning problem while taking into

account uncertainty in the problem’s parameters may render

the problem itself intractable, as the size of the associated

optimization problem (number of variables and constraints)

may increase substantially.

A common way to model this problem is Robust Optimiza-

tion [5]: the traffic demand is not known a priori, but a set

S of possible traffic matrices is given. The capacity installed

and the routing paths then must accommodate, at any time, all

traffic matrices in S . There are other approaches for solving

optimization problems under uncertainty, the most important

being Stochastic Programming [6], in which a probability dis-

tribution is associated with each element of the uncertainty set

[7, 8, 9, 10, 11]. Although Stochastic Programming can exploit

statistical information on uncertain parameters, it produces

even more difficult optimization problems. As a consequence,

depending on the size of the network, this modeling framework

can be impractical.

We use robust optimization for modeling our network design

problem. From now on, we consider the problem of installing

capacity on the network nodes in order to accommodate all

traffic matrices in a given set S , known a priori.

C. Previous work The application of robust optimization to network design

under traffic uncertainty has been a hot topic in the last decade.

Duffield et al. [12] introduce the hose model, an uncertainty set

where an upper bound for the incoming and outgoing demand

is defined for each node, i.e., the uncertainty set S is the

following set of vectors (vq)q∈Q:

S = {v ∈ R|Q|+ :

q∈Q:sq=ivq ≤ b′i ∀i ∈ V,

q∈Q:dq=ivq ≤ b′′i ∀i ∈ V }.

Gupta et al. [13] study a particular case of this problem where

the capacity must be installed so that the edges with nonzerocapacity form a spanning tree of the original network topology.

They prove that this version of the problem can be solved

in polynomial time and propose an algorithm to solve the

problem (see also [14, 15]).

Ben Ameur and Kerivin [16] propose an uncertainty model

where the set S of traffic matrices is a non-empty, bounded

polyhedron in |Q| dimensions (|Q| is the number of traffic de-

mands) defined by a set of linear inequalities. This polyhedral

uncertainty model encloses the hose model as a special case.

This leads to a semi-infinite linear programming (LP) problem,

an optimization problem with finitely many variables and

infinitely many constraints: given that there is one constraint

for each traffic matrix, which in turn is a point of a non-empty,bounded polyhedron, the number of constraints is infinite.

Rather than solving the semi-infinite LP explicitly, they

describe an iterative method that considers an initial set

of demands. At each step, after solving the current LP, a

separation problem is solved to generate a traffic demand that

cannot be routed. This demand is used to increase the total

capacity, and the algorithm continues until separation gives

no new violated demands, proving that all demands in the

polyhedron are satisfied.

Ben Tal and Nemirovski [17] introduce a class of robust

optimization problems where the uncertainty set is identified



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by a vector of mean values and a covariance matrix, which

define a confidence ellipsoid that includes a “safe” portion

of all vectors of uncertain parameters. Optimizing over this

uncertain set amounts to solving a second order conic opti-

mization problem [18]. Although more difficult than Linear

Programming problems, second order conic problems are

convex optimization problems and therefore admit an efficient

solution method.

Bertsimas and Sim [19] introduce an uncertainty model

where all parameters are allowed to take on a lower or an upper

value, while imposing a maximum number K of demands that

are at the upper value simultaneously. This is a compact yet

meaningful way to express a large uncertainty set.

Belotti et al. [20] deal with a network design problem using

the aforementioned uncertainty set in the very special case

where K = 1. At any point in time, all demands are at their

lower value except for one, which is at its upper value. As a

consequence, the maximum total traffic on each link is equal

to the sum of all lower values plus the peak volume of the

largest demand routed on that link. Riis et al. [21] discuss

another application of robust optimization to network design.

D. Robust network design

As anticipated above, the problem at hand asks to dimension

the capacity at each node of G. Each node has to be equipped

with a set of ports, corresponding to interfaces of a network

node with the incident links. A port is capable of routing traffic

to and from that node, and its capacity defines the maximum

traffic in either direction. For example, if a 10 Gbps port is

installed on node i as an interface for link ij, that port will

be able to handle at most 10 Gbps traffic from i to j and 10

Gbps in the opposite direction. Hence, for a traffic of 10 Gbps

i → j and 7 Gbps j → i there will have to be ports on bothi and j capable of 10 Gbps.

When installing capacity on OTN switches, each demand

is associated with a port and shares the port with no other

demand (see Section II). Hence, we do not need to take into

account demand uncertainty: each demand can be assumed to

be at its maximum value, and therefore traffic uncertainty has

little or no impact on OTN-only network design.

On MPLS links, however, statistical multiplexing substan-

tially reduces the amount of capacity to install. Because more

than one traffic demand is routed on a port, the amount of

capacity to be allocated on each link strongly depends on the

variability of the demand and on the uncertainty model. Hence,

MPLS links are best suited to route uncertain demands as longas the demands’ dynamic values can be exploited to reduce

the total traffic w.r.t. the total traffic routed on OTN links, on

which demands are assumed to be fixed at their peak value.

We use the uncertainty model of Bertsimas and Sim [19] to

describe the set of traffic matrices. For every q ∈ Q we are

given an average demand volume vavgq and a peak volume,

vpeak q . Every link (i, j) is used by a set Qij of demands,

of which at most K ij can be at their peak value while the

remaining are at average value.

The parameters vavgq , vpeak

q , and K ij are input to our problem

and can be estimated by the network planner based on network

measurements. The accurate estimate of the traffic demands’

variability, which is crucial to a correct outcome of the network

planning process, is difficult and has been the subject of

extensive research—see e.g. the Rocketfuel project [22]. We

assume these parameters to be an input to our problem, and

are likely to be the result of another, equally complex (if not

more complex) analysis task.

The choice of uncertainty model is critical in the design

problem we describe. We have chosen Bertsimas and Sim’s

model because we make no assumption regarding the traffic

matrix in our experiments, and because one parameter, K ij,

allows us to simplify the comparison of different technologies.

The optimization model we propose, nevertheless, admits

a straightforward extension to other uncertainty models as

general as those explained by Ben Ameur and Kerivin [16].

The capacity allocation on all nodes is determined by the

total traffic on the incident links. Consider the set Qij of

demands routed on link ij. Denote as Qij the subset of the

largest K ij demands in Qij , or Qij itself if |Qij| ≤ K ij . In

order to route all traffic matrices in S , the capacity on ij must

be at least the maximum traffic allowed by S on ij, orq∈Qij

vpeak q +

q∈Qij\Qij

vavgq ,

which is the worst-case total traffic among all subsets of traffic

demands routed on ij.

For simplicity, let us assume that uncertainty affects all links

equally, and use a single value K for all K ij. Throughout the

paper and in the experiments, we use relatively small values

of K ranging from 1 to 32. We provide in Section III-B an

explanation as to why these small values can tackle realistic

uncertain demands.

E. Sample Network and Traffic

In order to assess the validity and practicality of our

approach, we have created a sample network that closely re-

sembles, in size and demand distribution, a modern backbone

network. The nodes of this network are subdivided into core

nodes and edge nodes. There are 127 nodes in total: 20 core

nodes and 107 edge nodes. Each edge node is connected to

one core node (called its home node) through an edge link ,

while each core node is incident to one or more core links.

Figure 1 shows a small sample network with some core and

edge nodes and links.

