ILP formulations and solution techniquesFor Optical network design problems

Supervised byDr. Abd ElKarim Shaban OmrProfessor, Faculty of Engineering, Cairo University

Dr. Khaled Mohamed Fouad ElsayedProfessor, Faculty of Engineering, Cairo University

ByZein ElAbedin Mohamed Wali

Agenda

Optical Networks (why?) Routing and Wavelength Assignment (what?) Solution approaches (how?) Proposal

Agenda

Optical Networks (why?) Need for new network solution Optical Networks Advantages Multiplexing techniques WRON’s Lightpath

Routing and Wavelength Assignment (what?) Solution approaches (how?) Proposal

Optical Networks(Need for new network solution)

The need for new network solution emerges from the following reasons:

More users More bandwidth-intensive networking

applications (voice, video, ….) New generation networks involving HD-TV,

Video mail, …etc

Optical Networks (Cont.) (Optical Networks Advantages)

Based on fibers, Optical networks can be best suited for the above demands:

1. huge bandwidth (nearly 50 terabits per second (Tbps)2. low signal attenuation  (as low as 0.2 dB/km)3. low signal distortion (immune to electromagnetic interference)4. low power requirement5. low material usage6. small space requirement, and 7. low cost.

Optical Networks (Cont.) (Multiplexing techniques)

Different Multiplexing techniques may be used to efficiently utilize the huge bandwidth provided by optical networks:

Space-division multiplexing (SDM) Frequency/Wavelength-division multiplexing

(FDM/WDM) Time-division multiplexing (TDM) Code-division multiplexing (CDM)

Optical Networks (Cont.)(WRON’s)

WDM Routed Optical Networks WRON’s:(Fiber bandwidth is divided between several

independent logical channels each carried on different wavelength)

Fiber

Tx

Tx

Tx

Rx

Rx

Rx

Example WRON:

Optical Networks (Cont.)(WRON’s)

Optical Networks (Cont.)(WRON’s)

Example WRON (OXC structure)

Optical Networks (Cont.)(Lightpaths)

A lightpath is the basic mechanism of communication in WRON.

lightpath (also referred to as -channel), is a clear optical path –alternatively referred to as an all-optical communication channel -between two edge nodes, it bypasses electronic packet processing at intermediate nodes.

It is realized by finding a physical path and allocating a free wavelength on each link of that path

Agenda

Optical Networks (why?) Routing and Wavelength Assignment (what?)

Problem statement Wavelength conversion Classification

Solution approaches (how?) Proposal

Routing and Wavelength Assignment(RWA)(Problem statement)

Problem statement: Given:

– Set of lightpaths demands that need to be established.– A constraint on the number of wavelengths.

Required: – To determine the routes over which these lightpaths should

be set up.– Also to determine the wavelengths that should be assigned.

RWA (Cont.) (Problem statement)

Example:

RWA (Cont.) (Problem statement)

Constraints:1. Wavelength continuity constraint:A lightpath must use the same wavelength on

all the links along its path from source to destination edge node

2. Distinct wavelength constraint: All lightpaths using the same link (fiber) must

be allocated distinct wavelengths

RWA (Cont.) (Problem statement)

Illustration (wavelength continuity):

RWA (Cont.) (Wavelength Conversion)

The OXCs may be equipped with wavelength converters.

If all the OXC have such capability, the wavelength continuity constraint is relaxed, and the RWA problem is reduced to classical routing problem (in Circuit-switched networks)

RWA (Cont.) (Wavelength Conversion)

Illustration:

B

DC

A

E

2

Wavelength converter 1

2

1

1

Lightpath

RWA (Cont.) (Classification)

Classification: Traffic type:

Static Incremental Dynamic

Wavelength-conversion capability: Full-wavelength conversion Sparse wavelength conversion No wavelength conversion

Objective function: Min-RWA Max-RWA

RWA (Cont.) (Classification)

Fiber multiplicity Requests multiplicity formulation structure:

