Effective Traffic Grooming in WDM Rings
Abdur R. B. Billah , Bin Wang , Abdul A. S. Awwal
Dept. of Computer Science & Engineering, Wright State University, Dayton, OH 45435
Lawrence Livermore National Laboratory, P.O. Box -808, Livermore, CA 94551-0808
Abstract—
Much work has focused on traffic grooming in WDM ring networks. Previous work has considered many aspects of traffic
grooming, including minimizing number of ADMs, minimizing
number of wavelengths, single hub and multiple hub architectures, switching capabilities etc. In this work, we derive general
and tighter bounds on the number of wavelengths in WDM ring
networks that may be capable of switching streams across wavelengths. We also study traffic grooming in unswitched UPSR rings
and derive a more general lower bound for the number of ADMs
required in traffic grooming. This bound reduces to special cases
first obtained in [1]. A cost-effective multi-phase traffic grooming
algorithm is proposed and studied for UPSR rings. Our numerical results show that this algorithm outperforms existing traffic
grooming algorithms, resulting in traffic grooming that uses lower
number of ADMs. Our algorithm in many cases also reaches the
general lower bound derived.
I. I NTRODUCTION
Optical wavelength division multiplexing (WDM) network
technologies promise to offer vast amount of bandwidth by delivering data over multiple channels using multiple wavelengths
simultaneously [2]. Currently, dense WDM (DWDM) technology can already achieve up to wavelengths per fiber with
each wavelength carrying Gb/s, resulting in a total transmission capacity of up to Tb/s [3].
There is a growing mismatch between the transmission capacity of optic fibers and the electronic switching capability.
The transmission capacity of each wavelength is expected to
increase further with the advancement of technology. Although
electronic processing and switching are struggling to keep pace
with the transmission capacity of fiber optics, it is just a matter of time for electronics to become the bottleneck. Moreover,
the cost of high speed electronics is becoming economically
unattractive. The design and operation of revenue generating
optical WDM networks require to reduce cost. One important part of the cost is the cost of the network itself. It is thus
imperative to reduce costly electronics by enabling optical bypassing as much as possible. Wavelength ADMs (WADMs)
are capable of dropping (and adding) only the wavelengths
carrying traffic destined to (and originated from) a node. At
the same time, services requiring sub-wavelength capacity are
necessary. Individual traffic streams are likely to have small
bandwidth requirements compared to the bandwidth available
in a single wavelength. To efficiently utilize the bandwidth
on a wavelength, low rate traffic needs to be properly multiplexed at the ingress point and demultiplexed at the proper
Abdur R. B. Billah, Bin Wang: E-mail: abillah,bwang @cs.wright.edu.
Abdul A.S. Awwal: E-mail: awwal1@llnl.gov. The work reported in this paper
was supported in part by an ITEC-Ohio TAF grant and a DAGSI scholarship.
The work of Abdul A. S. Awwal was performed under the auspices of the U.S.
Department of Energy by the University of California, Lawrence Livermore
National Laboratory under contract No. W-7405-Eng-48.
egress point. Synchronous Optical Network (SONET) rings are
widely used in today’s network infrastructure. Each SONET
ring is constructed by using fibers to connect SONET ADMs.
An ADM can multiplex multiple lower rate streams to form
higher rate ones. The desire to cost-effectively utilize bandwidth in a WDM system gives rise to the concept of traffic
grooming which is defined as techniques used to combine lower
rate traffic streams onto available wavelengths in order to meet
user service requirements and to minimize cost.
Much work has focused on traffic grooming in WDM ring
networks [1, 4–13]. Previous work has considered many aspects
of traffic grooming, including minimizing number of ADMs,
minimizing number of wavelengths, single hub and multiple
hub architectures, switching capabilities etc. In this work, we
derive general and tighter bounds on the number of wavelengths
in WDM BLSR networks with wavelength conversion capability. We also study traffic grooming in unswitched UPSR rings
and derive a more general lower bound for the number of ADMs
required in traffic grooming. This bound reduces to special
cases first obtained in [1]. A cost-effective traffic grooming
algorithm is proposed and studied for UPSR rings. Our numerical results show that this algorithm outperforms existing
traffic grooming algorithms, resulting in traffic grooming that
uses lower number of ADMs. Our algorithm in many cases
also reaches the general lower bound derived.
