Download RWA Problem formulations

RWA Problem formulations
& Solution approaches
Optical Network Problems classification:
There is a wide variety of problems arising in the domain of optical network design they
may be generally classified according to the following:
1. Traffic type:
a. Static 
b. Incremental
c. Dynamic
2. Network topology:
a. Ring
b. Mesh 
c. Trees
d. Other
3. No. of fibers per span K  1
4. No. of wavelengths per fiber W
5. Wavelength conversion capability?
6. Objective function:
a. Min Cost (or minimize number of wavelengths), referred to as Min-RWA
b. Max carried traffic, referred to as Max-RWA
c. Min blocking probability
d. Others
7. Whether each O-D pair may have multiple connections or not?
8. Another important classification, which is more specific, divides the problems into
two main categories:
a. Routing and wavelength assignment (RWA)
b. Grooming, Routing and wavelength assignment (GRWA )
Depending on whether each connection is assigned a complete wavelength's bandwidth
or only a fraction of it, the former is named RWA and the latter is the GRWA problem.
We are generally interested in the class of problems (marked above); static traffic, mesh
topology as a generalization of both ring and tree topologies and generally consider the
objective function of minimizing the network cost. But in this survey we will provide a
presentation for all previous work in the domain.
We will have different sections to present the two categories of problems RWA and
GRWA, each with the proposed formulations, and solutions approaches as available from
the literature.
RWA
The Routing and Wavelength Assignment (RWA) Problem:1
In this section, we address the static routing and wavelength assignment (RWA)
problem, also known as the Static Lightpath Establishment (SLE) problem. In SLE,
lightpath requests are known in advance and the routing and wavelength assignment are
performed off-line. The typical objective is to minimize the number of wavelengths
needed to set a certain set of lightpaths for a given physical topology. As an alternative to
minimizing the number of wavelengths in the network, the dual problem is to maximize
the number of connections that can be established (minimize blocking) for a given
number of wavelengths and a given set of connection requests. This dual to the SLE
problem raises the issue of fairness, in that solutions to this problem will tend to establish
more short connections, which traverse fewer links than long connections, which traverse
a greater number of links.
The RWA problem can be formally stated as follows.
Given:
 Set of connection requests that need to be established on the network.
 Network topology: number of Nodes, Links Matrix
 An optional constraint on the number of wavelengths.
We are required to determine the different light paths to be established, the routes over
which they are set up and also determine the wavelengths that should be assigned to these
lightpaths so that the maximum number of lightspaths may be established. Or to achieve
a minimum number of wavelengths if that is not given as a constraint.
Different solutions are proposed in the literature to solve the RWA problem; they mainly
depend on composing the problem on two stages:
1. Routing
2. Wavelength assignment
Each stage can be solved with an efficient heuristic algorithm as shown below:
Routing:
 Fixed routing
 Alternate routing
 Adaptive routing
 Fault-tolerant routing
Another algorithm:
 Randomized rounding & path stripping
Wavelength assignment (WA):
1
. See Ref. [1]

Graph coloring problem2
Or others used only in dynamic and incremental traffic case
 Random WA
 First-fit
 Least-used/SPREAD
 Most-used/PACK
 Min-product
 Least-loaded
 Max-Sum
 Relative-capacity loss (RCL)
In the following sections we will present different formulations (possibly variations) of
the RWA problem, each with its proposed solution.
Min-RWA With no wavelength conversion capability:3
Description:
This formulation of the RWA problem considers the wavelength continuity
constraint, where no nodes in the network have any wavelength converters..
It should be noted that the wavelength continuity constraint is not explicitly mentioned in
the following formulation, because the formulation only addresses the Routing subproblem and the wavelength continuity constraint is handled in the Wavelength
Assignment sub-problem
Formulation:
Objective Function:
Minimize Fmax
F
Such that
  F ij
s ,d
Flow conservation constraints:
F
i
sd
ij
i, j
sd
max
  F jk 
sd
k


0
sd
sd
if s=j
if d=j
otherwise
Wavelength-continuity constraints:

sd
sd
F
ij
= 0,1
= 0,1
where
denote the traffic (in terms of a lightpath) from any source s to any destination d.(we
sd