For all traffic demands, both source and destination are edge

nodes. The routing path of each traffic demand will thereforecontain: one edge link from the source to its home core node;

zero or more core links (zero if the source and the destination

nodes have the same home core node), and an edge link to

the destination node.

We focus on the problem of allocating capacity on core

nodes only, and ignore the capacity to be installed on edge

links and nodes. As will follow from the discussion over

technologies below, the problem of allocating capacity on

edge links admits only one solution and hence is trivial.

Consequently, demands whose source and destination nodes

have the same home core node are ignored in this work.



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Edge node

Edge link

Core node

Core link

Fig. 1. A two-level network topology.

There are 39 core links, hence there are 146 links in total

(the network has 107 additional edge links, connecting each

edge node with its home core node). Normally, optimization

models on such a large network can be prohibitive, but, since

we can ignore capacity allocation on edge links, we are solving

a problem on a topology of 20 nodes and 39 edges.

The average traffic demands vavgq have been selected by

combining real traffic data from mid-size networks, hence it

represents well the type of traffic that is routed on a network

of this type. There are 4451 traffic demands. The peak value is

assumed, only for our experiments, to be vpeak q = αvavg

q , where

α is a constant. Figure 2 shows a distribution of the average

traffic volume of the matrix we have used.

The traffic characteristics used in the model were influ-

enced from different traffic profiles from various large service

provider (SP) networks. The traditional SPs have a top heavy

core where 10% of the largest demands occupies about 85% of

the network capacity. On the other hand, cable operators tend

to have flatter characteristics with 20% of the largest demandsoccupying about 40% of the network traffic. For the purposes

of our model we have the following traffic profiles (which

were “top heavy” but at the same time applicable for different

service providers):

• top 10% of demands holds 58% of total traffic;

• bottom 40% of demands holds 8% of total traffic;

• the total demand is 8 Tbps.

It is worth noting that, although we are interested in the

capacity allocation on core nodes and links, we still consider a

set of 4451 demands and do not group them in macro-demands

defined by a source core node and a destination core node.This would significantly reduce the size of the problem, but

would also restrict the set of feasible solutions and therefore

potentially increase the network cost. In fact, consider two

demands q ′ and q ′′ of Q whose sources and destinations differ

but have the same corresponding source and destination core

nodes: by considering a grouped traffic matrix, all feasible

paths for demands q ′ and q ′′ would have to be routed as

one single demand and not split on multiple ports, thereby

introducing an implicit and unintended constraint given that

each demand may be routed on either the OTN or the MPLS

layer or in different (smaller) OTN ports.

02

3

4

5

610

10

10

10

101000 2000 3000 4000 5000

Demand

Volume [Mbps]

Fig. 2. The demand volumes (in logarithmic scale) of the 4451 origin-destination pairs used in our experiments. Note that a large portion of thetraffic is concentrated on a small percentage of the traffic demands.

F. Outline of the paper

Our robust optimization model allocates capacity in a net-

work that uses MPLS and OTN nodes; the network can use

these nodes both standalone and combined. The key question

here is the optimal technology that yields a minimum cost

telecommunication network, given that certain technologies

can take advantage of statistical multiplexing. By applying an

exact solver to the optimization problem, we can get precise

estimates on the cost effectiveness of each technology when

only a limited knowledge of the traffic demand is available.

Our optimization model, while borrowing from other un-

certainty descriptions, is unique in that it addresses a specific

network provisioning problem, where the routing of eachdemand is fixed in advance but the layer (MPLS or OTN)

where each demand is routed has to be decided.

In the next section, we describe the problem of allocating

capacity on networks that have either MPLS or OTN nodes,

but not both, under the uncertainty model described earlier.

Section III introduces the problem of designing a network

by using both OTN and MPLS nodes and describes a robust

optimization model. In Section IV, we present extensive com-

putational results. We provide conclusions and open questions

in Section V.

I I . NETWORK PROVISIONING

We consider four types of capacity, and model how these

technologies result in different cost networks given the as-

sumptions in Section I-A. We then determine the cost-

optimized network based on a combination of the technologies.

Figure 3 shows the structure of the nodes considered: OTN-

only, MPLS-only, combined MPLS+OTN, and bypass.

A. OTN Switching

Optical transport network (OTN) switching technology is

the next generation SONET/SDH (synchronous optical net-

work / synchronous digital hierarchy) technology for providing



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point-to-point connectivity in a transport network. The OTN

architecture is specified in ITU-T Rec. G.872 and the frame

formats and payload mappings are specified in G.709. OTN is

a circuit switched technology ideal for managing end-to-end

circuit-switched services. The benefits of the technology are

that dedicated end-to-end bandwidth is allocated per circuit

offering a single level (but reliable) quality of service (QoS).

As specified in the G.709 specifications, individual client

demands are mapped to outer and inner containers called

Optical Data Unit (ODU) and Optical Transport Unit (OTU)

layers for transmission on the transport network. ODU frame

formats depend on the capacity of the physical link (for

example an ODU2 frame is used for 10 Gbps links). Within an

ODUk frame there can be multiple OTUk frames to provide

end-to-end circuit connectivity for different sized circuits.

These can be a combination of OTU0 (1.25 Gbps), OTU1

(2.5 Gbps), OTU2 (10 Gbps), OTU3 (40 Gbps) or OTU4 (100

Gbps). Thus two flow demands of 1 Gbps and 2 Gbps would

map to an OTU0 and an OTU1 frame respectively and then

be wrapped into an ODU2 frame for transmission on the link.

However, in the presence of variation and bursts of flowdemands, circuit-switched networks have to conservatively

allocate bandwidth to handle peak capacity of the flow. This

leads to lower utilization of the circuit and physical link

capacity.

B. Compensating potential aggregation issues with OTN

Smaller-sized flows (less than 1.25 Gbps) are mapped to

ODU0 containers for transport resulting in an acute underuti-

lization of capacity. In the presence of a large number of small

flows, this would lead to a severely inefficient network due

to a large amount of capacity being unused. Filling ODU0s

efficiently can only be achieved using an external aggregation

device (adding cost) or via proprietary sub-ODU0 mapping.

The model assumes that such a capability exists in the network,

although this would have to be provided separately.

Another consideration with OTN implementations is how

to reduce the risk of many larger containers (for instance,

ODU1 or ODU2) remaining empty. There is a mechanism

called ODUFlex that some implementations may provide to

concatenate ODUs to create N×ODU0 up to 100G. The model

accounts for this possibility as well.

C. MPLS routers

Multi-Protocol Label Switching (MPLS) is a packet-

switched technology used for transporting any type of trafficby grouping it into suitable sized packets. Service levels and

priorities can be set for different types of traffic such as wire-

less, wireline, video and data traffic. A connection oriented

path in a packet-switched network can be set up by means of a

Label Switched Path (LSP) which can be variable-sized based

on traffic needs; thus it is possible to achieve fine-grained

as well as coarse-grained switching of traffic. The existence

of buffers and queuing allows for prioritization of delay and

time sensitive traffic during bursts and statistical variances in

the flow demands. The multiplexing gains thus achieved by

integrating multiple statistically varying demands on a link

provide an efficient utilization of the link bandwidth. Packet-

switched networks are thus most advantageous in networks

where peak demands include multiple higher-than-average

demands.

