May 2006
Phys. Chem. News 29 (2006) 50-55
PCN
AN ALGORITHM FOR SOLVING THE MAX-RWA PROBLEM
IN WIDE ALL-OPTICAL WDM NETWORKS
A. Zyane1,2, 3*, S. Pierre2, Z. Guennoun3
1
Département de maintenance industrielle, Ecole Supérieure de Technologie de Safi – E.S.T,
Université Cadi Ayyad, B.P. 89, Route Dar Si Aissa, Safi,- 46000 - Maroc
2
LARIM, Laboratoire de recherche en réseautique et informatique mobile, Département de génie informatique,
Ecole Polytechnique de Montréal, Montréal - Canada
3
L.E.C, Laboratoire d’électronique et communication , Ecole Mohammadia d’ingénieurs –
E.M.I, Université Mohammed V, Rabat, Maroc
* Corresponding author. E-mail: [email protected], [email protected], [email protected]
Received: 02 August 2003; revised version accepted: 10 February 2006
Abstract
This paper proposes a new approach for routing and wavelength assignment in wide all-optical WDM
networks with wavelength continuity constraint. Given a number of available wavelengths on each optical
fiber, we seek to maximize the number of satisfied requests for connections. This is known as Max-RWA
problem. In our approach, called MLL (RWA with Minimum Loaded Link), the routing is based on the
search for the optimal path while trying to minimize the maximum load on the links of the network in
order to minimize the maximum links capacity. The wavelength assignment is based on a graphs coloring
method using tabu-search. We tested our algorithm on known networks like the EONNET and NSFNET.
Generally, our results are better with those provided by the existing solving approaches taken as
reference.
Index terms- RWA, routing and wavelength assignment, WDM networks, optical routing.
1. Introduction
In the new millennium, the telecommunications
industry is abuzz with megabits per second
(Mbps), gigabits per second (Gbps), terabits per
second (Tbps), and so on. The continuously
growing
demand
on
high
bandwidth
communication requires new area networks which
are able to provide the necessary transmission
capacity. In this context, Wavelength division
multiplexing (WDM) technology [1] has been
improving steadily in the recent years, with
existing systems capable of providing huge
amounts of bandwidth in a single fiber. The rapid
advancement of evolution of optical technologies
makes it possible to move beyond point to point
WDM transmission systems to an all-optical
backbone network that can take full advantage of
the bandwidth available by eliminating the need
for per-hop packet forwarding. In these networks,
transmission and switching are both implemented
in optical technology without the bottleneck of
intermediate optical-to-electrical conversions.
Multiple Wavelength division multiplexed
channels can be operated on a single fiber
simultaneously;
however,
a
fundamental
requirement in fiber-optic communication is that
these channels operate at different wavelengths
(non-overlapping wavelength channels). These
channels can be modulated independently to
accommodate dissimilar data formats, including
some analog and some digital, if desired.
In all-optical WDM network, each channel
request requires the establishment of an optical
path from the source to the corresponding
destination node in the network. Generally, there
exist two different kinds of optical paths: The first
one, called virtual wavelength path (VWP),
assumes that the signal is not necessarily carried
by the same wavelength during its travel through
the network. In a transparent optical network, this
concept requires the possibility of optical
wavelength conversion in the node of the network.
However, such all-optical wavelength converters
are very expensive. The availability of full
wavelength conversion simplifies the management
of the network, the wavelength assignment
algorithm in such a network becomes simpler
because all the wavelengths can be treated
equivalently, and the wavelengths used on
successive links along a path can be independent
of one another. However, the benefits of
wavelength conversion in reducing blocking and
improving other performance metrics are not
nearly as universal or apparent [13]. So in this
paper, we want to eliminate the need of
wavelength conversion at all. This strategy to
install optical paths is known as the concept of
wavelength paths (WP) and requires the additional
constraint that each connection is assigned just
50
A. Zyane et al, Phys. Chem. News 29 (2006) 50-55
assignment this leads in general to worse solutions
of the RWA problem. In order to get solutions of
high quality by keeping the dimension of the
related optimization problem at an acceptable
level, we propose another way to take the
correlation of routing and wavelength assignment
into consideration.
Generally, the routing scheme has much a
higher impact on the performance of the
connection blocking probability in the networks
than the wavelength assignment scheme [5].
