OPTICAL NETWORKS - Anasayfagencata/courses/ON/ONslides/... · 2013. 9. 13. · Optical Networks...

A. GençataİTÜ, Dept. Computer Engineering

2005

OPTICAL NETWORKS

Routing and Wavelength Assignment in WDM Wide Area Networks

Optical NetworksAyşegül Gençata, İTÜ, Dept. Computer Engineering

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Wide-area Optical NetworksWide-area (long-haul) optical networks

nodes employ optical cross-connects (OXCs) WDM channels called lightpaths are established between node pairs.

The terms lightpath and connection are interchangeable. To establish a “connection”between a source-destination pair, we need to set up a “lightpath” between them.


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LightpathA lightpath may span multiple fiber links.It provides a “circuit switched” interconnection between two nodes that:

have a traffic flow between them and are located “far” from each other in the physical fiber network topology.

Each intermediate node in the lightpath provides a circuit-switched optical bypass facility to support the lightpath.The network should be scalable.Transceivers are expensive so that each node may be equipped with only a few of them.Number of wavelengths is limited.Therefore, only a limited number of lightpaths may be set up on the network.


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RWAThe offered traffic is “circuit-oriented”

the offered traffic consists of a set of connectionseach connection requires the full bandwidth of a lightpath to be routed between its corresponding source-destination pair.

Once a set of lightpaths has been chosen or determined, we need to:

route each lightpath in the network, and assign a wavelength to it.

This is referred to as the routing and wavelength assignment (RWA) problem.


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Problem StatementFormally, the RWA problem can be stated as follows. Given:

a set of lightpaths that need to be established on the network, a constraint on the number of wavelengths,

Determine:the routes over which these lightpaths should be set up the wavelengths which should be assigned to these lightpaths

so that the maximum number of lightpaths may be established.Lightpaths that cannot be set up due to constraints on routes and wavelengths are said to be blocked, the corresponding network optimization problem is to minimize this blocking probability.


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Wavelength ContinuityNormally, a lightpath operates on the same wavelength across all fiber links that it traverses.

the lightpath is said to satisfy the wavelength-continuity constraint.

Thus, two lightpaths that share a common fiber link should not be assigned the same wavelength.If a switching/routing node is also equipped with a wavelength converter facility, then the wavelength-continuity constraints disappear.

a lightpath may switch between different wavelengths on its route from its origin to its termination.


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Connection RequestsConnection requests may be of three types:

Static: The entire set of connections is known in advance.Set up lightpaths for the connections in a global fashion while minimizing network resourcesKnown as static lightpath establishment (SLE) problem

Incremental:Connection requests arrive sequentially and remains in the network indefinitely.

Dynamic:A lightpath is set up for each connection request as it arrives, and it is released after some amount of time.Known as dynamic lightpath establishment (DLE).


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Solving the Static RWAPhysical topology and lightpath requests are known.Offline RWAThe objective is to minimize the number of wavelengths.SLE can be formulated as an integer linear program (ILP).

Objective: minimize the flow in each linkmeans minimizing the number of lightpaths passing through a link.

The general problem is NP-complete.


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Solution to SLEApproximation algorithms to solve the RWA problem to solve the RWA problem for large network sizes. The RWA problem can be decomposed into different sub-problems, and each sub-problem can be solved independently.

A linear program (LP) relaxation (using the idea of multi-commodity flow in a network)A general-purpose LP solver to derive solutions to this problem.Graph coloring algorithms to assign wavelengths to the lightpaths.

This method of subdividing the overall problem into smaller sub-problems, allows practical solutions to the RWA problem for networks with a large number of nodes.


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FormulationThe RWA problem, without the wavelength-continuity constraint, can be formulated as a multi-commodity flow problem with integer flows in each link. Let λsd denote the traffic (in terms of a lightpath) from any source s to any destination d.

at most one lightpath from any source to any destination; λsd = 1 if there is a lightpath from s to dotherwise λsd = 0.

Let Fsdij denote the traffic (in terms of number of

lightpaths) that is flowing from source s to destination d on link ij.


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FormulationThe ILP is as follows:


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Problem SizeIf we consider the general multi-commodity formulation, the number of equations and the number of variables in the formulation grow rapidly with the size of the network. For example, assume that there are:

10 nodes30 physical links (ij pairs), an average of 4 connections per node, 40 connections (sd pairs)

In the general formulation, number of λsd variables: 10 × 9 = 90number of Fsd

ij variables: 90 sd pairs × 30 ij pairs = 2,700. number of equations will be 3,721

Even for a small problem, the number of variables and equations are very large.These numbers grow proportionally with the square of the number of nodes.