The functionality we are talking about in Label Switching

Routers (LSRs) is bare label switching and is therefore roughly

equivalent to MPLS Transport Profile (MPLS-TP). These

ports are usually more expensive than OTN ports, therefore

a tradeoff arises between relatively expensive equipment that

can exploit statistical multiplexing and cheaper equipment,

which is unable to take advantage of statistical multiplexing

and therefore has to be installed in larger quantities. Hence

we are interested to know, for given uncertainty sets and an

initial network topology, which technology among MPLS or

OTN is more cost effective.

D. Bypass

In a typical network of MPLS routers, traffic enters the

network through the ingress router and transits over zero or

more routers in reaching the egress router. In a fully meshednetwork, all nodes are connected to each other and thus

traffic from an ingress router directly travels to the egress

router. However, in a typical network, as the number of nodes

increases it is quite complex to connect each node to every

other node; as a result traffic transits over some routers en

route from the ingress to the egress node. Each node thus has

a portion of traffic that is local (i.e. is sourced or destined to

that node) and the rest of the traffic is transit traffic (destined

to another node). Since router ports are needed for both local

and transit traffic, one can reduce the number of router ports

for transit traffic by introducing an OTN layer between the

routers. The MPLS routers switch only the local traffic; transit

traffic at a node bypasses the router and is switched by theOTN layer. Since the OTN layer aggregates local and transit

traffic, additional trunking ports are needed between the OTN

and MPLS switches.

E. Provisioning at the MPLS, OTN, and Bypass layers

Before introducing the optimization model used in our

experiments, we show how to allocate capacity on MPLS-

only, OTN-only, or bypass networks. This is straightforward

and does not need any optimization process. In the following,

for each traffic demand q ∈ Q we define the path pq as the set

of directed links on which q is routed, determined by solving

a shortest path problem on the OSPF weights.We first show how to compute the capacity of an OTN-

only network. We assume that 1.25 Gbps, 2.5 Gbps, and 10

Gbps ports can be installed at the OTN layer. As mentioned

in Section II-A, 10 Gbps links have to be installed to accom-

modate the set of ports. Each demand q ∈ Q, whose volume

must be assumed to be fixed at vpeak q , is routed on a port with

an equal or larger capacity. Therefore, the number of ports

can be determined immediately. Consider the vector of OTN

port types, cotn = (1.25, 2.5, 10). The number of OTN ports of

capacity cotnk , with k ∈ {1, 2, 3}, to be installed at the interface

of node i to node j in V is the maximum between the ports



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(a) OTN switch (b) MPLS router (c) Combined MPLS+OTN (d) Bypass router

Fig. 3. The structure of OTN, MPLS, combined OTN and MPLS, and bypass nodes.

used for incoming traffic and those for outgoing traffic: for

each k ∈ {1, 2, 3},

notnij,k = max{|{q ∈ Q : (i, j) ∈ pq ∧ cotn

k−1 < v peak q ≤ cotn

k }|,

|{q ∈ Q : ( j, i) ∈ pq ∧ cotnk−1 < v peak

q ≤ cotnk }|},

where cotn0 = 0. In order to find the amount of 10 Gbps

OTN links to be allocated, note that in our problem definition

the port capacities are multiples of one another, and are sub-

multiples of the 10 Gbps link capacity. This simplifies the

computation of the number N otnij of links:

N otnij = ⌈ 1

10 max{1.25notnij,1.25 + 2.5notn

ij,2.5 + 10notnij,10,

1.25notnji,1.25 + 2.5notn

ji,2.5 + 10notnji,10}⌉.

It is then straightforward to compute the total number of 10

Gbps links at each node i ∈ V , which is given as the sum of

N otnij over all j adjacent to i.

For allocation of small demands into ODU0, the number

of ports of 1.25 Gbps is determined as follows. Consider all

demands whose volume vpeak q is below 1.25 Gbps, and routed

on 1.25 Gbps ports on link ij in the direction from i to j,

and denote it as Q1.25ij . A set of ports have to be allocated

so as to accommodate all demands of Q1.25ij , in such a way

that multiple demands of Q1.25ij can be routed on the same

port, although they must be entire demands. Note that this

a significant change with respect to the OTN ports, each of

which is instead dedicated to one demand.

Therefore, one has the problem of minimizing the number

of 1.25 Gbps ports that can accommodate some or all demands

below 1.25 Gbps. This is equivalent to solving a bin packing

problem [23] on every link. More specifically, Q1.25ij has to

be partitioned into H subsets Q1.25ij,h , for h ∈ 1, 2 . . . , H , such

thatH h=1 Q1.25

ij,h = Q1.25ij and

q∈Q1.25

ij,hvpeak q ≤ 1.25 Gbps for

each h = 1, 2 . . . , H . Given that we need H such ports, a small

value of H is desirable to keep the network cost limited. The

design process will thus require the solution of one bin packing

problem for each link ij. We do not provide further details for

the sake of conciseness, but point out that these problems are

relatively easy for the instance size we are considering.

Once this problem is solved, the number of ports for each

link ij on both directions i → j and j → i is known and

denoted as noduij and nodu

ji respectively. The number of ODU0

ports to be installed on node i is given by

j∈V :(i,j)∈E

max{noduij , nodu

ji }.

In the MPLS case, the capacity depends on the average

and peak values of the demands. We assume that only 10

Gbps ports are available at the MPLS level, and compute the

number of such ports based on the set of demands routed

on link ij in the direction i → j, denoted Qij , and in the

opposite direction, denoted Qji . We recall that these subsets

of demands are known in advance. The total MPLS traffic on

a link ij under our uncertainty scenario has been described in

Section I-D, and we report the number of 10 Gbps ports:

n

mpls

ij = 1

10

q∈Qij v

peak

q +

q∈Qij\Qij v

avg

q

.

The bypass case is slightly more complex: the incoming

and outgoing traffic at each node i are sent onto the network

through an MPLS router that allows statistical multiplexing.

That traffic is then forwarded on the network by means of OTN

switches. At each node, transiting traffic is handled by OTN

switches only, while incoming and outgoing traffic traverses

MPLS routers. Suppose Qsrci is the set of demands with origin

i, and Qdsti the set of demands with destination i. Denote as

Qsrci the set of K largest demands among Qsrc

i , or Qsrci itself

if |Qsrci | ≤ K , and analogously define Qdst

i . The total capacity



7

to be installed is the number of ports for the local traffic

nbyp,locali =

110 max

q∈Qsrc

ivpeak q +

q∈Qsrc

i\Qsrc

ivavgq ,

q∈Qdsti

vpeak q +

q∈Qdst

i\Qdst

ivavgq

,

which has to be considered both at the MPLS layer and at the

OTN layer (i.e., the cost of each such port will be that of an

OTN port plus an MPLS port) plus the number of OTN ports

for the transit traffic, n

byp,tr

ij . This, in turn, is the total, amongall adjacent nodes j, of the ports to be installed at node ifor traffic flowing on link ij in both directions. Let us denote

Qtrij the set of traffic demands routed on ij from i to j not

originating in i, and Qtrij ⊆ Qtr

ij a subset containing the K largest demands (or, again, Qtr

ij itself if |Qtrij| ≤ K ). Denote

Qtrji the set of traffic demands routed on ji from j to i not

ending in i, and define Qtrji analogously. Then

nbyp,trij =

110 max

q∈Qtr

ijvpeak q +

q∈Qtr

ij\Qtr

ijvavgq ,

q∈Qtrji

vpeak q +

q∈Qtr

ji\Qtr

jivavgq

.