Link-Based Path-Based

Agenda

Optical Networks (why?) Routing and Wavelength Assignment (what?) Solution approaches (how?) Proposal

Solution approaches(1)

Min-RWA, link-based, no conversion, unique requests, single fiber:

Problem decomposition into:– Routing sub-problem– Wavelength Assignment sub-problem

Solution approaches(1)

Routing:

Solution approaches(1)

Wavelength Assignment: using Graph Coloring

Solution approaches(2)

Min-RWA, link-based, no conversion, multiple requests, single fiber (routing)

Solution approaches(3)

Min-RWA, link-based, full-wavelength conversion, unique requests, single fiber

This case reduces the RWA problem to the classical routing problem

once lightpaths has been established, any wavelength available on any link may be used

not of much commercial importance, since in most cases full wavelength conversion in the network is not preferred and not even necessary due to high costs and limited performance gains.

Solution approaches(4)

Max-RWA, path-based, no conversion, multiple requests, single-fiber (Selection & WA)

Solution approaches(4)

Illustration:

P(sd1)

P(sd2)

P(sdR)

SD1

SD2

SDR

Connection

requestsCandidate paths Links

Capacity constraints are applied for each link, such that each wavelength is used at most once

AMatrix

BMatrix

Solution approaches(5)

Max-RWA, path-based, no conversion, multiple requests, single-fiber

Solution approaches(5)

Illustration:

P(sd1)

P(sd2)

P(sdR)

SD1

SD2

SDR

Connection

requestsCandidate paths

AMatrix

W1

Set of path-flow variables

Set of wavelengths

W2

Ww

f1

f2

fR

DMatrix

fVector

Solution approaches(6)

Max-RWA, link-based, no conversion, multiple requests, single-fiber (Routing & WA)

Solution approaches(7)

Max-RWA, link-based, no conversion, multiple requests, single-fiber (Routing & WA)

Solution approaches(8)

Max-RWA, path-based, no conversion, multiple requests, mutli-fiber (Selection & WA)

Solution approaches(8)

Applied Heuristic

Start

End

All connections

satisfied?

No. of used Wavelengths = No. of used Wavelengths +1

No

Yes

Report solutio

n

Call Greedy Algorithm for maximum coverage

No. of used Wavelengths = 0

Assign the paths for the satisfied connection the current wavelength

Solution approaches(9)

Greedy Heuristic Approach

Maximum Edge Disjoint Paths problem:

Given: a graph and a set of source-destination pairs are given and the requirement is to find Edge disjoint paths for as many of the pairs as

possible

Start

End

All connections

satisfied?

No. of used Wavelengths = No. of used Wavelengths +1

NoYes

Report solution

Call Greedy Algorithm for EDP

No. of used Wavelengths = 0

Assign the paths for the satisfied connection the current wavelength

Solution approaches(10)

Min-RWA, path-based, full conversion, unique requests, single-fiber (Selection & WA)

Solution approaches(10)

Objective function used

The cost function of every link is convex, monotonically increasing, and piecewise linear.

The breakpoints of each piecewise linear link cost function occur at the integer points The cost for flow larger than W is , thereby imposing a link capacity constraint.

Solution approaches(11)

Min-RWA, path-based, no conversion, unique requests, single-fiber (Selection & WA)

Solution approaches(12)

Min-RWA, path-based, sparse conversion, unique requests, single-fiber (Selection & WA)

W Converter

No W Converter

Solution approaches(12)

Approach:

Solution approaches(13) Min-RWA, path-based, sparse conversion, multiple

requests, multi-fiber (Selection & WA)

Solution approaches(13) Min-RWA, path-based, sparse conversion, multiple

requests, multi-fiber (Selection & WA)

Solution approaches(14)

Tabu Search Heuristic Approach

Agenda

Optical Networks (why?) Routing and Wavelength Assignment (what?) Solution approaches (how?) Proposal

Motivation Proposed Model Network growing problem TU-Based solution technique (in progress)

Proposal(Motivation)

Motivation:– Only few models addressed the Min-RWA

problem.– Mostly all the approaches presented ILP models,

but relied on approximation or heuristic algorithms to solve the problem especially for large size networks.

