Optical Networks Virtual Topology Design - UFPA · Optical Networks, VTD, Cicek Cavdar, KTH General Problem Statement ! Number of wavelengths carried by each fiber, M. An N × N traffic

Optical Networks Virtual Topology Design

5 September 2012, Belem, Para, Brazil Dr. Cicek Cavdar, [email protected]

Optical Networks Lab (ONLab) Royal Institute of Technology, Stockholm, Sweden

Special thanks to Biswanath Mukherjee from UC-Davis, Aysegul Yayimli from ITU, for the class material.

Optical Networks, VTD, Cicek Cavdar, KTH

Virtual Topology n  A lightpath provides “single-hop” communication between

any two nodes, which could be far apart in the physical topology.

n  However, having limited number of wavelengths, it may not be possible to set up lightpaths between all user pairs.

n  Multi-hopping between lightpaths may be necessary. n  The virtual topology consists of a set of lightpaths.

¨  packets of information are carried by the virtual topology “as far as possible” in the optical domain using optical circuit switching

¨  packet forwarding from lightpath to lightpath is performed via electronic packet switching, whenever required.

n  Lightpaths in the virtual topology is set up using RWA techniques.

n  The virtual topology is also referred to as Lambda Grid, or just Grid.


Problem n  An optimization problem to optimally select a virtual

topology subject to ¨  transceiver (transmitter and receiver) ¨  wavelength constraints

n  with one of two possible objective functions: 1.  for a given traffic matrix, minimize the network-wide average

packet delay. 2.  maximize the scale factor by which the traffic matrix can be

scaled up (to provide the maximum capacity upgrade for future traffic demands).

n  Since the objective functions are nonlinear and since simpler versions of the problem have been shown to be NP-hard, we shall explore heuristic approaches.


NSFNET Backbone


NSFNET (1991) n  Information is transferred over this backbone as packets

at a rate of 1.544 Mbps per link. n  The backbone consists of over a dozen nodes, each

containing one or more computers operating as electronic packet switches.

n  These switches are connected with one another via optical fibers to form an irregular mesh structure.

n  Although fiber is employed to connect the nodes, the fiber’s tremendous transmission bandwidth is not exploited. ¨  Data transmission on each fiber link is performed only at T1

(1.544 Mbps) rate on a single wavelength.


General Problem Statement Problem: Embedding a desired virtual topology on a given

physical topology (fiber network). We are given: n  A physical topology Gp = (V ,Ep) consisting of a weighted

undirected graph, where ¨  V is the set of network nodes, ¨  Ep is the set of links connecting the nodes.

Undirected means that each link in the physical topology is bidirectional. A node i is equipped with a Dp(i) × Dp(i) WRS, where Dp(i) is the number of physical fiber links emanating out of (as well as terminating at) node i.


General Problem Statement n  Number of wavelengths carried by each fiber, M. n  An N × N traffic matrix, where

¨ N is the number of network nodes, ¨ The (i, j)-th element is the average rate of packet

traffic flow from node i to node j. The traffic flows may be asymmetric.

n  The number of wavelength-tunable lasers (transmitters) and wavelength-tunable filters (receivers) at each node.


General Problem Statement The goal is to determine: n  A virtual topology Gv = (V ,Ev) as another graph

where: ¨  the out-degree of a node is the number of transmitters at that

node ¨  the in-degree of a node is the number of receivers at that node. ¨  The nodes of the virtual topology correspond to the nodes in the

physical topology. ¨  Each link in the virtual topology corresponds to a lightpath

between the corresponding nodes in the physical topology. ¨  Each lightpath may be routed over one of several possible paths

on the physical topology.


General Problem Statement n  A wavelength assignment for lightpaths.

¨  If two lightpaths share a common physical link, they must necessarily employ different wavelengths.

n  The sizes and configurations of the WRSs at the intermediate nodes. ¨ Once the virtual topology is determined and the

wavelength assignments have been performed, the switch sizes and configurations follow directly.


Packet Communication n  Communication between any two nodes takes

place by following a path (a sequence of lightpaths) from the source node to the destination node on the virtual topology.

n  Each intermediate node in the path must perform:

1.  an opto-electronic conversion, 2.  electronic routing (or packet switching in the

electronic domain), and 3.  electro-optic forwarding onto the next lightpath.


