ILP formulations and solution techniques
For Optical network design problems
Supervised by
Dr. Abd ElKarim Shaban Omr
Professor, Faculty of Engineering, Cairo University
Dr. Khaled Mohamed Fouad Elsayed
Professor, Faculty of Engineering, Cairo University
By
Zein 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.
2.
3.
4.
5.
6.
7.
huge bandwidth (nearly 50 terabits per second (Tbps)
low signal attenuation (as low as 0.2 dB/km)
low signal distortion (immune to electromagnetic interference)
low power requirement
low material usage
small space requirement, and
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
Rx
Tx
Tx
Rx
Tx
Rx
Optical Networks (Cont.)
(WRON’s)

Example WRON:
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:
Lightpath
D
C
E
1
2
2
B
A
Wavelength converter
1
1
RWA (Cont.)
(Classification)
Classification:

Traffic type:




Wavelength-conversion capability:




Static
Incremental
Dynamic
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)
SD1
P(sd2)
SD2
A
Matrix
SDR
Connection
requests
Capacity constraints are
applied for each link, such
that each wavelength is used
at most once
B
Matrix
P(sdR)
Candidate paths
Links
Solution approaches(5)

Max-RWA, path-based, no conversion, multiple requests,
single-fiber
Solution approaches(5)

Illustration:
P(sd1)
SD1
f1
W1
W2
P(sd2)
SD2
A
Matrix
SDR
Connection
requests
f2
f
Vector
D
Matrix
P(sdR)
fR
Candidate
paths
Set of path-flow
variables
W
w
Set of
wavelengths
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)
Start
Solution approaches(8)

No. of used Wavelengths
=0
All
connections
satisfied?
Applied Heuristic
No
No. of used Wavelengths =
No. of used Wavelengths +1
Yes
Call Greedy Algorithm
for maximum
coverage
Assign the paths for the
satisfied connection the
current wavelength
Report
solutio
n
End
Start
Solution approaches(9)
Yes

Greedy Heuristic Approach
No. of used Wavelengths = 0
All
connections
satisfied?
No
No. of used Wavelengths =
No. of used Wavelengths +1
Maximum Edge Disjoint Paths
problem:
Given: a graph and a set of sourcedestination pairs are given and the
requirement is to find Edge disjoint
paths for as many of the pairs as
possible

Call Greedy Algorithm for
EDP
Assign the paths for the
satisfied connection the
current wavelength
Report
solution
End
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 multigraph instead of a simple undirected graph

0
1
2
0
1
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:
W
P

m 
c
ki a kj
i 1 k 1
j

j  1,2,...R
The capacity constraint:
P
1

c
kib kj
k 1
i  1, 2,...W , j  1, 2,...L
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.
Start
Read input
data from file
Proposal (Cont.)
(Network Growing Problem)
Build the LP model
Solve the model
•
Solution approach:
No
Yes
Feasible?
/*Values obtained from the 2nd run */
W = the number of available wavelengths per link.
W(L) = the needed number of wavelengths on link L
Build new model with worst case
no. of wavelengths
For each node pairs (i,j)
For each link Lij (between the nodes (i,j)
If ( W (Lij) < W ) freeWaves = freeWaves + WW(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
Solve the model
Calculate modifications
Report
solution
End
References
1.
2.
3.
4.
5.
6.
Biswanath. Mukherjee, "Optical communication networks", McGraw-Hill
Publishers, 1997
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
R. Ramaswami and K. Sivarajan, “Routing and wavelength assignment in alloptical networks”, IEEE/ACM Trans. Networking, vol. 3, October 1995.
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.
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.
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.
8.
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.
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!