– No model addressed the Min-RWA problem with multi-fiber links case.

Proposal(Proposed Model)

Min-RWA, Path-based, no conversion, multiple requests, Multiple-fiber

Handling multiple-fibers:

Network is modeled to an undirected multi-graph instead of a simple undirected graph

10 2 10 2

Proposal (Cont.)(Proposed Model)

ILP formulation (Selection & WA)

Proposal (Cont.)(Proposed Model)

Weights selection: Lemma 1:At optimality, the traffic demand must be

satisfied at equality.Moreover, using any monotonically increasing

weights for the increasing index wavelengths will ensure that the minimum number of wavelengths is used.

Proposal (Cont.)(Proposed Model)

Proof: ( By contradiction) The traffic demand constraint:

The capacity constraint:

1 1

1,2,...j

W P

ki kji k

j Rm c a

1

1 1,2,... , 1,2,...P

ki kjk

i W j Lc b

Proposal (Cont.)(Network Growing Problem)

Given:– Set of lightpaths demands that need to be

established.– A constraint on the number of wavelengths.

When the current network topology and resources does not satisfy the demanded requests, it is required to obtain the minimum set of modifications (in terms of additional resources) to satisfy the connection requests.

Our assumption: the suggested modifications are only the addition of fibers to already existing links.

Proposal (Cont.)(Network Growing Problem)

Read input data from file

End

Feasible?

Build new model with worst case no. of wavelengths

Solve the model

NoYes

Report solution

Calculate modifications

Solve the model

Build the LP model

Start

/*Values obtained from the 2nd run */W = the number of available wavelengths per link.W(L) = the needed number of wavelengths on link L

For each node pairs (i,j)For each link Lij (between the nodes (i,j) If ( W (Lij) < W ) freeWaves = freeWaves + W-

W(Lij)EndFor

For each link Lij (between the nodes (i,j) If (( W(Lij) > W ) && ( freeWaves < W(Lij) – W) ) Fibers to add = ceil

((W(lij)-W-freeWaves)/W) EndFor

EndFor

•Solution approach:

References

1. Biswanath. Mukherjee, "Optical communication networks", McGraw-Hill Publishers, 1997

2. D. Banerjee, and B. Mukherjee, “A practical approach for routing and wavelength assignment in large wavelength-routed optical networks," IEEE Journal on Selected Areas in Communications, Vol. 14 No. 5, 1996

3. R. Ramaswami and K. Sivarajan, “Routing and wavelength assignment in all-optical networks”, IEEE/ACM Trans. Networking, vol. 3, October 1995.

4. R.M. Krishnaswamy, K.N. Sivarajan, “Algorithms for Routing and Wavelength Assignment Based on Solutions of the LP-Relaxation”, IEEE Communications Letters, vol. 5, no. 10, October 2001.

5. Mohamed Saad, Zhi-Quan Luo, "On the Routing and Wavelength Assignment in Multifiber WDM Networks", IEEE Journal on Selected Areas in Communications (special series on optical communications and networking), vol. 22, no. 9, November 2004.

6. A.E. Ozdaglar, D.P. Bertsekas, “Routing and Wavelength Assignment in Optical

Networks”, IEEE/ACM Transactions on Networking, vol. 11, no. 2, April 2003.

References

7. Steven S. W. Lee, Maria C. Yuang, Po-Lung Tien, and Shih-Hsun Lin, "A Lagrangean Relaxation-Based Approach for Routing and Wavelength Assignment in Multigranularity Optical WDM Networks", IEEE Journal on Selected Areas in Communications (special series on optical communications and networking), vol. 22, no. 9, November 2004.

8. Christiane Dzongang, Philippe Galinier, and Samuel Pierre, "A Tabu Search Heuristic for the Routing and Wavelength Assignment Problem in Optical Networks", IEEE Communications letters, Vol. 9, No. 5, May 2005.

Thank You!