The rest of this paper is organized as follows. Section II derives general bounds for the number of wavelengths needed in
BLSR networks. Section III gives a summary on known bounds
on the number of ADMs required for traffic grooming in WDM
rings. A new and more general lower bound is derived for
unswitched UPSR rings. The proposed traffic grooming algorithm is described in Section IV. Performance evaluation and
comparisons with existing algorithms are reported in Section V.
The paper concludes in Section VI.
II. N EW B OUNDS ON THE N UMBER OF WAVELENGTH
The lower bounds on the number of wavelengths necessary
for traffic grooming in both WDM UPSR and BLSR/2 networks
were studied in previous work including [4, 8, 12]. The lower
bounds derived in [12] are also applicable in WDM BLSR/4
networks. The lower bounds in [8] have been calculated algorithmically with no mathematical formulation. The bounds
for WDM BLSR networks in [8] is applicable only to BLSR/4
and can easily be modified to be applied on WDM BLSR/2 networks. In this paper, taking a more general perspective, we
derive the lower bounds on the number of wavelengths needed
in BLSR ring networks under static uniform traffic (full-duplex
and all-to-all). We believe all the bounds presented in this paper
are achievable.
We use , and to represent the number of nodes, the
granularity
of a wavelength, and the size of a node-traffic
in terms of low rate tributary streams (e.g., OC-3), respectively. We considered the impact of traffic splitting on the lower
bounds on the number of wavelengths. Our bounds for WDM
BLSR networks are found to be tighter than those obtained in
[12] when is greater than .
A. BLSR Rings
In WDM BLSR networks, a traffic stream from node to
node is carried through the shortest path route, i.e., with the
minimum number of hops. The maximum number of hops is,
therefore, limited to for an -node ring. However, each
wavelength in a BLSR/2 ring can carry only up to of its
full capacity. The rest of its capacity is reserved for carrying
backup traffic for protection [4]. We calculate the lower bound
for BLSR rings separately for odd and even number of nodes.
We first compute the number of “full circles” (FCs) that are required to support all-to-all uniform traffic of low rate streams
in one direction. The traffic in the reverse direction is carried
through another fiber in exactly the same way. The number of
wavelengths is then calculated in terms of FCs, and .
1) FCs in BLSR Rings with Odd Number of Nodes: In a
BLSR ring with odd number of nodes, there is always a single
shortest path from a source to a destination. The number of FCs
can be determined by observing the following properties:
The maximum node-distance between any pair of nodes
is "! . So, traffic from a source node # to every other
destination node $ goes through %&
'('
' "! hops.
The number of paths with node-distance )*+)
,
%-('
'('./ "! is equal to the number of nodes in the
network.
The number of FCs consisting of paths with node-distance
) (or ) number of hops) is always ) , where )0,
%-('
'('/ 1! .
For example, only one FC is required to carry all nodetraffics between adjacent nodes in one direction. Two FCs are
required to carry traffic streams between nodes that are two
hops apart, and so on. The total number of FCs is therefore
1! , :9(; 1! .
given by: 324526 27'('
' /8
2) FCs in BLSR Rings with Even Number of Nodes: The
lower bound on wavelength required in the case of even number of network nodes given below is tighter than that derived in
[14], and is achievable. In a WDM BLSR ring with even number of nodes, there is always a single shortest path between a
pair of nodes at a distance less than or equal to / . However,
there are two shortest paths between a pair of nodes that are
at a distance of . The number of FCs can be determined by
observing the following properties:
The maximum node-distance between any pair of nodes is
. So, traffic from a source node # to every other destination node $ goes through %&
'('
' hops.
The number of paths with a node-distance ) is +)<,
%-('
'(' / .
The number of FCs with node-distance ) (or ) number of
hops) is always )*%)=,<%-('
'('/ .
The number of paths with a node-distance is .
Fig. 1. BLSRs with even number of nodes,three paths can be accommodated
on two wavelengths if nodes are capable of wavelength switching.
As stated above the number of paths between a pair of nodes
at node-distance is . By carefully assigning each traffic
to an FC, it is possible to accommodate them in > ?A@ 2B FCs.
We note that this assignment requires wavelength conversion
(switching) capability in some of the nodes (e.g., Figure 1).