In summary, in the OTN-only, bypass, and the MPLS-only

scenarios the capacity allocation can be carried out by a simple

procedure which does not require solving an optimization

problem. Relatively small optimization problems have to be

solved for the allocation of demands to the single OTU0 links.

One such problem must be solved per link in order to compute

the total capacity on that link.

III. OPTIMIZATION MODELS FOR COMBINED MPLS-OTN

SOLUTIONS

In order to cut costs even more, one could think of building

a network with both MPLS and OTN ports that can route

all traffic matrices in the uncertainty set. Hence, the network

admits both MPLS routers and OTN switches at the same

hierarchy, and the problem is that of determining the numberof MPLS and OTN ports for all nodes of the network.

Because we are dropping the restriction to use only one

technology, the resulting network, excluding additional costs

to interface, if necessary, MPLS routers to OTN switches, will

be less expensive than both the OTN-only and the MPLS-

only network. The question is then how much cheaper this

combined network would be.

This network optimization problem can be shown to be NP-

hard, i.e., it is very unlikely that there be an algorithm that

solves it in a number of elementary steps that is polynomial

in the size of the problem (number of nodes and links of G).

The proof of NP-hardness is by reduction from the Subset

Sum problem [24]: consider a set S and a function c : S → R.Define c(S ′) =

i∈S ′ c(i) for any S ′ ⊆ S . The Subset Sum

problem asks to partition S into two sets S 1 and S 2 (i.e.,

S 1∪S 2 = S and S 1∩S 2 = ∅) in such a way that |c(S 1)−c(S 2)|is minimized. Our network design problem is NP-hard because

any instance of Subset Sum problem on a set S and a function

c can be transformed in an instance of our problem, where the

network G has only one link, the number of traffic demands

equals |S |, α = 1, K = 0, and the traffic value of each demand

i ∈ S is ci. We omit the details for the sake of conciseness.

This general design problem can be formulated as a Mixed

Integer Linear Programming (MILP) model [25]. The MILP

model illustrated in this section generalizes the design method

outlined above for OTN-only and MPLS-only networks: by

forbidding either the MPLS or OTN variables in the model

below, we obtain a much simpler problem that admits the

solution outlined above.

A large portion of demands have routing path spanning

two or more core links. These demands, in principle, could

use the MPLS layer on certain links of the routing path and

the OTN layer on the remaining links. This would imply

that some intra-node, inter-layer capacity is needed on one

or more nodes. However, we have strong empirical evidence

suggesting that solutions where a demand uses both layers

is sub-optimal. More specifically, preliminary experiments on

the aforementioned 20-node network, for multiple values of K and α, have shown that there is at least one optimal solution

where each demand is routed on one layer. A consequence of

this is a major simplification of the optimization model and

thus shorter solution times. Therefore, from now on we assume

that each traffic demand is entirely routed on either the MPLS

layer or the OTN layer and a key decision to make is which

layer each demand is routed on.Suppose that Qmplsij is the set of demands routed on the

MPLS layer of link ij, and analogously Qotnij for the OTN

layer. In this case, the amount of ports to be allocated for i and

j is given in Section II-E. Because this quantity depends on

Qmplsij and Qotn

ij , which are not known in advance, the decision

on each demand is a variable in an optimization model.

For each traffic demand q ∈ Q, we define binary variables

f otnq , f odu

qh , and f mplsq . Variables f otn

q and f mplsq are one if demand

q is routed on the OTN or MPLS layer, respectively, and

zero otherwise. Variable f oduqh is one if demand q uses the

subset Q1.25ij,h (see definition in the previous section), and zero

otherwise. Here the number H of 1.25 Gbps ODU ports is

computed so that enough capacity is available in case all of the demands in Q1.25

ij are routed on the ODU layer, but we

can expect that not all of the variables f oduqh are nonzero. These

binary variables are subject to the following constraint:

f otnq + f mpls

q +h∈H

f oduqh = 1 ∀q ∈ Q, (1)

which forces exactly one of these variables to one and the

remaining to zero. There are |Q|(H + 2) such variables.

We also define a set of integer variables for the ports on

the OTN layer: xotnij,t is the number of OTN ports on link ij

(i.e., installed at i and j), where t indexes the capacity vector

cotn = (1.25, 2.5, 10). Also, we define binary variables xoduij,h,

which are one if 1.25 Gbps ODU0 ports are required to supportthe combination of ports in h for aggregated demands. Finally,

the integer variable xmplsij is the number of 10 Gbps MPLS ports

allocated on link ij.

These variables are related to flow variables through the

capacity constraints. At the OTN layer, we need one OTN port

of capacity 1.25 Gbps on link ij for each demand q routed

on ij whose peak value is at most 1.25 Gbps and such that

f otnq = 1. Similarly, we need one 2.5 Gbps OTN port on link

ij for each demand q routed on ij whose peak value is strictly

greater than 1.25 Gbps and not larger than 2.5 Gbps, and such

that f otnq = 1, and similar considerations apply to 10 Gbps



8

OTN ports. We model this with the constraint

xotnij,t ≥

q ∈ Q : ij ∈ pq∧

cotnt−1

< vq ≤ cotnt

f otnq ∀ij ∈ E, t ∈ {1, 2, 3}, (2)

where cotn0 = 0. The capacity constraint for ODU ports requires

that a port be present if there is nonzero traffic on it:

1.25 xodu

ij,h ≥

q∈Q:ij∈ pq

vpeak

q f odu

qh ∀h = 1, 2 . . . , H , i j ∈ E.

(3)

Finally, the capacity on each link of the MPLS layer is defined

as follows:

10 xmplsij ≥

q∈Q

vavgq f mpls

q + zij ∀ij ∈ E, (4)

where zij is a variable describing the peak flow, and is

defined by a special port capacity constraint. Let us define

the peak-average demand difference as δ q = vpeak q − vavg

q .

Then the peak flow is given by the optimal solution value of

another optimization model which yields the worst-case (i.e.,

maximum) traffic on ij as a function of the set of demandsrouted at the MPLS layer of ij:

∀ij ∈ E zij ≥ max

q∈Q:ij∈ pqδ qf mpls

q

s.t. Q ⊆ Q|Q| ≤ K.

(5)

Given that only a subset of demands of cardinality K is al-

lowed to have peak value, the right-hand side of this constraint

can be rewritten as

max

q∈Q:ij∈ pqδ qf mpls

q µqs.t.

q∈Q µq ≤ K

µq ∈ {0, 1}∀q ∈ Q,

which is an optimization model on binary variables µq definingthe set of demands at peak value. In order to evaluate constraint

(5) one has to solve, for each link ij, a maximization problem

in variables µq where f mplsq are taken as parameters.