2
3
. See Ref. [1] and [2]
. See Ref. [1] and [2]
consider at most one lightpath from any source to any destination; hence
a lightpath between s to d
F
sd
jk


sd
=1 if there is
=0 otherwise.
sd
denote the traffic (in terms of lightpaths) that is flowing from source s to destination
d on link ij.
Complexity:
This problem is NP-complete.
Solution approach:
This problem is solved using a technique called "randomized rounding and path
stripping" developed by B. Mukherjee in his book "Optical communication networks"
The WA sub-problem is solved by formulating it as "graph coloring problem" and using
largest-fit algorithm.
Min-RWA With no wavelength conversion capability & multiple connection
requests:4
Description:
This problem is a generalization of the previous one, where each O-D pair can have more
than a single connection request. As above this formulation covers only the Routing subproblem.
Formulation:
Objective Function:
Minimize Fmax
F
Such that

max
F
s ,d , w
Flow conservation constraints:
F
i
sdw
ij
i, j
sdw
ij

  F jk 
sdw
  sdw
k
sdw
0
Wavelength-continuity constraints:

sdw
w
Wavelength uniqueness:
if s=j
if d=j
otherwise
 sd
sdw
F = 0, 1
F 1
ij
sdw
ij
(or = 0,1 ?)
s ,d
where
denote the traffic (number of connection requests) from any source s to any
sdw

4
. See Ref. [3]
destination d on any wavelength w.(we assume that two or more lightpaths may be set up
between the same source-destination pair, if necessary, but each of them must employ a
distinct wavelength; hence sdw  1

F
sd
jk
denote the traffic (number of connection requests) from source s to destination d on
F
link ij and wavelength w.
one path.
 traffic matrix in which
sdw
ij

sd
 1 since a wavelength on a link can be assigned to only
denotes the number of connections needed between
source s and destination d.
Complexity:
This problem is a generalization to the previous and thus an NP-complete problem.
Solution approach:
This formulation as presented for the illustration purpose5, but could be ( with appropriate
modification) reformulated as the previous one and solved using the same techniques.
Min-RWA With full-wavelength conversion capability:
Description:
In a wavelength-routed WDM network, the wavelength-continuity constraint can be
eliminated if we are able to use a wavelength converter to convert the data arriving on
one wavelength on a link into another wavelength at an intermediate node before
forwarding it on the next link.
Formulation:
Objective Function:
Minimize Fmax
F
Such that
  F ij
s ,d
Flow conservation constraints:
F
i
sd
ij
i, j
sd
max
  F jk 
sd
k


sd
sd
if s=j
if d=j
otherwise
0
Where
denote the traffic (in terms of a lightpath) from any source s to any destination d.(we
sd

consider at most one lightpath from any source to any destination; hence
a lightpath between s to d
5
. See Ref. [3]

sd
=0 otherwise.

sd
=1 if there is
F
sd
jk
denote the traffic (in terms of lightpaths) that is flowing from source s to destination
d on link ij.
Complexity
This problem reduces to the transportation problem and hence can be solved in
polynomial time. Although being an easy problem, it is not of much commercial
importance, since in most cases full wavelength conversion in the network is not
preferred and may not even be necessary due to high costs and limited performance
gains.
The above reason gives rise to the following related problems which aims to efficiently
utilize available converters in the network:
1. Sparse location of wavelength converters in the network
2. Sharing of converters
3. Limited-range wavelength conversion
Max-RWA, Mutli-Fiber WDM networks, with multiple requests:6
Description:
This problem considers the case when more than a single fiber forms the physical link
between the networks nodes
Formulation:
Objective Function:
Such that:
Physical constraint:
Maximize