Especially, routing under consideration of the
network status is important to improve wavelength
utilization [8, 9]. However, existing routing
algorithms referring only to current network
conditions can still cause high blocking
probability at a later time.
This paper proposes a new approach of routing
and wavelength assignment in wide all-optical
WDM networks with wavelength continuity
constraint. Section 2 states the problem and the
existing solving approaches. Section 3 details the
new solving approach proposed. Section 4
presents and analyzes numerical results. Section 5
concludes the paper and presents further works.
one wavelength for its transmission. This
limitation is known as the wavelength continuity
constraint which makes the wavelength-routed
networks different from the traditional circuitswitched public switched telephone network
(PSTN).
The design of a wide all-optical WDM network
with the already mentioned properties is a long
process based on a previously determined traffic
matrix, which contains information about the
number of channel demanded between each pair of
source and destination node. It includes the
topology design, the link dimensioning and the
routing problem [2, 11]. Given a network topology
with a set of stations, a set of bi-directional links, a
list of demands and a number of wavelengths, the
routing and wavelength assignment problem
(RWA) consists of deciding the lightpath to be set
up in terms of their source and destination nodes
and to assign a wavelength to that path so as to
minimize the number of routed connections,
subject to both a wavelength continuity constraint
and a distinct wavelength constraint [5, 10]. A
lightpath must use the same wavelength on all the
links along its path from source to destination
node. The second constraint states that two
lightpaths using the same link (fiber) must be
allocated distinct wavelengths. We distinguish
between static and dynamic ligthpaths, depending
on the nature of the traffic in the network. When
the nature of the traffic pattern is static, a set of
ligthpaths is established all at once that remain in
the network for a long period of time. Such static
lightpath establishment [5] is relevant in the initial
design and planning stage and the objective is to
maximize the network throughput (i.e., maximize
the number of ligthpaths established with given
network resources). For the dynamic traffic
scenario where the traffic pattern changes rapidly,
the network has to respond to traffic demands
quickly and economically. In such a dynamic
traffic case, a ligthpath is set up for each
connection request as it arrives, and the lightpath
is released after some finite amount of time. This
is called dynamic lightpath establishment [9]. In
this paper, we restrict ourselves to static lightpath
establishment.
As explained before, the RWA problem consists
of two different underlying problems: routing and
wavelength assignment. To get really optimal
solutions, it is necessary to solve both problems
simultaneously. Nevertheless in some publications
as for example in [7] both tasks are treated
separately. That strategy has the advantage of
keeping the dimension of the related optimization
problems at a lower level. But due to the
correlation of optimal routing and wavelength
2. Problem statement and related work
2.1. Problem formulation: Max-RWA
The routing and wavelength assignment
problem (RWA) has been studied widely in the
literature [3, 5, 6, and 10]. Different formulations
of the routing and wavelength assignment are
possible. Recently, Krishnaswamy et al. [10] have
considered the Max-RWA problem, which consist
of maximizing the number of lightpaths that may
be established in a wavelength routed optical
network (WRON), given a connection matrix, i.e.,
a static set of demands, and the number of
wavelengths the fiber supports. They have
considered WRON’s with no wavelength
conversion capabilities.
To formulate the Max-RWA problem, we use
the following notation. s and d and denote
originating and terminating node of a lightpath. m
and n denote the endpoints of a physical link. k
when used as a superscript denotes the wavelength
number. q used as superscript or subscript denotes
the qth lightpath between a source–destination
pair.
a) Parameters:
N = Number of nodes in the network. λ = the
traffic matrix, i.e., λsd is the number of
connections that are to be established between
node s and node d. Plm denotes the existence of a
link in the physical topology. If Plm =1 then there
is a fiber link between node m and n, otherwise
51
A. Zyane et al, Phys. Chem. News 29 (2006) 50-55
Plm is 0. W = the number of wavelengths the fiber
can support.
∑∑
q
b) Variables:
between (s, d), else bq ( s, d ) = 0 .
( k ,q )
( s, d ) = 1 , if the qth lightpath between
node s and node d uses wavelength k; else
• Conservation of Wavelength constraints:
(6)
C ( k , q ) ( s, d ) = 0 .
• C l ,m ( s, d ) = 1 , if the qth lightpath between
( k ,q )
For all (s, d), k, q and m:
node s and node d uses wavelength k and is routed
through
physical
link
(l,
m);
∑C
(k , q)
l, m
else C l ,m ( s, d ) = 0 .