Another class of variables represents the number of links

needed at all nodes, which constitutes the ultimate decision

variable on which the total network cost is computed. We

define the variables yotnij as the number of 10 Gbps links to

be installed on both ends of link ij at the OTN layer. The

constraints relating link variables to port variables are easily

defined, for each ij ∈ E , as follows (note that OTN links must

accommodate OTN and ODU ports):

t∈{1,2,3}

cotnt xotn

ij,t + 1.25

H h=1

xoduij,h ≤ 10 yotn

ij ∀ij ∈ E. (6)

Each 10 Gbps OTN link must be assigned a set of whole

OTN ports, i.e. traffic on an OTN port cannot be shared

between two OTN links. In general, a more complicated set

of constraints, resembling the bin packing set of constraints

for ODU port/flow capacity, would be necessary. However, the

link capacity, 10 Gbps, is a multiple of the largest OTN port

capacity, and each OTN port capacity cotnt is a multiple of the

preceding OTN port capacity, cotnt−1, thus resulting in another

bin packing problem that is implicitly solved in (6).

This set of variables and constraints is only defined at the

OTN layer, as the number of MPLS links is equal to that of

MPLS ports xmplsij , hence no additional variable is needed.

Let us assume that the cost of a single link is gotn at the OTN

layer and gmpls at the MPLS layer. It is realistic to assume that

gmpls = βgotn with β ≥ 1, i.e. the cost of MPLS links is greater

than that of OTN links. We have set β = 1.3 in most of our

experiments, which reflects a cost differential that seems more

realistic for the kind of functionality required, at the MPLS

level, from this type of network. This introduces a trade off

between the cheaper (but under-utilized) OTN links and the

more expensive MPLS links, which use statistical multiplexing

and can hence make a better use of the total capacity. The

objective function is the total core network cost, i.e., the total

OTN link cost plus the total MPLS link cost

2ij∈E

(gotnyotnij + gmplsxmpls

ij ), (7)

where all quantities are multiplied by two because the number

of ports on link ij is installed at both end nodes i and j.

To recap, the problem P of determining the minimum cost

capacity on the nodes of G to route all demands in the

uncertainty set S requires to minimize the objective function

(7) subject to constraints (1,2,3,4,5,6). This is a nonlinear

integer optimization problem, the only nonlinear constraint

being (5).

A. Robust optimization: an opponent’s view

Problems like P are in general classified as bilevel program-

ming problems as the evaluation of a constraint is associated

with solving another optimization problem [26]. Our model is

representative of a smaller class of bilevel programming, that

of robust optimization. The key component is the right-hand

side of constraint (5), which, as said before, is evaluated bysolving a linear optimization problem. The interpretation of

this inner optimization problem is functional to understanding

the advantages and the meaning of robust optimization.

Let us suppose an external player (such as nature, competi-

tors, or customers) is given the solution to our optimization

model (1,2,3,4,5,6,7) (i.e., the decisions on the layer on which

every demand is routed) and has the intent of maximizing

the nuisance on the network we have designed. This external

player, which is in practice our opponent , has the control over

the uncertainty set, i.e., decides which demands in a subset of

cardinality K are at their peak value. The opponent will then

determine a subset of demands that maximizes the load on the

installed capacity. In order to anticipate the opponent’s move,we must implicitly solve the optimization model.

We begin by relaxing integrality over variables µq and

constrain them in the interval [0, 1] rather than in the non-

convex set {0, 1}. This is a relaxation of the above problem

in that all solutions that are feasible for the initial problem

are also feasible when requiring that µq be continuous. Also,

because the coefficient matrix of the problem (5) is totally

unimodular , any optimal solution of the relaxed problem is

optimal for the original problem as well [25].

Now the maximization problem on the right-hand side of

constraint (5) is a Linear Programming (LP) problem, which



9

is much easier to handle for our purposes. We rewrite it here

for the sake of clarity:

P ij : max

q∈Q:ij∈ pqδ qf qµq

s.t.

q∈Q µq ≤ K

0 ≤ µq ≤ 1 ∀q ∈ Q.

In order to get rid of the maximization sign, we use weak

duality and apply a well known procedure in robust optimiza-

tion [27]. Any LP problem max{c⊤x : Ax ≤ b}, with an

n-vector of variables x ∈ Rn and m constraints, with c ∈ Qn,

A ∈ Qm×n, and b ∈ Qm, admits a dual LP problem

min{u⊤b : u⊤A = b, u ≥ 0}

with a vector u ∈ Rm of variables. Weak duality dictates that

for any vector x satisfying Ax ≤ b and for any non-negative

vector u satisfying u⊤A = b, we have [28]

u⊤b ≥ c⊤x.

As a consequence, u⊤b provides an upper bound to the right

hand side of constraint (5). Replacing the right-hand side of

(5) with u⊤b and adding the dual constraints u⊤A = b, u ≥0 to the original problem allows us to implicitly solve the

maximization problem in (5) and obtain an upper bound on the

total traffic on link ij. It also yields a set of linear constraints,

thus eliminating the nonlinearity introduced by the max sign.

Consider each link ij ∈ E and the associated problem P ij .

Denote ωij the dual variable of constraint

q∈Q µq ≤ K and

λqij the dual variable of the upper bounding constraint for µq.

Then the dual of problem P ij is the minimization problem

Dij : min Kωij +

q∈Q λqijs.t. ωij + λqij ≥ δ qf q ∀ij ∈ E, q ∈ Q

ωij ≥ 0 ∀ij ∈ E

λqij ≥ 0 ∀ij ∈ E, q ∈ Q.

This yields a model where constraints (5) are replaced by the

following sets of constraints:

zij ≥ Kωij +

q∈Q λqij ∀ij ∈ E

ωij + λqij ≥ δ qf q ∀ij ∈ E, q ∈ Qωij ≥ 0 ∀ij ∈ E λqij ≥ 0 ∀ij ∈ E, q ∈ Q.

Thus, we replace the set of |E | nonlinear constraints by

introducing |Q||E | + |E | new variables and |E | + |Q||E | con-

straints. Although this increases the model size substantially,

eliminating the nonlinear constraint (5) reduces the problem

to an Integer Linear Programming (ILP) problem, thus moretractable. Also, it can be shown that only a small subset of

these constraints are necessary, by replacing Q with Qij in

the above models P ij and Dij . We omit the details for the

sake of conciseness, but we did implement this reduction in

our experiments.

As an interpretation of this procedure, consider the initial

problem and remove the max term in constraint (5), and

suppose that µq instead are variables of P . This would mean

that we, not the opponent, control µq. Then we no longer

have to explicitly solve an optimization model, and can decide

which demands assume peak value. Solving such a (nonlinear)

problem yields an unrealistic best-case solution, in which the

most optimistic set of demands assume peak value: the empty

set. In fact, it is easy to prove that any optimal solution of such

a problem would have all µq set to zero, therefore obtaining

a null value of zij, which corresponds to a no-peak scenario.

Because we use the dual of this problem, by eliminating

the minimization sign we still impose the dual constraints,

thus forcing the variables λq

ij

and ωij to be dual-feasible. This

in turn ensures that, for any feasible solution of the overall

problem, the right-hand side of constraint (5) is an estimate

from above of the maximum capacity allocation, hence forcing

zij to a value that is no smaller than the optimal solution of

problem P ij . This yields a safe estimate (from above) of the

total peak traffic on link ij. Eliminating the minimization sign

puts the dual variables λqij and ωij in our control, albeit forcing

them to model a worst-case scenario.

B. Statistical multiplexing and robust allocation

The fundamental assumption of the model outlined above

is that the uncertainty in the traffic demand can be dealt with

by allocating capacity to accommodate a set of demands attheir average value plus a worst-case extra capacity defined by

the largest K demands. Given that this worst-case total traffic

is computed independently at every link, the total capacity

allocated on the network may be overly conservative even

for small values of K . This suggests that an appropriate

value of K is not easy to find. As discussed above, the

availability of statistical data for traffic matrices may provide

more information, though it would be difficult to incorporate

it in an optimization model as simple as the one described.