Illustrative Example n  How WDM can be used to upgrade an existing

fiber-based network. n  Example:

¨ A hypercube to be embedded as a virtual topology over the NSFNET physical topology.

n  We assume an undirected virtual topology. ¨ bidirectional lightpaths

n  In general, the virtual topology may be a directed graph.

n  The physical topology is enhanced by adding two fictitious nodes, AB and XY.


Illustrative Example n  The switching architecture of nodes consists of:

¨ An optical component. n  a wavelength-routing switch (WRS) n  can switch some lightpaths, n  can locally terminate some other lightpaths by directing them

to node’s electronic component.

¨ An electronic component. n  an electronic packet router

(may be an IP router: IP-over-WDM) n  serves as a store-and forward electronic overlay on top of the

optical virtual topology.


Example n  The virtual topology chosen is a 16-node

hypercube.


Example: A Possible Embedding


Example n  This solution requires 7 wavelengths. n  Each link in the virtual topology is a lightpath with electronic

terminations at its two ends only. Example: n  The CA1-NE lightpath could be set up as an optical channel on one

of several possible wavelengths on one of several possible physical paths: ¨  CA1-UT-CO-NE, or CA1-WA-IL-NE, or others.

n  According to the solution, the first path is chosen on wavelength 2 for CA1-NE lightpath.

n  This means that the WRSs at the UT and CO nodes must be properly configured to establish this CA1-NE lightpath.

n  The switch at UT must have wavelength 2 on its fiber to CA1 connected to wavelength 2 on its fiber to CO.

n  Since connections are bidirectional, the CA1-NE connection implies two lightpaths, one from CA1 to NE and one from NE to CA1.


UT Node


UT Node n  The switch UT has to support four incoming fibers plus

four outgoing fibers, ¨  one each to nodes AB, CA1, CO, and MI, as dictated by the

physical topology.

n  In general, each switch interfaces with four lasers (inputs) and four filters (outputs), ¨  each laser-filter pair is dedicated to accommodate each of the

four virtual links on the virtual topology.

n  The labels 1l 2b 3d 5l on the output fiber to CO: ¨  The UT-CO fiber uses four wavelengths 1, 2, 3, and 5. ¨  Wavelengths 2 and 3 are “clear channels” through the UT switch

and directed to the physical neighbors CA1 and MI, respectively. ¨  Wavelengths 1 and 5 connect to two local lasers.


UT Node n  The box labeled Router is an electronic switch which

takes information from: ¨  terminated lightpaths (1c 4b 5c) ¨  a local source

and routes them – via electronic packet switching – to: ¨  the local destination ¨  the local lasers (lightpath originators)

n  The router can be any electronic switch. ¨  e.g., an IP router.

n  The non-router portions of the node architecture are the optical parts that may be incorporated to upgrade the electronic switch to incorporate a WRS.


Formulation of the Optimization Problem Notation: n  s and d used as subscript or superscript to

denote source and destination of a packet, respectively.

n  i and j denote originating and terminating nodes, respectively, in a lightpath.

n  m and n denote endpoints of a physical link.


Formulation: Given Given: n  Number of nodes in the network: N. n  Maximum number of wavelengths per fiber: M n  Physical topology Pmn

¨ Pmn = Pnm = 1 if there is a direct physical fiber link between nodes m and n

¨ Pmn = Pnm = 0 otherwise ¨ The problem can be generalized to accommodate

multi-fiber networks, where Pmn can take integer values.


Formulation: Given n  Distance matrix whose elements are fiber

distance dmn from node m to node n. ¨ For simplicity in expressing packet delays, dmn is

expressed as a propagation delay (in time units). ¨ dmn = dnm ¨  dmn = 0 if Pmn = 0.

n  Number of transmitters at node i = Ti (Ti ≥ 1). n  Number of receivers at node i = Ri (Ri ≥ 1). n  Capacity of each channel: C

¨ normally expressed in bits per second, but converted to units of packets per second by knowing the mean packet length.


Formulation: Given n  Traffic matrix λsd

¨  the average rate of traffic flow from node s to node d ¨ λss = 0 ¨ Additional assumptions:

n  Packet inter-arrival durations at node s and packet lengths are exponentially distributed.

n  So standard M/M/1 queuing results can be applied to each network link (or “hop”) by employing the independence assumption on inter-arrivals and packet lengths due to traffic multiplexing at intermediate hops.

n  By knowing the mean packet length (in bits per packet), the λsd can be expressed in units of packets per second.