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Problem Size ReductionA smarter solution can be obtained by only considering the λsd variables that are 1,

reduces the number of λsd variables from 90 to 40. eliminates all of the third set of equations.reduces the number of Fsd

ij variables to 40 × 30 = 1,200.

This approach is more specific to the particular instance of lightpaths that need to be set up.It only takes into account the lightpaths that need to be established.


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Problem Size ReductionAssume that a particular lightpath will not pass through all of the ij links. If we can determine the links which have a good probability of being in the path through which a lightpath may pass, we can only consider those links as the Fsd

ijvariables for that particular sd pair. Thus, if on an average, a lightpath sd passes through seven links, there will be approximately 40 × 7 = 280 Fsd

ijvariables.We can find a set of alternate, shortest paths between a given source-destination pair.The links constituting these alternate paths can then be used as part of the LP formulation.


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Problem Size ReductionWe can relax the integrality constraints, getting rid of all of fourth set of equations.Using this approach, we will be left with a total of 351 equations.

1 objective function30 instances of first set 320 instances of second set.

Since there are on an average of seven links considered per connection, we need to enumerate equations in the second set for

eight nodes (6 intermediate and 2 end nodes) per connection, for each of 40 connections.

Hence, using knowledge which is specific to a particular set of lightpaths, we can drastically reduce the size of the LP problem formulation and make it tractable for large networks.


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Randomized RoundingUsed to construct a good solution to the original ILP using the information derived from the solution of the relaxed problem.The randomized rounding technique is applicable to a class of 0-1 ILPs.The technique is probabilistic.

With high probability, the algorithm will provide an integer solution in which the objective function takes on a value close to the optimum of the relaxation.

The optimal value of the objective function in the relaxed version is better than the optimal value of the objective function in the original 0-1 integer program. This technique has been effectively used in multi-commodity flow problems.


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Multi-commodity FlowIn a general, multi-commodity flow problem, we are given:

an undirected graph G(V,E)k commodities that need to be routed.

In an instance of the problem, various vertices are the sites ofsources and sinks for a particular commodity. One unit of flow is to be conveyed from each source s to each destination d through the edges in E. Each edge e has a capacity c(e)

it is an upper limit on the total amount of flow in E. The flow of each commodity in each edge must be 0 or 1. The objective is to minimize the common capacity in each link. The general integral problem is known to be NP-complete.The non-integral version can be solved using linear programming methods in polynomial time.


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Multi-commodity FlowIn the formulation of the problem:

Each commodity corresponds to a lightpath from a source node to a destination node. The capacity is the number of wavelengths supported in each fiber.The objective function is to minimize the number of wavelengths needed to accommodate all of the requests.

The algorithm consists of the following three phases:solving a non-integral multi-commodity flow problempath strippingrandomized path selection


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Non-integral Multi-commodity FlowWe relax the requirement of the 0-1 flows to allow fractional flows in the interval [0,1]. The relaxed capacity minimization problem can be solved by a linear programming method. If the flow for each commodity i on edge e is fi(e),

a capacity constraint of the form:

is satisfied for each edge in the network.C is the optimal solution to the non-integral optimization problem.

The value of C is a lower bound on the best possible integral solution.

1

( )k

ii

f e C=

≤∑


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Path StrippingThe main idea is to convert the edge flows for each commodity i into a set τi of possible paths which could be used to route the flow of that commodity.Initially, τi is empty. For each commodity i do:A. Discover a loop-free, depth-first, directed path

e1, e2, . . . , ep from the source to the destination.B. Let fm = min fi(ej ), where 1 ≤ j ≤ p.

For 1 ≤ j ≤ p, replace fi (ej) by fi(ej) − fm. Add the path e1, e2, . . . , ep to τi along with its weight fm.


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Path StrippingC. Remove any edges with zero flow from the set of

edges that carry any flow for commodity i. If there is nonzero flow leaving si, repeat Step B.Otherwise, continue for next commodity i.

Upon termination, the sum of the weights of all the paths in τi is 1. Path stripping gives us a set of paths τi that may carry the flow for commodity i in the optimal case.


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Randomized Path SelectionFor each i, cast a |τi| die with face probabilities equal to the weights of the paths in τi. Assign the path whose face comes up, to commodity i.As a summary:

The formulation of the problem allows the Fsdij

variables to take on fractional values. These values are used to find the fractional flow through each of a set of alternate paths. A coin-tossing experiment is used to select the path over which to route the lightpath λsd based on the probability of the individual paths.