( k ,q )
l
{
the
flow
conservation
equations
multicommodity flow problems.
As mentioned before, the problem of routing
and of wavelengths assignment RWA is classified
NP-hard [12]. Therefore, the proposed methods in
the literature are based on heuristics that allow
obtaining a feasible solution in acceptable
calculation time.
Different approaches were proposed in the
literature to resolve the Max-RWA problem.
Recently, in [10], Krishnaswamy et al. have
formulated the Max-RWA problem when no
wavelength conversion is allowed as an integer
linear programme (ILP) which may be solved to
obtain an optimum solution. They solved the ILP
exactly for small size networks (few nodes). For
moderately large networks (tens of nodes) they
developed algorithms, which we called KS in
table.1 based on solutions obtained by solving the
LP-relaxation of the ILP formulation. Dzongang et
al. [4] proposed a more effective method, called
TABU-RWA and based on the graph colouring
using tabu search. The general idea of TABURWA algorithm consists of converting the RWA
problem into a graph colouring problem and of
using a neighbourhood heuristic (tabu-search) to
solve this colouring problem.
q
For all (s, d); bq ( s, d ) ∈ {0,1} :
q
(2)
s ,d
q
Remark: Ensures that number of connections
established is at most λ s,d .
• Wavelength continuity constraints:
For all q and (s, d):
W −1
∑
k =0
C ( k ,q ) ( s, d ) = bq (s, d )
(3)
Remark: This asserts that if lightpath bq (s, d )
exists then only one wavelength is assigned to it,
among the W possible choices.
For all q, (s, d ), (l , m ) and k:
C l(,km,q ) ( s, d ) ≤ C ( k ,q ) ( s, d )
(4)
Remark: expression (4) ensures that only those
C l(,km,q ) ( s, d ) could be nonzero for which the
corresponding C
( k ,q )
in
2.2. Related work
(1)
d) Constraints:
∑ b (s, d ) ≤ λ
m ≠ d.
Remark: The above constraints enforces that the
same wavelength is reserved at every node for a
lightpath bq (s, d ) . Note that they analogous to
by
c) Objective:
Maximize the number of carried connections,
that is,
max ∑∑ bq (s, d )
l
⎧ C (s, d) , if m = s
⎪ (k, q)
⎨− C (s, d) , if m = d
⎪
0
, if m ≠ s and
⎩
}
The Max-RWA problem is defined
Krishnaswamy et al. in [10] as follows:
(s, d) ⋅ Pl, m − ∑ Cm(k,l, q) (s, d) ⋅ Pm,l =
(k , q)
Note that k ∈ {0,1,2,3,..., W − 1}, where W is
the number of wavelengths the fiber can support
and for a given (s, d), q ∈ 1,2,3..., λ( s ,d ) .
s ,d
i, j
m)
Remark: By the above constraint we ensure that
no two lightpaths traversing through the physical
link (l, m) will have the same wavelength assigned
to them.
• bq ( s, d ) = 1 , if there exists a qth lightpath
•C
C l(,km,q ) ( s, d ) ≤ 1 , for all and q and (l,
( s, d ) variables are nonzero.
3. The Solving approach Proposed
Observing the fact that TABU-RWA does not
take into account the link load at the first step
• Wavelength clash constraints: (5)
52
A. Zyane et al, Phys. Chem. News 29 (2006) 50-55
wavelengths). Column UB indicates an upper
bound of the number of demands that can be
satisfied. This upper bound was computed in [10]
by using a LP-relaxation. Each column indicates
the performance (number of satisfied demands)
obtained by the KS algorithm proposed in [10],
the Tabu-RWA algorithm [4], and the MLL
algorithm.
(step of routing), we tried to study the traffic
distribution on different links of the network.
After several TABU-RWA executions, we
observed more loaded links than others and links
under-used or completely unused. Therefore, we
have a bad distribution of load at different links of
the network. This increases the blocking
probability in the network. This remark allows us
to bring a new approach for the routing while
trying to minimize the number of paths passing
through the same link; this will decrease the
utilization of the network links. Our solving
approach is based on this idea.