Formulation: Variables Variables: n  Virtual topology Vij:

¨ 1 if there is a lightpath from i to j in the virtual topology ¨ 0 otherwise. ¨ The formulation is general since lightpaths are not

necessarily assumed to be bidirectional. n  Vij = 1 ⇏ Vji = 1.

¨ Further generalization of the problem can be performed by allowing multiple lightpaths between node pairs, i.e., Vij > 1.


Formulation: Variables n  Traffic routing variables λsd

ij ¨ denotes the traffic flowing from s to d and employing

Vij as an intermediate virtual link. ¨  traffic from s to d may be “bifurcated” with different

fractions taking different sets of lightpaths. n  Physical-topology route variables pij

mn is: ¨ 1 if the fiber link Pmn is used in the lightpath Vij ¨ 0 otherwise.

n  Wavelength color variables cijk is:

¨ 1 if a lightpath from i to j is assigned the color k ¨ 0 otherwise.


Formulation: Constraints Constraints: n  On virtual-topology connection matrix Vij :

n  These equations ensure that: ¨ The number of lightpaths emanating out of and ¨  terminating at a node

are at most equal to that node’s out-degree and in-degree, respectively.


Formulation: Constraints n  On physical route variables pij

mn:

n  First two equations constrain the problem so that pijmn

exist only if there is a fiber (m,n) and a lightpath (i,j). n  The remaining equations are the multi-commodity

equations that account for the routing of a lightpath from its origin to its termination.


Formulation: Constraints n  On virtual-topology traffic variables λsd

ij :

Equations for the routing of packet traffic on the virtual topology.

They take into account that the combined traffic flowing through a channel cannot exceed the channel capacity.


Formulation: Constraints n  On coloring of lightpaths cij

k :

n  First equation requires that a lightpath be of one color only.

n  Second equation ensures that the colors used in different lightpaths are mutually exclusive over a physical link.


Formulation: Objective 1 Delay Minimization:

n  The innermost brackets: ¨  the first component corresponds to the propagation delays on

the links mn which form the lightpath ij ¨  the second component corresponds to delay due to queuing and

packet transmission on lightpath ij. n  If we assume shortest-path routing of the lightpaths over

the physical topology, then the pijmn values become

deterministic. n  If, in addition, we neglect queuing delays, the

optimization problem reduces to minimizing the first component.


Formulation: Objective 2 Maximizing Load (Minimizing Maximum Flow):

n  Also nonlinear n  Minimizes the maximum amount of traffic that

flows through any lightpath. n  Corresponds to obtaining a virtual topology

which can maximize the offered load to the network if the traffic matrix is allowed to be scaled up.


Algorithms for VT Design n  The problem of optimal virtual-topology design

can be partitioned into the following four sub-problems, which are not necessarily independent:

1.  Determine a good virtual topology. n  Which transmitter should be directly connected to which

receiver?

2.  Route the lightpaths over the physical topology. 3.  Assign wavelengths optimally to the various

lightpaths. 4.  Route packet traffic on the virtual topology.


Solutions n  Several heuristic approaches have been employed to

solve these problems. ¨  Labourdette and Acampora, “Logically rearrangeable multihop

lightwave networks,” IEEE Transactions on Comm., Aug. 1991. ¨  I. Chlamtac, A. Ganz, and G. Karmi, “Lightnets: Topologies for

high speed optical networks,” IEEE/OSA Journal of Lightwave Technology, May/June 1993.

¨  B. Mukherjee, S. Ramamurthy, D. Banerjee, and A. Mukherjee, “Some principles for designing a wide-area optical network,” Proceedings, IEEE INFOCOM ’94, June 1994.

¨  R. Ramaswami and K. Sivarajan, “Design of logical topologies for wavelength-routed all-optical networks,” Proceedings, IEEE INFOCOM ’95, April 1995.

¨  Z. Zhang and A. Acampora, “A heuristic wavelength assignment algorithm for multihop WDM networks with wavelength routing and wavelength reuse,” IEEE/ACM Transactions on Networking, vol. 3, pp. 281–288, June 1995.