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Assigning WavelengthsOnce a path has been chosen for each connection, we need to assign wavelengths to each lightpath.

any two lightpaths that pass through the same physical link are assigned different wavelengths.

If the intermediate switches do not have the capability to perform wavelength conversion, a lightpath is constrained to operate on the same wavelength throughout its path.Assigning wavelength colors to different lightpaths, to minimize the number of wavelengths under the wavelength-continuity constraint, reduces to the graph coloring problem.


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Graph ColoringConstruct a graph G(V,E)

Each lightpath in the system is represented by a node in graph G. There is an undirected edge between two nodes if the corresponding lightpaths pass through a common physical fiber link.

Color the nodes of the graph G such that no two adjacent nodes have the same color.This problem has been shown to be NP-complete The minimum number of colors needed to color a graph G is difficult to determine. However, there are efficient sequential graph coloring algorithms.


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Physical Topology for Simulations

A randomly-generated physical topology:100 nodesEach node having a physical nodal degree uniformly distributed between two and five. All links are unidirectional 357 directed links


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Traffic ModelA set of lightpaths to be established between randomly-chosen source-destination (sd) pairs

An sd pair can have zero or one lightpath, and all sdpairs are treated equally.

Associated with each node, we identify d:the average number of lightpath connections the nodes will source.

Thus, in an N-node network, the probability that a node will have a lightpath with each of the remaining (N − 1) nodes equals d/(N − 1). d is the average “logical degree” of a node.


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Results


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ResultsThe maximum network congestion gives the number of wavelengths we would need, if the intermediate switching nodes were equipped with wavelength converters.The time taken to solve the LP increases rapidly as the number of connections increase (corresponding to larger problem formulations).The table entries which are nil correspond to the case when the LP solver failed to give a solution.


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Routing Algorithms for DLEIn case of dynamic traffic, LP is not an option.Heuristic methods are used to solve the DLE problem.The general RWA is divided into:

Routing sub-problemWavelength assignment sub-problem

Basic routing heuristics are:Fixed routingFixed-alternate routingAdaptive routing


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Fixed RoutingThe most straightforward approachAlways choose the same fixed route for a given s,d pair.Example: Fixed shortest path routing

The shortest paths are calculated in advance for each source-destination pair.Any connection between a specified node pair is established using the pre-determined route.

Disadvantage:If the wavelengths along the fixed path are busy, the connection is blocked.


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Fixed Alternate RoutingEach node maintain a routing table that contains a list of fixed routes to each destination.

Example: shortest path, second shortest path, third shortest path

Alternate routes are link disjoint.When a connection request arrives:

The source node attempts to establish the lightpath on alternate path 1.If no wavelength on the route is available, then the second alternate path is tried.Connection is blocked if a lightpath cannot be established on any of the alternate paths.


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Adaptive RoutingThe route between two nodes are calculated dynamically.Ongoing connections are taken into account.Network state must be represented in some form.Each time a connection request arrives, the route must be determined according to free wavelengths on the physical links.


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Wavelength Assignment for DLESeveral heuristics are used:

RandomFirst-fitLeast used / SPREADMost used / PACKMin productLeast loadedMax-SumRCLetc.


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Random Wavelength Assignment

Determine the set of all available wavelengths on the required path.Choose one randomly (with uniform probability).


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First-fit WAAll wavelengths are numbered.A lower-numbered wavelength is considered before a higher-numbered one.That is, the first available wavelength is selected.Compared to random WA lower computation cost.Main idea: Pack all of the in-use wavelengths toward the lower end of wavelength space.

Continuous longer paths toward the higher-numbered end will have a good chance of being available.


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Least Used (SPREAD)Select the wavelength that is least used in the network.Attempts to balance the load, among all the wavelengths.Breaks the long wavelength paths quickly.The performance is worst than random WA.


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Most Used (PACK)Attempts to select the most used wavelength in the network.Outperforms LU significantly, and slightly better than FF.Packs connections into fewer wavelengths.


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Min ProductUsed in multi-fiber networks.The goal is to pack wavelengths into fibers, thereby minimizing the number of fibers.It computes:

π(p) is the set of links of path pDlj is number of fibers on link l and wavelength j

Chooses the lowest numbered wavelength that minimizes the above value.In a single-fiber network, becomes FF.

( )lj

l p

Dπ∈∏


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Least LoadedDesigned for multi-fiber networks.Selects the wavelength that has the largest residual capacity on the most loaded link along the route.In single-fiber networks it reduces to FF.Outperforms LU and FF in multi-fiber networks.

OPTICAL NETWORKS - Anasayfagencata/courses/ON/ONslides/... · 2013. 9. 13. · Optical Networks...

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