pP ( e )
Traffic constraint:
  x( p, w)
wW pP
x( p, w)  K e
 
wW pP ( s , d )
e, w
x( p, w)  r s ,d
x( p, w)  o , integer
s, d
p, w
where
K e denote the number of fibers deployed on link e  E .
W = {w1,w2, …..wF} is the set of wavelengths a fiber can support, where | W | =F
P denote the set of paths in the network. Moreover let P(e) ( respectively P(s,d)) denote
the subset of paths in the network that use link ( respectively join nodes s and d).
r
s ,d
denote the number of connection requests between nodes s and d.
x( p, w) denote the number of lightpaths successfully routed over physical path p  P
and assigned wavelength w W .
6
. See Ref. [4]
Note:
a connection request is realized by finding a physical path p  P ( s, d ) and a
wavelength
lightpath.
w W that is assigned to every link on p. The pair (p,w) is called a
Complexity:
NP-Hard problem.
Solution approach:
The solution approaches depends on formulating the problem as a maximum coverage
problem and using a greedy algorithm use for it to solve this RWA problem.( guaranteed
to be within a factor of ( 1-1/e) of the optimal.
Max-RWA Path-Based ILP formulation with explicit wavelength constraint(1):7
Description:
This problem considers maximizing the carried traffic over WDM network, with the
formulation based on flow variables for the available paths. Moreover the wavelength
continuity constraint is explicit in the formulation
Formulation:
N
Objective Function:
Maximize
m
i 1
Such that:
m 0
c 0
i
ij
i
, integer
i=1,….,N
,integer
i= 1,….,P, j=1,…..,W
wavelength constraint:
C B 1
Traffic demands:
m 1 C A
m  p
T
WxM
T
W
i
i
i=1,….,N
where
N: denote the number of source-destination (s-d) pairs.
M: denote the number of links.
W: denote the number of wavelengths per link
m = {mi}, i=1,2….,N : number of connections established for source-destination pair i
 : offered load ( total number of connection requests to be routed).
P={Pi}; i=1,2,…,N: fraction of the load which arrives for source-destination pair I (thus,
Pi  = number of connections to be set up for source-destination pair i)
7
. See Ref. [5]
A={aij}: PxN (path- s-d pair incidence) matrix in which aij = 1 if path i is between sourcedestination j, and aij =0 otherwise.
B={bij}: PxM (path-edge incidence) matrix in which bij = 1 if link j is on path i, and bij =0
otherwise.
C={cij}: PxW (path-wavelength assignment) matrix in which cij = 1 if wavelength j is
assigned to path i, and cij =0 otherwise.
Complexity:
Solution approach:
This formulation as presented for the purpose to provide tight bounds on the solution of
the RWA algorithms
Max-RWA Path-Based ILP formulation with explicit wavelength constraint(2):8
Description:
This problem considers a new view of the RWA problem (using the concepts of graph
coloring problem used before in the WA sub-problem)
By creating a new graph Gp where each node corresponds to a path in G and two in Gp
are adjacent iff the corresponding two paths in G share a common link. This means a set
of paths in G can be assigned a common wavelength only if the corresponding nodes in
Gp form and independent sets in Gp.
Formulation:
N
Objective Function:
Maximize
m
i 1
Such that:
i
w 0
, integer
i
i=1,….,L
L
wavelength constraint:
 w W
i 1
Traffic demands:
i
f  wD
mf A
m  p
T
i
i
i=1,….,N
where
N,M,W, m ,  , P, A,B as previous problem
L: the number of maximal independent sts in Gp.
W = {wi}, i=1,2….,L : the number of wavelengths that are assigned to the nodes in
independent set i.
8
. See Ref. [5]
D={dij}: PxL (path-independent set incidence) matrix in which dij = 1 if independent set j
contains path i, and dij =0 otherwise.
f ={fi}the vector of path-flows.
Complexity:
Solution approach:
This formulation as presented for the purpose to provide tight bounds on the solution of
the RWA algorithms
Max-RWA Link-Based ILP formulation with explicit wavelength constraint(1):9
Description:
This problem explicitly considers the wavelength-continuity constraint. Moreover the
wavelength number is explicitly assigned to each ligthpath in the formulation
Formulation:
Objective Function:
 b (i, j )
Maximize
q
i, j
Such that:
Traffic demands:
b
q
q
(i, j ) 

q
F 1
C
Wavelength continuity:
( k ,q )
k 0
(i, j )b q (i, j ) {0,1}
i, j
(i, j )  b q (i, j )
C (i, j)  C (i, j)
 C (i, j)  1
( k ,q )
Physical Links:
( k ,q )
l ,m
( k ,q )
Wavelength continuity:
l ,m
q(i, j )
q(i, j ), (l , m), k
k (l , m)
q (i , j )
Flow conservation:
C
l
( k ,q )
l ,m
(i, j ) P l ,m   C m,l (i, j ) P m,l
( k ,q )
l