Thus, the total number of FCs is given by: CEDGF:HJIK,<L2MN2
2O'
'(' / 2P> ? @ 2O
SR
, QJ ; 2T> ? @ 27 . In other words,
U
G; 9 2O if is a multiple of 4,
CEDGF:HJIK,
V
if is even, not a multiple of 4.
;K9 W
(1)
3) Lower Bounds on Wavelength in BLSR/2 Rings: The
minimum number of wavelengths required in a WDM BLSR/2
network for various scenarios can be formulated in terms of the
FCs determined above.
Case 1: If traffic splitting is allowed or X is a multiple of
Y Y
( ,7 mod X ), Z[FGHJI\,
_
]^^
R
Red
V
Qc:; 9."! 2
Qc:9.? "! W
;R
;
X Rgd
V
^^`ba Qc:; 9f
Qc:9? f
W
2
V iSj km9 l X d
h V
a :;K9 W 2
W
X
if is odd,
if is a multiple of 4,
if is even, not a multiple of 4,
(2)
Case 2: If traffic splitting is not allowed and X is not a
Y
multiple of , ZnF:HJIK,
]^^
R
R
V
^
;oL
p ! W
Qc:; 9."! 2
:
9."
if is odd,
_
;R
V 9rqs ; R
; i 9pf l W if is a multiple of 4,
^^ a Qc:; 9f
Qc:
2
^`
V i j km9 l 9tq
ha V
o p W
:; 9 W 2
if is even, not a multiple of 4,
r9 qs
(3)
Y
where ,7 div X and ,u mod X .
The above bounds can be extended easily for calculation of
lower bounds for BLSR/4 networks, and is omitted here.
h
III. T RAFFIC G ROOMING AND N EW ADM B OUNDS
In this section, we discuss the number of ADMs required for
traffic grooming in WDM UPSR ring. It has been shown in [1,
8] through examples that it is not always possible to minimize
the number of wavelengths and the number of ADM’s simultaneously.
The bounds on the number of ADMs in WDM rings have
been addressed in many papers including [1, 4, 6–8, 12]. For
uniform all-to-all traffic, lower bounds on numbers of ADMs
required for UPSR and BLSR/2 for wvx have been formulated in [4]. The bounds assume wavelength switching (conversion) capability in the network. Elaborate discussions on stream
switching can be found in [4]. Lower bounds on the number of
ADMs for specific cases (y,z6,|{ and },z6,z~ )
for uniform
traffic in UPSR rings have also been derived in [1].
€
The bounds in [4] is tighter than the corresponding ones in [1],
as the latter did not assume switching capability of the network.
The lower bound on the number of ADMs for distance dependent traffic (for the case in which 6,{ ) in UPSR rings was
given in [1]. Lower bounds on the number of ADMs have been
calculated algorithmically for both unidirectional and bidirectional rings in [8] with uniform and nonuniform traffic streams.
However, no closed form expressions have been derived. Note
that the lower bounds derived in [1] for unswitched UPSR rings
is loose.
In this section, we derive a general lower bound on the number of ADMs for uniform all-to-all traffic in UPSR WDM rings
without wavelength switching (conversion). We note that with
switching, the bound is tighter, but the wavelength continuity
constraint is not preserved and the cost of switching is also not
negligible [6].
A. General Lower Bound for Unswitched UPSR Rings
Theorem 1: When ƒ‚„ , the minimum number of ADMs in
a UPSR WDM ring with no wavelength switching for uniform
all-to-all traffic (where each pair of nodes have low rate traffic
streams between them) is at least
Y
V
}†-‡
6g‘w’“g”•2
’
Y
Y W (4)
,uˆŠ‰Œ‹"Ž
”–—”M4Ž’“˜™
’N24N>š-›œ @
o p£
;
Y
if
f
Ÿ
f
I s y”[¤¥
X
ž 9 ž @ and ,¢¡
where ”Š,>“ž
"!
- otherwise.
ž
Proof. The first term in the maximum is trivial, so for this proof
we will only consider the second term. We first define a cluster
as a group of nodes whose pairwise all-to-all traffic streams will
be groomed on or supported entirely by the same wavelength
¦
. We then determine the cluster size ” by noting that the numI I R
ber of pairwise low rate streams among ” nodes is Q 1! ž .