Let us consider, in the remainder of this section, a different

problem: how does a value of K compare to a random

set of demands at their peak values? In other words, for a

capacity allocation that includes extra ports for supporting thelargest K peaks, can we estimate the number T of demands

that, on average, will be satisfied by the resulting capacity

allocation? Note the contrast between K and T : while the

former determines the set of largest demands routed on a

link, the latter is the expected number of demands, randomly

selected from Q, that can have peak value and still are routed

on every link.

Suppose that each demand can take its peak value with

equal probability, that is, each demand has probability T /|Q|of being at peak value. The total extra capacity can be

computed as the expected value of the extra demand using

the uniform distribution pq = T /|Q| for all q ∈ Q, and is

D =

q∈Q δ q T |Q| . In order to compare D with the worst-case

amount of extra capacity to install, we must have

q∈Q

δ qT

|Q|=q∈Q

δ q,

where Q is the set of K largest demands. An estimate of T is therefore

T =|Q|

q∈Q δ q

q∈Q δ q.

Suppose now that the probability that a demand is at peak

value grows with the demand volume, and let us denote this



10

probability distribution as progressive. In particular, assume

w.l.o.g. that vavg1 ≤ vavg

2 ≤ ... ≤ vavg

|Q|. Suppose, furthermore,

that the probability pq that a demand q ∈ Q be at its peak

value is proportional to its index q , i.e. pq = γq , where γ is

such that the expected number of demands at peak values is

T , i.e.

q∈Q γq = T and hence

γ =T

q∈Q q

=2T

|Q|(|Q| + 1).

Again, we obtain T as a function of K by solving the equationq∈Q

pqδ q =q∈Q

δ q,

which yields

T =|Q|(|Q| + 1)

q∈Q δ q

2

q∈Q δ q.

The following table provides values of T for small values of

K for both the uniform and the progressive case using demand

data from the instance we have used.

K 1 2 4 8 16 32

T (unif.) 160 210 300 446 642 895

T (prog.) 96 126 180 268 385 587

When allocating capacity, the worst-case scenario, even for

small values of K , can cover a number of demands at their

peak value on a single link ij that is much larger than K , and

that can be up to one fifth of |Q|, for K = 32 in the uniform

case. This table would clearly look different if the demand

distribution were different: the instance we have chosen has

a number of very large demands, while the volume is slowly

degrading for the smaller ones.

In practice, the number of simultaneous peaks in a real

network would likely be up to 10. However, for the purposeof meaningful analysis, we have considered K up to 32 peaks.

Note that a larger K means that more bandwidth needs to be

provisioned in an MPLS network and thus a higher cost.

IV. COMPUTATIONAL RESULTS

We present below the results of a set of experiments

conducted on the robust optimization models described in this

paper. The main purposes of these experiments are:

1) to assess the utility of the proposed robust optimization

models as a means to obtain networks that are resilient

to at most K simultaneous peaks;

2) to compare networking technologies in terms of totalnetwork costs; and

3) to quantify the dependence of the network cost on the

number K of simultaneous peaks allowed;

K takes values in the set {1, 2, 4, 8, 16, 32}. These values are

only apparently small in comparison with |Q| = 4451: apart

from the reasons pointed out in the previous section, we also

have observed, in preliminary tests, that a further increase of

K does not change the results significantly: in several cases,

increasing K from 32 to 64 yields an increase of 1% or less

on the network cost, while changing K from 1 to 2 leads to

a steeper increase. Furthermore, numerous other experiments

have been conducted previously with smaller values of K , and

in practice even 32 is considered very conservative.

As we assume that the ratio between peak and average

volume is a constant α ≥ 1, i.e., vpeak q = αvavg

q ∀q ∈ Q,

all traffic matrices are described by the set

S = {vavgq (1 + (α − 1)µq), q ∈ Q :

q∈Q µq ≤ K, µq ∈ {0, 1}∀q ∈ Q}.

This definition of uncertainty is uniquely determined by αand K . Wrapping the definition of uncertainty around just

two parameters may seem too simplistic, but this allows us

to formulate a compact optimization model for the capacity

allocation problem. In order to accurately describe the real

uncertainty set, appropriate values of K and α are necessary.

It is barely worth noting here that the larger K and α, the more

conservative the result, i.e., the more expensive the network:

for very large K , we are allowing many traffic demands to

be at their peak value, therefore increasing the potential total

traffic on each link. If K = |Q|, the uncertainty set can

actually be reduced to a single point, as S is dominated by

a single traffic matrix: one in which all demands are at theirpeak value (for a detailed description of dominance in traffic

demands, see [29]). The opposite case is one in which K = 0(or equivalently α = 1), where S is given by the traffic matrix

in which all traffic demands are equal to their average value.

A. Implementation details

We have implemented our optimization models using the

AMPL modeling language [30], and solved them with the

Gurobi Mixed Integer Linear Programming solver1, a state-of-

the-art commercial solver which implements a parallel branch-

and-bound algorithm. All experiments were conducted on aLinux 64 bit machine with four processors, which are used

in parallel by the branch-and-bound algorithm, i.e., when the

branch-and-bound solver is operative, at most four branch-and-

bound nodes are solved simultaneously. All solver parameters

were set to their default value.

Integer Linear Programming problems are difficult, and

no polynomial-time solver is known for them [25] . As a

consequence, if the time allowed is limited, ILP solvers can

solve to optimality – i.e., provide a solution with cost zopt and

a proof that it is optimal – only instances of a certain size.

If the time allotted to search for an optimal solution is

scarce, ILP solvers attempt to provide a feasible, sub-optimal

solution, with cost zfeas ≥ zopt, and are always able to providea lower bound zlb ≤ zopt of the optimal solution. Depending

on the quality of the solver, the optimality gap zfeas−zlb

zlb(which

is zero when an optimal solution is found) provides a measure

of the quality of the solution found. In all our experiments, we

have imposed a time limit of two hours to all ILP instances

solved, namely those for the combined MPLS+OTN network.

We have observed that in all of these tests, either an optimal

solution was found or a relatively good one was found with

an optimality gap below 1%.

1See http://www.gurobi.com for more information.

http://www.gurobi.com/

http://www.gurobi.com/



11

B. Capacity allocation

Table I shows the results for the four technology combi-

nations considered: Bypass, Optimized MPLS+OTN, MPLS

only, and OTN only. For each value of K ∈ {1, 2, 4, 8, 16, 32}and α ∈ {1.5, 3, 4.5}, we report the network cost (7) for all

four alternatives and the corresponding number of 10 Gbps

links that need to be installed on each network. For the bypass

network, we also report the local traffic and the transit traffic(columns 3, 4, and 5, respectively). It is evident that MPLS

links, which fully exploit statistical multiplexing, use network

resources more efficiently and hence limit the increase in

network cost when the uncertainty increases (i.e. for large

values of K and α).