(i, j )
C
( k ,q )

= C (i, j )
0

( k ,q )
if m=i
if m=j ; for all(i,j), k, q and m
if m  i and m  j
where
notation:
i and j denote the originating and terminating node of a lightpath.
l and m denote the endpoints of a physical link
k: denotes the wavelength number
9
. See Ref. [6]
q: used as subscript or superscript denotes the qth lightpath between a source-destination
pair.
N: the number of nodes in the netwok.
= {  } the traffic matrix, i.e  is the number of connections that are to be

ij
ij
established between node I and node j.
P lm denotes the existence of a link in the physical topology. If
fiber link between node l and m, otherwise P lm =0.
F: the number of wavelengths a fiber can support.
Variables:
b q(i, j ) =1, if there exists a qth lightpath between (i,j), else
C
( k ,q )
( k ,q )
l ,m
lm
=1 then there is a
b (i, j ) =0.
q
(i, j ) =1, if there exists a qth lightpath between node i and node j uses wavelength
k, else
C
P
C
( k ,q )
(i, j ) =0.
(i, j ) =1, if there exists a qth lightpath between node i and node j uses wavelength k
and is routed over physical link (l,m), else
C
( k ,q )
l ,m
(i, j ) =0.
Complexity:
Solution approach:
This formulation is used with combination to an LP-relaxtaion to form a suitable
heuristic.
Max-RWA Link-Based ILP formulation with explicit wavelength constraint(2):10
Description:
This problem uses the fact that each fiber link with F wavelengths can be viewed as F
fibers each one supporting a single wavelength.
Formulation:
Objective Function:
Maximize
 b (i, j )
i
j
Such that:
Traffic demands:
 
q
10
. See Ref. [6]
m
(i , j )
( i , m ), q
 
q
l

(i , j )
( l , j ), q
 b (i, j ) 

(i , j )
(i, j )

Flow balance:
(i , j )
( l , m ), q
l
wavelength continuity:
P
 
i
   (l ,m),q P (l ,m),q  0
(i , j )
( l , m ), q
l
(i , j )
( l , m ), q
q(i, j ) and m  i, and m  j
1
(l,m),q
j
where
notation: same as the previous problem
P(l ,m),q =1 denotes the existence of the qth multiple link of the physical link (l,m),
otherwise
P
( l , m ), q
=0
Variables:
(i , j )
 (l ,m),q =1, if a lightpath setup between node I and node j uses the qth multiple link of
edge (l,m) else

(i , j )
( l , m ), q
=0
b (i, j) denotes the number of lightaths established between node I and node j.
Complexity:
Solution approach:
This formulation is used with combination to an LP-relaxtaion to form a suitable
heuristic.
References:
1. “Optical Communication Networks'' by Biswanath Mukherjee
2. "A Practical Approach for Routing and Wavelength Assignment in Large
Wavelength-Routed Optical Networks"
Dhritiman Banerjee and Biswanath Mukherjee, Member, IEEE
IEEE Journal on selected areas in communications, vol. 14,No.5, JUNE 1996
3. “A review of routing and wavelength assignment approaches for wavelength-routed
optical WDM networks,” H. Zang, J. P. Jue, and B. Mukherjee, Optical Network
Mag., vol. 1, no. 1, pp. 47–60, Jan. 2000.
4. “On the Routing and Wavelength Assignment in Multifiber WDM Networks”,
Mohamed Saad, Student Member, IEEE, and Zhi-Quan Luo, Senior Member, IEEE
5. “Routing and Wavelength Assignment in All-Optical Networks”, Rajiv Ramaswami
Member, IEEE and Kumar N. sivarajan, Member, IEEE.
IEEE/ACM transactions on Networking Vol.3, No.5 ,Oct 1995
6. “Algorithms for Routing and Wavelength Assignment Based on Solutions of LPRelaxations”, Rajesh M. Krishnaswamy and Kumar N. Sivarajan