I I R
Thus, ” can be determined from Q 1! ž ‚T . Solving for ”
;
as an integer from the inequality, we have ”4,§>“ž f Ÿ ž 9 f ž X @ .
The number of ADMs necessary to support the trafficž within
the cluster is ” , one at each node. Therefore, the average
number of low rate
traffic streams (e.g., OC-3) supported by
I R
one ADM is Q " ! ž within the cluster. The amount of bandwidth left (inR terms of low rate streams) on the wavelength
I I
Q 1! ž . It is then possible to groom more internode
is Š
traffic onto this wavelength. However, more ADMs may be
required.
To maintain the efficiency of ADM utilization at
I R
Q 1 ! ž low-rate streams per ADM or improve the ADM utilization efficiency, internode
trafficR streams will be groomed onto
¦
I I R
I
Q
"
!
ž ¤ Q " ! ž . That is, one more ADM is
only if •
R
¦
I
added with at least Q 1 ! ž or more streams groomed onto if
oSp£
of ADMs is mainI s O”¨¤© . Therefore, the efficiency
o p£
"!
tained or even improved. Otherwise, if I s ª”[v¥ , no more
"! ¦ . We then define
additional traffic U streams are groomed onto
o p£
Y
Y
if I s [”y¤«
an integer ,
As a result, ”n2
1!
otherwise.
I I R
I I R Y
Q 1! ’ internode trafADMs can groom Q 1! 2P
> X @ 
ž
fics. The total number of internode
traffics (in terms of ) in
R
an -node ring is /Qc/ 1! . The minimum number of ADMs
R
d
V
†-‡
/Qc/ "! ' ¬­®¬œ¯Œ°e± oI p£ f ¬Ž­²¬¯Œ°g± Rgd W
,
required is then
fAQ
R I dR
V
9
s
9
d W .
µ
,
I I QJR "! dR Q f o
Y
Q 1! Q³!. f X“´ ž
s R M,¶{ , then
Remarks.
R ”ª,< and ,· . There…}†-‡ If =? ,¢ and
œQJ ¸ ? "! , /Qc/ "! , which is the lower bound
fore,
,
Y
obtained in [1]. If ¹,b andRS¸-º
,
~ , then ”4R ,T
~ and ,· .
»
V
V
}†‡
W ,
/Qc/»œ1
/Qc/(1
¸&¼!
½ ¼ ! ´ W which is exTherefore,
,
R V
½ ¼ ! ´ W , in [1] .
µ
actly the bound, QJ ("
IV. P ROPOSED T RAFFIC G ROOMING A LGORITHMS
In order to minimize the number of ADMs, heuristic algorithms have been proposed for grooming uniform and nonuniform traffic streams in UPSR and BLSR WDM networks in
several papers including [1, 5, 8]. Mathematical expressions
on the number of ADMs for presumably BLSR/4 networks are
presented in [5] using super-node approximation for both uniform and distance-dependent (nonuniform) traffic. The lower
bounds presented in [5] are essentially for networks without
switching capabilities and is, therefore, slightly looser than
those mentioned in the previous section. Algorithms and closed
form expressions for specific cases (e.g., O,¾¿,©{ and
¹,‘º,~ ) have been derived in [1] for uniform traffic in
UPSR networks. These traffic grooming algorithms are based
on forming multiple groups of nodes. The number of nodes in
each group is decided by the value of - the granularity of the
wavelength. A heuristic greedy algorithm is also proposed for
distance dependent traffic model in [1, 5]. Heuristic algorithms
proposed in [8] are more general and comprehensive in dealing
with both uniform and arbitrary non-uniform traffic in UPSR as
well as BLSR networks. The traffic can be sub-wavelength, full
wavelength, and super-wavelength streams.
In this section, we propose an algorithm for grooming uniform all-to-all traffic streams in UPSR WDM rings to cover the
entire spectrum of traffic stream sizes from 4,À to 4,Á
regardless of whether and are multiples. It does not require
any switching capability [4] in the network. We believe that
the proposed algorithm is superior to those described above in
terms of minimizing the number of ADMs as well as dealing
with traffic streams of various sizes. In many cases it results in
the number of ADMs equal to the lower bound derived in the
previous section as well as in [1] (i.e., the case in which traffic
streams are not switched).