In this paper, we propose a new approach of
routing and wavelengths assignment in wide alloptical WDM networks with wavelengths
continuity constraint. Given a number of available
wavelengths on each optical fiber, we seek to
maximize the number of satisfied requests for
connections. This is known as Max-RWA
problem. In our approach, the routing is based on
the research of the optimal path while trying to
minimize the maximum load on the links of the
network in order to minimize the maximum links
capacity and then minimize the utilization of the
network links; which will decrease the blocking
probability in the network. After an initial step of
shortest path routing, there is a second step of
rerouting when we attempts to reroutes the most
congested paths in the network. The wavelengths
assignment is based on a graphs coloring method
using tabu-search. To solve the max-RWA
problem, we propose the MLL algorithm (RWA
with Minimum Loaded Link), which consist of
four steps as described in Figure 1.
4. Numerical results
We compare the satisfied traffics of our
approach MLL to KS and Tabu-RWA algorithms.
Each link is bi-directional and consists of two
fibers, with one fiber in each direction.
To evaluate these algorithms, we use two wellknown networks which are the NSFNET network
and EONNET network. NSFNET has 14 nodes
and 21 edges whereas EONNET has 20 nodes and
39 edges. For each network, there are set of
demands - 268 connections for NSFNET and 374
connections for EONNET. For each network, we
consider 15 different values of W (the number of
wavelengths) ranging from 10 to 24.
In Table 1, each line corresponds to a fixed
number W of wavelengths (from 10 to 24
// Step 1 (Routing) Selection of paths:
- For every request i, select the shortest path joining
S(i) to D(i) according to the number of hops using
Dijkstra algorithm.
// Step 2 (Re-routing) Re-routing of paths
passing through the congested links:
- Classify the links of the network according to their
load (number of paths passing through it); we
define the maximum load C of the network as the
load of the most congested link (link L in figure.2);
- Next, try to re-route the paths (classified according
to their number of hops) that pass through the most
congested link in order to minimize its load and
consequently the maximum load of the network;
Rerouting procedure is performed in three steps as
explained in figures 2, 3 and 4 (do Step 2.1; if it
does not succeed, do Step 2.2, otherwise step 2.3).
- Re-routing procedure will stop itself when it will
not be able to minimize again the maximum load of
the network.
// Step 3: Construct the auxiliary graph Gt using
the selected paths at Step 2.
To construct the auxiliary graph, we have to respect
the following remarks:
- The nodes correspond to all the determined paths
for all the traffics;
- The nodes are grouped together according to the
traffic to which they correspond;
- Two nodes are connected if the paths that they
represent are adjacent;
// Step 4 (Wavelength assignment):
- Colour the graph Gt while trying to maximize the
number of coloured nodes using tabu-search. A
resolution of the colouring problem corresponds to
a routing plan.
Figure 1: MLL algorithm.
53
A. Zyane et al, Phys. Chem. News 29 (2006) 50-55
D
S
Xi-2
Link L
Xi
Xi-1
Xi+2
Xi+1
Figure 2. Step 2.1 Attempt of re-routing of link L
in the direction Xi ÆX i+1
D
S
Xi-2
Xi
Xi-1
Xi+2
Xi+1
Link L
Figure 3. Step 2.2 Attempt of re-routing of link L in the
direction XK ÆX i+1 with K< X i
Xi+1
S
Xi-2
Xi-1
Xi
D
Xi+2
Link L
Figure 4. Step 2.3 Attempt of re-routing of link L in
the direction Xi ÆX k with K> X i+1
NSFNET Network
EONNET Network
Wavelength W
UB
KS
Tabu-RWA
MLL
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
198
218
238
248
258
263
267
268
268
268
268
268
268
177
184
195
205
215
226
239
250
255
263
265
267
267
268
268
197
208
218
228
237
245
252
258
261
264
265
266
267
268
268
198
210
218
230
237
246
257
262
267
268
268
268
268
268
268
UB
285
301
317
329
337
344
350
356
362
367
370
373
374
374
374
KS
TabuRWA
MLL
262
274
284
295
310
316
319
333
339
340
343
347
355
361
367
281
294
307
318
328
338
345
352
356
361
366
370
372
374
374
283
298
313
325
333
341
347
354
361
366
370
373
374
374
374
Table 1: Number of satisfied demands for KS, TABU-RWA and MLL on NSFNET and EONNET.
Note: There are two missing values in the Table (column UB), because they are clearly erroneous [4].
54
A. Zyane et al, Phys. Chem. News 29 (2006) 50-55
[3] S. Chamberland, D. Oulaï, and S. Pierre, “Joint
Routing and Wavelength Assignment in WDM
Networks for Permanent and Reliable Paths”,
Computers and Operations Research, 3, No.