In Table II we provide similar results with one parameter

change: the cost ratio β between OTN and MPLS is set to one,

i.e., OTN and MPLS links are assumed to have equal cost in

this case. It comes as no surprise that the tradeoff between

MPLS routers and OTN switches favors MPLS networks even

more: when MPLS links and OTN links have equal cost, a

network composed solely of MPLS links is much cheaper

than the corresponding OTN network. We omit the results forthe optimized (combined) MPLS+OTN network as they are

unnecessary in this case: the combined network has an optimal

solution equal to that of the MPLS-only network.

Bypass MPLS OTN

Linksα K transit local cost cost links cost links

1.5 1 767 1750 32840 17420 1742 38460 38462 787 1820 33940 18100 18104 816 1898 35300 19040 19048 854 1986 36940 19740 1974

16 906 2098 39100 21060 210632 958 2218 41340 22060 2206

3 1 846 1996 36880 19900 1990 69340 69342 927 2236 40900 22260 2226

4 1048 2556 46520 25560 25568 1201 2930 53320 29340 2934

16 1398 3370 61660 33700 337032 1614 3836 70640 38300 3830

4.5 1 927 2246 41000 22380 2238 98980 98982 1074 2672 48200 26720 26724 1282 3232 57960 32320 32328 1549 3890 69880 38940 3894

16 1894 4646 84340 46460 464632 2269 5482 100200 54640 5464

TABLE IICOMPARISON OF NETWORK COSTS FOR SEVERAL TRAFFIC UNCERTAINTY

SETS, WHERE MPLS AND OTN LINKS HAVE THE SAME COST.

Finally, in Table III we summarize the previous two tablesby pointing out the percentage difference in cost obtained

when using MPLS nodes rather than OTN nodes. Note that

the the cost difference is large when β =gmpls

gotn= 1, for

higher values of α and low values of K (see e.g. the 77.3%

highlighted in the table), and viceversa the difference is

relatively small (although still more than 30%) when the cost

ratio is 1.3 and for small values of α and large values of K .

C. Approximating worst-case scenarios using fixed peaks

Another perspective on the robust optimization model pre-

sented above arises from considering a single scenario, albeit

OTN MPLS

K = 1 K = 2 K = 4

β α cost cost % cost % cost %

1.3 1.5 38460 22466 41.5 23170 39.7 23958 37.71.3 3 69340 24928 64.0 28784 58.4 32872 52.51.3 4.5 98980 29020 70.6 33424 66.2 41608 57.9

1 1.5 38460 17420 54.7 18100 52.9 19040 50.41 3 69340 19900 71.3 22260 67.8 25560 63.11 4.5 98980 22380 77.3 26720 73.0 32320 67.3

K = 8 K = 16 K = 32

β α cost cost % cost % cost %

1.3 1.5 38460 24396 36.5 25382 34.0 25980 32.41.3 3 69340 37072 46.5 41834 39.6 45984 33.61.3 4.5 98980 49340 50.1 58034 41.3 65848 33.4

1 1.5 38460 19740 48.6 21060 45.2 22060 42.61 3 69340 29340 57.6 33700 51.3 38300 44.71 4.5 98980 38940 60.6 46460 53.0 54640 44.7

TABLE IIICOST REDUCTION OF AN MPLS NETWORK WHEN COMPARED TO AN OTN

NETWORK. NOTE THAT β =gMPLS

gOTN.

a very conservative one: the K largest demands are set to their

peak value, while the remaining ones are at average value.

This is equivalent to restricting the uncertainty set S toone single traffic matrix, which is very limiting from the

opponent’s standpoint given that many scenarios are ruled out.

The clear advantage is that robust optimization is not necessary

to model this, as the traffic matrix simply needs to be set

according to the single scenario.

Extensive tests performed for all values of K to compare

the total network cost, in both cases, indicate that assuming

a single, although pessimistic, scenario produces very cheap

networks compared to the more conservative robust optimiza-

tion model. This can be explained by projecting the K largest

demands onto their routing paths, which probably do not cover

the whole set of links E and contribute a very small increase

of capacity on each link. This heuristic method can prove very

effective in larger networks if a solution is needed in shorter

time, but clearly no guarantee is provided on all other traffic

matrices of the uncertainty set. Other realistic scenarios, where

a disjoint set of smaller demands are at their peak value, would

overload the capacity installed on those links that are not used

in the routing paths of the largest K demands.

D. Equal Cost Multi-Path (ECMP) routing

Up to this point, we have assumed that all routing paths

follow the OSPF protocol and are established as shortest paths

according to weights provided as input. In order to obtain

Tables I-III, we have performed tests with single routing paths,i.e., we did not impose the Equal Cost Multi-Path (ECMP),

which splits part of the traffic if more than one shortest path

to destination is available.

We have also performed some tests (not reported here) by

first imposing ECMP at the MPLS layer only and then at both

layers, to address the effects on the network cost, and we

have observed no substantial difference. The OSPF weights

provided with the instance were such that few point-to-point

demands admit more than one shortest path. More precisely,

there are 20(20 − 1) = 380 pairs of core nodes in the core

network. Of these, only 10 pairs admitted more than one



12

Bypass Optimized MPLS OTN

Local traffic (Mbps) Links Linksα K in out transit MPLS OTN cost cost OTN MPLS cost Links cost Links

1.5 1 7248.3 7236.1 8595.9 767 1750 35141 22466 86 1662 22646 1742 38460 38462 7451.6 7423.2 8893.2 787 1820 36301 23170 458 1430 23530 18104 7738.3 7698.4 9289.9 816 1898 37748 23958 828 1206 24752 19048 8079.6 8077.0 9754.9 854 1986 39502 24396 994 1112 25662 1974

16 8534.0 8559.4 10295.2 906 2098 41818 25382 1038 1154 27378 210632 9051.8 9068.0 10894.2 958 2218 44214 25980 1116 1140 28678 2206

3 1 8037.1 7988.5 9831.1 846 1996 39418 24928 106 1836 25870 1990 69340 69342 8850.5 8737.0 11017.0 927 2236 43681 28784 86 2148 28938 22264 9997.2 9837.4 12607.5 1048 2556 49664 32872 240 2344 33228 25568 11362.5 11351.8 14472.7 1201 2930 56923 37072 868 2184 38142 2934

16 13179.7 13281.1 16637.3 1398 3370 65854 41834 1482 2078 43810 337032 15250.6 15315.4 19023.3 1614 3836 75482 45984 2482 1628 49790 3830

4.5 1 8826.0 8740.8 11067.6 927 2246 43781 29020 94 2160 29094 2238 98980 98982 10249.5 10050.8 13146.3 1074 2672 51422 33424 134 2468 34736 26724 12256.1 11976.4 15964.7 1282 3232 61806 41608 344 2936 42016 32328 14645.4 14626.7 19263.0 1549 3890 74527 49340 1190 2880 50622 3894

16 17825.6 18003.1 23053.9 1894 4646 90022 58034 2010 2918 60398 464632 21449.7 21563.1 27219.6 2269 5482 107007 65848 3470 2396 71032 5464

TABLE ICOMPARISON OF NETWORK COSTS FOR SEVERAL TRAFFIC UNCERTAINTY SETS . THE MPLS/OTN RATIO OF LINK COST IS 1.3, I.E., MPLS PORTS COST

30% MORE THAN OTN PORTS. NOTE THAT THE DATA FOR OTN NETWORKS DOES NOT DEPEND ON K .

shortest path. Because the experiments under ECMP did not

show big changes and because ECMP is outside the scope of

this work, we omit the full results.