Our experience and analyses show that properly balancing
the utilization of ADMs across the nodes tends to minimize
the total number of ADMs. On the other hand, in a unidirectional ring network without switching capabilities, for any
traffic grooming solution with split traffic, there exists a corresponding solution without split traffic that uses the same number of or fewer ADMs [1]. Based on the above two notions,
we mainly focused on balancing the ADM utilization at every
node and not splitting a traffic stream into more than one wavelength. The algorithm is divided into four phases. The first
phase, Phase 0, is to construct full circles - one for the traffic
streams between each pair of nodes. The other three phases are
used for grooming the circles constructed in Phase 0. We note
that not all the phases are required for grooming each nodetraffic, and the phases are not always necessarily in sequence.
TABLE I
a) Phase 0: This is the initial phase of constructing cir- C OMPARISON OF NUMBER OF ADM
S : ÄÆÅ[ÇÈ , ÉLÊ IS THE LOWER BOUND
cles from the traffic matrix. We borrow the idea of circle conOBTAINED IN THIS PAPER , ÉLÊ–Ë IS THE LOWER BOUND OBTAINED
ALGORITHMICALLY IN [15], ÌŠÍ IS THE PROPOSED MULTI - PHASE
struction from [8]. However, each of our circles contains trafALGORITHM , ËAÎ IS THE GROUPED ALGORITHM PRESENTED IN [1], AND
fic streams equal to the node-traffic between a pair of nodes.
ËÐÏ IS THE GREEDY APPROACH ALGORITHM PRESENTED IN [15].
For UPSR WDM rings the circle construction algorithm is quite
r=1
r=4
N
LB
LBG
MP
GR
GA
LB
LBG
MP
GR
GA
straightforward. In our terminology, a full circle is formed out
4
4
4
4
4
4
6
6
7
8
7
5
5
5
5
5
5
10
10
11
13
11
of two traffic streams between two nodes. In UPSR rings, both
6
6
6
6
6
6
15
15
15
18
17
7
9
10
11
11
13
21
21
21
26
23
traffic streams traverse in the same direction, forming a full cir8
12
12
14
16
15
28
28
29
32
30
9
15
16
18
18
20
36
36
38
42
37
cle.
10
18
18
20
20
21
45
45
46
50
48
11
22
23
27
29
30
55
55
57
63
60
12
27
28
34
36
35
66
66
66
72
72
b) Phase 1: We divide this phase into two sub-phases.
13
32
33
39
39
42
78
78
78
87
84
Phase 1a: In this phase, we make use of the concepts of
14
37
37
42
42
48
91
91
91
98
99
15
42
42
55
55
58
105
105
105
116
112
16
48
48
62
64
65
120
120
120
128
130
grouping nodes introduced in [1] and super-nodes in [5].
17
55
68
68
136
136
148
18
62
72
72
153
154
162
19
69
89
89
171
172
185
We call it clustering to distinguish from grouping in which
20
76
97
100
190
191
200
the number of nodes in each group is taken equal to Ÿ [1]. The number of nodes in a cluster, however, is equal
to the number of a set of nodes whose all-to-all traffic can
be accommodated in one wavelength without splitting any Ñ
Ñ
stream. Thus, for O,©-S{+ , and ¥,Â~ , the cluster
size is ; for },Á the cluster size is { and for },© ,
it is ~ . This phase of grooming uses only one ADM per
node and supports at least one or more circles per ADM
depending on the value of . Each cluster requires one
(a)
(b)
new wavelength.
Fig. 2. Number of ADMs for uniform traffic in UPSR for Ä:Å}ÇÈ (a) Ò3ÅªÇ ;
Phase 1b: This sub-phase grooms cross-traffic [1] across (b) ÒAŊÓÔgҖÅMÕ .
the nodes from two different clusters. We call it crossis full.
grooming. We consider cross-traffic grooming more gen5) Step 5: If no circle is found to groom in Step 2, unerally to include both equal and unequal number of nodes
groom the circle groomed in Step 1 and release the
from two clusters. Grooming between disjoint nodes from
wavelength.
two clusters taking three nodes from one cluster and five
6) Step 6: If Phase 2 is not applied before for reasons
nodes from another in the case with },z , is an examexplained above, apply Phase 2 here. If there are
ple of unsymmtric cross-grooming. The algorithm crossmore circles to be groomed, go to Phase 3b.