(2005) 1073-1087.
[4] C. Dzongang, P. Galinier, S. Pierre, “A Tabu
Search Heuristic for the Routing and Wavelength
Assignment Problem in Optical Networks”, IEEE
Communications Letters, 9, No. 5 (2005) Page?.
[5] H. Zang, J. P. Jue, B. Mukherjee, “A review of
routing and wavelength assignment approaches for
wavelength routed optical WDM networks”,
Optical Networks Magazine, 1 No. 1 (2000) 4760.
[6] S. Ramamuthy, B. Mukherjee, “fixed alternate
routing and wavelength conversion in wavelength
routed optical networks”, Proceeding of IEEE
GLOBECOM 1998, Sydney, Austria, 4 (1998)
2295-2302.
[7] I. Chlamtac, A. Ganz, G. Karmi, “Ligthpath
communication: An approach to high bandwidth
optical WANs”, IEEE Trans. Communications,
40, No. 7, July (1992) 1171-1182.
[8] L. Li, A. K. Somani, “Dynamic wavelength
routing using congestion and neighborhood
information”, IEEE/ACM Transactions on
Networking, 7, No. 5 (1999) 779- 786.
[9] S. Xu, L. Li, S. Wang, “Dynamic routing and
assignment of wavelength algorithms in multifiber
wavelength division multiplexing networks”,
IEEE
Journal
on
Selected
Areas
in
Communications, 18, No. 10 (2000) 2130- 2137.
[10] R. M. Krishnaswamy, K. N. Sivarajan,
“Algorithms for Routing and Wavelength
Assignment Based on Solutions of LPRelaxations”, IEEE Communications Letters, 5,
No. 10 (2001) 435-437.
[11] B. Mukherjee, S. Ramamuthy, D. Banerjee,
A. Mukherjee, “Some principles for designing a
wide area optical network”, IEEE/ACM Trans.
Networking, 4 (5) (1996) 684-696.
[12] Harder E., Lee S., Choi H., “On Wavelength
Assignment in WDM Optical Networks”,
Proceedings of the Fourth International
Conference on Massively Parallel Processing
Using Optical Interconnections, (1997) pp. 32-38.
[13] S. Dixit, “IP OVER WDM: Building the Next
Generation
Optical
Internet”,
WILEYINTERSCIENCE, (2003).
From Table 1, we observe that when we have a
lot of wavelengths (W ≥ 23), the algorithms KS,
TABU-RWA and MLL have the same number of
satisfied demands (the upper bound UB) for both
networks. Otherwise, MLL finds the optimal
solution (the upper bound UB) on NSFNET when
w≥18, and on EONNET, when w≥20.
On NSFNET, both MLL and TABU-RWA
obtain the same performance in too cases, when
W=12 or 14. But, when we use a few number of
wavelengths (10≤W≤17), MLL outperforms KS
and TABU-RWA heuristic for both networks and
each value of W. On EONNET, MLL satisfies
more demands that KS and TABU-RWA when
10≤W≤19.
Comparisons with results of algorithms
proposed in the literature show that our algorithm
not only finds better solutions on the tested
instances in terms of number of routed
connections, but, also finds the optimal solution in
46% of the instances for NSFNET and 30% of the
instances for EONNET. We can also notice that
the results obtained by MLL are generally very
close to the upper bound.
5. Conclusion
Thus, the results substantiate our claim that our
heuristic achieves the objective of maximizing the
number of satisfied requests for connections, and
have shown that MLL outperforms all other
algorithms recently proposed in the literature, for
the routing and wavelength assignment in wide
all-optical
WDM
networks
considering
wavelength continuity constraint.
It would also be interesting to perform new
experiments with other types of networks like
multifiber networks and networks with links of
different capacities. We are now investigating the
way to consider the reliability and survivability
aspect in the routing and wavelength assignment
for the optical networks.
References
[1] T. E. Stern and K. Bala, “Multiwavelength
Optical Networks”, Prentice Hall PTR (1999).
[2] R. Krishnaswamy and K. Sivarajan, “Design of
logical topologies: A linear formulation for
wavelength-routed optical networks with no
wavelength
changes”,
IEEE/ACM
Trans.
Networking, 9 No. 2 (2001) 186-198.
55