Solutions n  Embedding of a packet-switched virtual topology

on a physical fiber plant in a switched network was first introduced in the second reference, and this network architecture was referred to as a lightnet.

n  The work in ref. 4 proposes a virtual-topology design where the average hop distance is minimized, which automatically increases the network traffic supported. This work uses the physical topology as a subset of the virtual topology.


Solution Approach n  To obtain a thorough understanding of the problem, we

concentrate on Sub-problems 1 and 4 above. ¨  the number of available wavelengths is not a constraint. ¨  In the expanded problem, both the number of wavelengths and

their exact assignments are critical.

n  An iterative approach consisting of “simulated annealing” to search for a good virtual topology (Sub-problem 1)

n  The “flow-deviation” algorithm for optimal (possibly “bifurcated”) routing of packet traffic on the virtual topology (Sub-problem 4).


Solution Approach n  We consider lightpaths to be bidirectional

¨ most (Internet) network protocols rely on bidirectional paths and links.

n  We consider Optimization Criterion (2) (maximizing offered load) for illustrative solution. ¨ mainly because we are interested in upgrading an

existing fiber-based network to a WDM solution.


Simulated Annealing n  Simulated annealing (along with genetic algorithms) has been found

to provide good solutions for complex optimization problems. n  In the simulated annealing process, the algorithm starts with an

initial random configuration for the virtual topology. n  Node-exchange operations are used to arrive at neighboring

configurations. n  In a node-exchange operation, adjacent nodes in the virtual

topology are examined for swapping. n  Example:

¨  if node i is connected to nodes j, a, and b, ¨  while node j is connected to nodes p, q, and i in the virtual topology, after the node-exchange operation between nodes i and j, ¨  node i will be connected to nodes p, q, and j, ¨  while node j will be connected to nodes a, b, and i.


Simulated Annealing n  Neighboring configurations which give better results

(lower average packet delay) than the current solution are accepted automatically.

n  Solutions which are worse than the current one are accepted with a certain probability. ¨  This probability is determined by a system control parameter.

n  The probability with which these failed configurations are chosen decreases as the algorithm progresses in time so as to simulate the “cooling” process of annealing.

n  The probability of acceptance is based on a negative exponential factor ¨  inversely proportional to the difference between the current

solution and the best solution obtained so far.


Simulated Annealing n  The initial stages of the annealing process examine

random configurations in the search space ¨  to obtain different initial starting configurations without getting

stuck at a local minimum as in a greedy approach.

n  However, as time progresses, the probability of accepting bad solutions goes down, ¨  the algorithm settles down into a minimum, after several

iterations.

n  The state become “frozen” when there is no improvement in the objective function of the solution after a large number of iterations.


Flow Deviation n  By properly adjusting link flows, the flow-deviation

algorithm provides an optimal algorithm for minimizing the network-wide average packet delay.

n  Traffic from a given source to a destination may be bifurcated. ¨  different fractions of it may be routed along different paths to

minimize the packet delay.

n  Idea: If the flows are not balanced, then excessively loading of a particular channel may lead to large delays on that channel ¨  and thus have a negative influence on the network-wide average

packet delay.


Flow Deviation n  The algorithm is based on the notion of shortest-path

flows. ¨  First calculates the linear rate of increase in the delay with an

infinitesimal increase in the flow on any particular channel. ¨  These “lengths” or “cost rates” are used to pose a shortest-path

flow problem (can be solved using one of several well-known algorithms such as Dijkstra’s algorithm, Bellman-Ford algorithm, etc.)

¨  The resulting paths represent the “cheapest” paths on which some of the flow may be deviated.

n  An iterative algorithm determines how much of the original flow needs to be deviated.

n  The algorithm continues until a certain performance tolerance level is reached.


Experimental Results n  The traffic matrix employed is an actual measurement of

the traffic on the NSFNET backbone for a 15-minute period. ¨  11:45 pm to midnight on January 12, 1992, EST.

n  The raw traffic matrix shows traffic flow in bytes per 15-minute intervals between network nodes.

n  Nodal distances used are the actual geographical distances.

n  Initially, each node can set up at most four bidirectional lightpath channels.

n  Later more experiments were conducted to study the effect of having higher nodal degree.

n  The number of wavelengths per fiber was assumed to be large enough. ¨  all possible virtual topologies could be embedded.


Traffic Matrix


Experimental Results n  For each experiment,

¨  the maximum scale-up achieved ¨  the corresponding individual delay components, ¨  the maximum and minimum link loading ¨  the average hop distance

is tabulated.