Phase 3b
grooms only if it results in effective use of ADMs, otherwise it goes to the next phase.
1) Step 1: Take a wavelength and groom a circle onto
c) Phase 2: This phase grooms more circles onto the
it.
wavelengths created in the previous phase only ’ if a wave2) Step 2: If the wavelength is full, start a new wavelength has enough capacity to accommodate one or more circles
length.
and e’ if the addition of an ADM does not decrease the factor
3) Step 3: Repeat Step 1 and 2 until there is no more
“circles per ADM” (i.e., the efficiency of ADM utilization). On
circles to be groomed.
the other hand, if the remaining capacity in the wavelength after
V. N UMERICAL R ESULTS
Phase 1 is such that addition of one more ADM creates a situaIn
this
section,
we present and analyze the results obtained
tion where more than one circles can be groomed onto the same
from
our
multi-phase
heuristic algorithm and compare them
wavelength in order to increase the factor “circles per ADM”,
with
the
lower
bounds
derived
in the previous section. We also
Phase 1 and Phase 2 can go in parallel rather than in sequence.
compare
the
results,
where
applicable,
with the bounds and reThis is true in cases with Ã,B and Ã,7 . This phase does not
sults
in
[1],
[15]
and
[4].
Table
I
compares
our results with
need to create any new wavelength.
1
those
in
[1]
and
[15]
for
two
cases
(
Ã
¶
,
m
7
,
d) Phase 3: This is the final phase that grooms any cir- ÖÆ×
ÖÆ×Ù{ Ø and =,¶
~ ).
is
the
lower
bound
obtained
in
this
paper,
is the lower
cles left ungroomed after the previous phases and completes the
¿Ú
bound
obtained
algorithmically
in
[15],
is
the
multi-phase
grooming process. It uses a simple heuristic approach to groom
ØÃÛ
is the grouped algorithm
circles which can not otherwise be groomed efficiently. We di- algorithm proposed in this paper,
ØÃÜ
presented
in
[1]
and
finally,
is
the
greedy approach algovide it into two sub-phases:
rithm presented in [15]. We analyze the results case by case
Phase 3a
based on the value of node-traffic .
1) Step 1 : Take a wavelength and groom a circle onto
A. Case m,<
it.
Figure 2(a) compares the number of ADMs resulted from
2) Step 2: Groom another circle, if any, with one overour multi-phase heuristic algorithm with corresponding lower
lapping end node.
bounds derived in Section III as well as with those obtained
3) Step 3: Groom as many circles as possible, if any,
in [1] and [15]. Our algorithm performs almost always better
with two overlapping end nodes.
Ý
4) Step 4: Repeat Step 2 and Step 3 until the wavelength
We obtained the results of [15] upto 16 nodes from [16].
Multi-phase Algorithm
r=1,g=16, Lower Bound
Grouped Algorithm
Greedy Approach
100
160
r=2
r=2 Lower Bound
r=3
r=3 Lower Bound
140
80
Number Of ADMs
Number Of ADMs
120
60
40
100
80
60
40
20
20
0
0
4
6
8
10
12
14
Number Of Nodes
16
18
20
4
6
8
10
12
14
Number Of Nodes
16
18
20
200
Multi-phase Algorithm
r=1,g=4, Lower Bound
Grouped Algorithm
Greedy Approach
Multi-phase Algorithm
r=5,g=16, Lower Bound Unswitched
Lower Bound Switched
SingleHub
200
Multi-phase Algorithm
SingleHub
Lower Bound Switched
r=8,g=16, Lower Bound Unswitched
350
300
150
Ñ
100
Number Of ADMs
Ñ
100
Number Of ADMs
Ñ
Number Of ADMs
150
250
200
150
100
50
50
50
0
0
4
6
8
10
12
14
Number Of Nodes
16
18
20
0
4
6
8
10
12
14
Number Of Nodes
(a)
16
18
20
(b)
4
6
8
10
12
14
Number Of Nodes
16
18
20
(c)
Fig. 3. Number of ADMs for uniform traffic in UPSR for Ä3ÅnÇÈ (a) ÒAÅMÞ ; (b)ҖÅMß ; (c) ÒAŊà .
than the greedy approach taken in [15]. It also exhibits as good
as or better performance in some instances than the “grouped”
algorithm presented in [1] (see left part of Table I).