E. Reducing the overall demand upon multiple peaks

The set S of potential traffic matrices, a crucial concept in

our model, is in general a polyhedron and can be modified if

needed. We consider an example of uncertainty set that avoids

over-conservativeness.

The model we have discussed so far assumes that at most

K demands may take on a peak value. At any moment, if

the number of demands at peak value is small compared toK , link usage (and thus congestion) is well below the limits

determined by the network capacity. However, if close to K demands are at peak value, congestion and capacity occupation

figures can be critical and network service might be at risk.

To prevent this risk, or to limit the increase in network cost

associated with large K , the network planner and the network

users may enter contractual agreements dictating that all net-

work users accept to reduce their demand, no matter whether

it is at peak value or not, by a given (small) percentage, which

varies with the number of current peak demands. In other

words, if, at a certain instant, few or none of the demands

are at peak value, the network functions normally and users

can communicate at the requested data rate. However, with agrowing number of peak demands, all users will transmit at

a slightly reduced rate. A reduction defined by a parameter ϑ(typically ϑ ∈ {5%, 10%}) can be considered, and it is fully

enforced when the number of peak demands is exactly K . As

a result, total network throughput can be decreased artificially

by reducing the volume of all demands, while, if all demands

are at average value, they are left unchanged.

The rationale behind this change in the uncertainty set S is

the need to reduce the cost of a network whose traffic demands

are, most of the time, all at average value. In the relatively rare

bursty periods, with K demands at peak value, it becomes

reasonable to decrease every demand for the (presumably

small) duration of the burst phase. If K ′ < K demands are at

their peak value, a smaller reduction of ϑK ′

K can be enforced.

This uncertainty set can be modeled with linear constraints.

Define a set of binary variables µq , for each q ∈ Q, indicating

whether demand q is at average value (µq = 0) or peak value

(µq = 1), and a variable Z equal to the ratio of the number of

demands at peak value to K . The uncertainty set is defined by

the set of vectors v ∈ R|Q|+ subject to the following constraints

(which will be replicated for all ij ∈ E ):

Z =

q∈Q µq/K (8)

vq ≥ (1 − ϑZ )vavgq ∀q ∈ Q (9)

vq ≤ α(1 − ϑZ )vavgq ∀q ∈ Q (10)

vq ≤ (1 + (α − 1)µq)vavgq ∀q ∈ Q (11)

q∈Q µq ≤ K (12)

µq ∈ {0, 1} ∀q ∈ Q.

These constraints reduce to the simpler set of constraints of

the maximization model in (5) when ϑ = 0. Analogously to

the robust optimization problem discussed above, the dual of

the problem of maximizing

q∈Q f qµqδ q subject to the above

constraints gives a set of linear constraints that can replace the

nonlinear capacity constraint.

The continuous LP relaxation of the above problem is ob-

tained by replacing the last constraint with µq ∈ [0, 1] ∀q ∈ Q.

Let us use the set of dual variables ξ ij , ρqij , σqij , ηqij , and ωij for

constraints (8)-(12) and λqij for the upper bound constraint on

µq, i.e., µq ≤ 1. As a consequence, in the robust optimization

model we must replace, for each ij ∈ E , the nonlinear



13

constraint (5) with the following set of constraints:

zij ≥

q∈Q vavgq (−ρqij+

ασqij + ηqij) + Kωij +

q∈Q λqij ∀ij ∈ E

ξ ij − ϑ

q∈Q vavgq ρqij+

α

q∈Q vavgq σqij = 0 ∀ij ∈ E

− 1K

ξ ij + ωij + λqij−vavg

q (α − 1)ηqij ≤ δ qf q ∀ij ∈ E, ∀q ∈ Q

−ρqij + σqij + ηqij = 0 ∀ij ∈ E, ∀q ∈ Qρqij , σq

ij , ηqij , ωij , λqij ≥ 0 ∀ij ∈ E, ∀q ∈ Q.

Note that the larger set of constraints in problem P ij results

in a larger set of dual constraints and therefore a larger overall

robust optimization problem. This results in a MILP problem

that is similar in structure to that presented in Section III, but

more difficult to solve.

We have conducted some experiments using this model

as well, but only report a summary here for the sake of

conciseness. For ϑ = 0.1, i.e., an allowed reduction of 10%,

we have observed a decrease in the network cost between 13%

and 15% for the MPLS-only network, and a decrease of about

10% for the MPLS+OTN network. For ϑ = 0.05, instead, thedecrease is between 8% and 9% for the MPLS-only network

and 6% for the MPLS+OTN network.

F. Transponders and network cost

Even though we have so far considered node equipment

only, when factoring link costs in the model the comparison

becomes even more in favor of MPLS. In fact, one transponder

for each 10Gbps link has to be added to the network (see RFC

3031 [31]), both at the MPLS and at the OTN level.

Given that the cost of transponders is comparable to that

of 10 Gbps links and that there is no cost differential for

transponders used at either level, the effect will be that of asmaller equivalent cost ratio between MPLS and OTN ports,

i.e., a smaller β . As a consequence, the savings obtained from

an MPLS-only network or an optimized MPLS+OTN network

will be even larger than those discussed above.

V. CONCLUDING REMARKS

We have introduced a set of optimization models for the

allocation of capacity on networks with multiple layers. Tech-

nologies vary in cost and capability, and when more than one

are available it is difficult to choose one or a combination of

them that provides guaranteed service at a low cost.

Our models yield the most appropriate combination of

technologies and result in a network that serves multiple traffic

matrices. This class of models handles traffic uncertainty, and

computational experiments on a realistic network instance

show that they can provide a provably optimal solution in

reasonable computing times. Also, these models are able

to guarantee that all demands in a given uncertainty set,

characterized by simple parameters, are satisfied.

A. Open questions

There are several possible extensions to this work. First,

we have focused mostly on node equipment and considered

only node costs, but it might be of interest to investigate

further the effect of link costs on the network provisioning

process, for instance by explicitly taking transponder costs into

consideration.

Second, while OSPF weights are given as an input in our

problem, the network cost might benefit from a scenario in

which the network operator that designs the network is also

allowed to set the OSPF weights. This would render the model

significantly harder, but would also push optimization even

further. Previous attempts [32, 33] have shown that OSPF

weights have a significant influence on some network perfor-

mance indicators (congestion and routing cost respectively).

Third, it might be of interest to check whether relaxing

the requirement that all routing paths be shortest with respect

to OSPF weights further improves the network performance

(again, see [32]).

Partially related to this is the question whether, in networks

that have a high degree of ECMP (unlike the network in-

stance we have used), multiple shortest paths would make

a significant difference in the network cost. In fact, in SP

networks, ECMP is very common, both for load balancingand for resilience (if a link fails, having another path with the

same metric leads to less churn in the network). As a result,

every pair of edge routers has at least two paths, and often

many more.

Lastly, we have not considered the costs of adding fault

tolerance (or high-availability) in all models. Circuit networks

require end-to-end redundant configuration for managing fail-

ures (more than twice the cost of the non-redundant network).

MPLS networks allow for fast detection and reroute of traffic

during failure with discriminatory treatment for high priority

traffic, thus allowing for more efficient local link protection

and correspondingly lower costs.

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http://www.ietf.org/rfc/rfc3031.txt

http://www.ietf.org/rfc/rfc3031.txt

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