Experimental Results n  The aggregate capacity for the carried traffic is fixed by

the number of links in the network. ¨  reducing the average hop distance can lead to higher values of

load that the network can carry. n  The queuing delay was calculated using a standard M/M/

1 queuing system. ¨  mean packet length calculated from the measured traffic:

133.54 bytes per packet. ¨  link speed is 45 Mbps.

n  Infinite buffers at all nodes. n  The “cooling” parameter for the simulated annealing is

updated after every 100 acceptances using a geometric parameter of value 0.9.

n  A state is considered “frozen” when there is no improvement over 100 consecutive trials.


Physical Topology as Virtual Topology (No WDM) n  Goal: to obtain a fair estimate of what optical hardware

can provide in terms of extra capabilities. n  Start off with just the existing hardware, comprising:

¨  fiber and point-to-point connections ¨  a single bidirectional lightpath channel per fiber link ¨  no WDM

n  The maximum scale-up achieved: 49 n  The load of the link with the maximum traffic: 98% n  The load of the link with the minimum traffic: 32%. n  These values serve as a basis for comparison as to what

can be gained in terms of throughput by adding extra WDM optical hardware: ¨  tunable transceivers ¨  wavelength routing switches.


Multiple Point-to-Point Links (No WRS) n  Goal: to determine how much throughput we could

obtain from the network: ¨  without adding any photonic switching capability at a node ¨  by adding extra transceivers (up to four) at each node

n  The initial network had 21 bidirectional links in the physical topology.

n  Using extra transceivers at the nodes, extra links are set up on the paths NE-CO, NE-IL, WA-CA2, CA1-UT, MI-NJ, and NY-MD.

n  These lightpaths are chosen manually. ¨  Different combinations were considered. ¨  The channels providing the maximum scale-up was chosen.


Arbitrary Virtual Topology (Full WDM) n  Full WDM with WRSs at all nodes.

¨  It is possible to set up lightpaths between any two nodes.

n  All lightpaths are routed over the shortest path on the physical topology.

n  Starting off with a random initial topology, simulated annealing is used to get the best virtual topology. ¨  shown in the table.

n  Provides a maximum scale-up of 106. n  The increased scale-up demonstrates

the benefits of the WDM-based virtual-topology approach.


Effect of Nodal Degree and Wavelength Requirements n  If we consider full WDM, and increase the nodal degree

to five and six, we find that the maximum scale-up increases nearly proportionally with increasing nodal degree.

n  In the experiments, the observed maximum scale-ups: ¨  for P = 5: 135 ¨  For P = 6: 163

n  As the nodal degree is increased, the average hop distance of the virtual topology is reduced. ¨  This provides the extra improvement in the scale-up.

n  Minimizing hop distance can be an important optimization problem.


Open Problems n  A significant amount of room exists for

developing improved approaches and algorithms.

n  An interesting avenue of research is to study how routing and wavelength assignment of lightpaths can be combined with the choice of virtual topology and its corresponding packet routing in order to arrive at an optimum solution.

n  Dynamic establishment and reconfiguration of lightpaths is an important issue which needs to be thoroughly studied.


A Linear Objective Function n  The objective functions in the previous ILP

formulations were non-linear. n  A new objective function can be derived:

n  This function minimizes the average packet hop distance in the network and it is linear.


Simplifying Assumptions n  The number of variables and equations in the

formulation can be reduced. ¨  This number, in the original problem formulation grows as O(N4),

and can easily overwhelm today’s state-of-the-art computing facilities.

¨  To make the problem more tractable, we can reduce the number of constraints by pruning the search space.

¨  Pruning is based on tracking a limited number of alternate shortest paths between source-destination pairs.

¨  The selected routes are within a constant factor (α ≥ 1) of the shortest-path distance between the given source-destination pair.


Illustrative Examples n  Numerical examples of the virtual topology

network design problem, using the NSFNET illustrative backbone as the physical topology. ¨ 14 nodes connected in a mesh network. ¨ Each of its links are bidirectional.

n  The number of transmitters is equal to the number of receivers, and is the same for all nodes.


Illustrative Examples n  The traffic matrix is randomly generated. n  A certain fraction F of the traffic is uniformly distributed

over the range [0,C/a]. n  The remaining traffic is uniformly distributed over [0,CΥ/

a]. n  C: the lightpath channel capacity, n  a : an arbitrary integer which may be 1 or greater, n  Υ denotes the average ratio of traffic intensities between

node pairs with high traffic values and node-pairs with low traffic values.

n  This model allows to generate traffic patterns with varying characteristics.


Illustrative Examples n  Figures plot system characteristics averaged

over 25 different virtual topologies. ¨ each corresponding to an independent traffic matrix.

n  Traffic parameters C = 1250, a = 20,Υ = 10, F = 0.7,K = 2, and α = 2.

n  Ti and Ri were allowed values between 4 and 8. n  W was allowed to take values between 1 (no

WDM) and 7.


Average Hop Distance n  The average hop distance in the network is a function of the number

of lightpaths set up in the network. ¨  which directly depends on the number of transceivers and wavelengths

supported. n  The case corresponding to one wavelength in the system

corresponds to a point-to-point network (no WDM). n  The average hop distance decreases with a balanced increase in

the number of transceivers and wavelengths in the network. n  Increasing transceivers without adding extra wavelengths marginally

improves the quality of the solution. n  For more than six transceivers, and more than four wavelengths, the

performance improvement is marginal for the chosen network. n  Ideally, we would like the AHD to be as close as possible to its lower

bound of unity.


Average packet hop distance for the optimal solution


Average transceiver utilization n  Next figure plots the transceiver utilization for

different values of the number of wavelengths in the system, and number of transceivers at a node.

n  As one expects, it quantitatively demonstrates that the transceiver utilization decreases as: ¨  the number of wavelengths is reduced ¨ and/or the number of transceivers is increased.


Average transceiver utilization


Average wavelength utilization n  Next figure plots the wavelength utilization for

the same set of experiments. n  It demonstrates that the wavelength utilization

decreases: ¨ when the number of wavelengths is increased ¨ and/or the number of transceivers is reduced.

n  These results confirm that it is necessary to obtain the correct balance between transceivers and wavelengths in the system to properly utilize both of these expensive resources.


Average wavelength utilization


Heuristic Approaches n  Two heuristic approaches that allow us to solve

large problem instances of the virtual-topology design problem.

n  Results from these heuristics compare favorably with the optimal result obtained by solving the exact problem formulation.

n  Maximizing Single-Hop Traffic. n  Maximizing Multi-hop Traffic.


Maximizing Single-hop Traffic n  Attempts to establish lightpaths between node pairs with

the highest Λsd values subject to: ¨  the number of transceivers at the end nodes, ¨  and the availability of a wavelength.


Maximizing Multi-hop Traffic n  In a packet-switched network, the traffic carried by a link

may include: ¨  forwarded traffic, and ¨  traffic originating from that node.

n  Intuitively, any lightpath establishment heuristic which accounts for the forwarded traffic that the lightpath will carry, should provide better performance than a heuristic which only tries to maximize the single-hop traffic.

n  The heuristic starts with the physical topology as the initial virtual topology.

n  It attempts to add more lightpaths one by one. n  The performance of this heuristic is found to be slightly

better than that of the previous heuristic.


Maximizing Multi-hop Traffic n  Hsd denotes the number of electronic hops

needed to send a packet from s to d. n  The heuristic attempts to establish lightpaths in

decreasing order of Λsd(Hsd − 1), subject to ¨  the number of transceivers at the two end nodes, and ¨  the availability of a wavelength.

n  After each lightpath is established, Hsd values are recalculated, as traffic flows might have changed due to the new lightpath.


Maximizing Multi-hop Traffic


Results n  The next table shows the average hop distance for the

two heuristic approaches as compared to the optimal solution.

n  The same sample of 25 traffic matrices were used to evaluate the performance of the heuristics.

n  The following figure plots these performance results for a four wavelength system.

n  The average hop distance decreases with an increase in the number of transceivers in the system.

n  The heuristics perform a little poorly relative to the LP’s optimal solution which can be treated as a lower bound.

n  Also, the heuristic which maximizes the multi-hop traffic performs slightly better than the heuristic which maximizes single-hop traffic for smaller number of transceivers.



Results

？

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Optical Networks Virtual Topology Design - UFPA · Optical Networks, VTD, Cicek Cavdar, KTH General Problem Statement ! Number of wavelengths carried by each fiber, M. An N × N traffic