B. Case E,7-m,«
Figure 2(b) illustrates the performance of our algorithm relative to the lower bounds for =,T and \,¶ . In general, the
algorithm exhibits good performances and the results are close
to the lower bounds. We do not have the exact numerical results of [8], but we believe, from rough comparison, that our
algorithm performs better than theirs (for the case of ª,á ).
We could not compare our results for E,« with other work as
we are not aware of any paper in the literature that illustrated
this case.
C. Case E,O{
This case is equivalent to the case \,B{SE,P as presented
in [1, 8, 15]. Figure 3(a) compares the number of ADMs obtained from our multi-phase algorithm with the lower bounds
derived as well as with the results from [1, 15]. It is evident
that our algorithm clearly outperforms the grouped algorithm
[1] in almost every cases. Table I (right part) shows that our
lower bound derived in Section III is exactly the same as that
calculated algorithmically by [15].
D. Case E,7
Figure 3(b) compares the number of ADMs resulted from
our multi-phase heuristic algorithm with corresponding lower
bounds derived in Section III. The algorithm performs very
well in general and uses ADMs close to the lower bounds. Included in this figure are the lower bound for switched UPSR
and results for single hub architecture obtained from [4]. The
multi-phase algorithm, which does not require any switching
capability, outperforms the single hub architecture in every instance. Our algorithm applies to networks without switching
capabilities, yet requires fewer number of ADMs as well as
fewer wavelengths. We could not find any similar cases from
[1, 8, 15].
E. Case E,Oâ
Figure 3(c) shows the results of comparison between the
number of ADMs used by the multi-phase heuristic algorithm
and the lower bound derived in Section III. As observed from
the figure, the algorithm results in the same number of ADMs
as the lower bound in all cases studied. We note that the lower
bound obtained in [8] is the same as ours and their heuristic algorithm (greedy approach) performs the same as ours. Included
in this figure are the lower bound for switched UPSR and the results for single hub architecture obtained from [4]. The multiphase algorithm, which does not require any switching capability, overwhelmingly outperforms the single hub architecture in
every instance.
VI. C ONCLUSIONS
In this work, we derive general and tighter bounds on the
number of wavelengths in WDM BLSR networks that may
be capable of switching streams across wavelengths. We also
study traffic grooming in unswitched UPRS rings and derive a
more general lower bound for the number of ADMs required
in traffic grooming. This bound reduces to special cases in
previous work. A cost-effective multi-phase traffic grooming
algorithm is proposed and studied for UPSR rings. Our numerical results show that this algorithm outperforms existing traffic grooming algorithms, resulting in traffic grooming that uses
lower number of ADMs. Our algorithm in many cases also
reaches the general lower bound derived. We are currently extending the techniques developed to groom more general traffic
patterns as well as study traffic grooming in other types of ring
networks with or without switching capabilities.
R EFERENCES
[1] E. Modiano and A. Chiu. Traffic grooming algorithms for minimizing
electronic multiplexing costs in WDM ring networks. IEEE Journal of
Light-wave Technology, January 2000.
[2] R. Ramaswami and K. N. Sivarajan. Optical networks: A practical perspective. Morgan Kaufmann Publisher, 1998.
[3] U. Black. Optical networks: Third generation transport systems. Prentice
Hall PTR, 2002.
[4] O. Gerstel, P. Lin, and G. Sasaki. Combined WDM and SONET network
design. Proceedings of INFOCOM, New York, pages 734–743, 1999.
[5] J. Simmons and A. Saleh. Quantifying the benefit of wavelength add-drop
in WDM rings with distance-independent and dependent traffic. IEEE
Journal on Light-wave Technology, 17:48–57, 1999.
[6] E. Modiano and R. Berry. The role of switching in reducing network port
counts. Proceedings of the 39th Annual Allerton Conference on Communication, Control, and Computing, Allerton, Illinois, September 2001.
[7] E. Modiano and R. Berry. Switching and traffic grooming in WDM networks. Joint Conference on Information Sciences (JCIS), Durham, North
Carolina, March 2002.
[8] X. Zhang and C. Qiao. An effective and comprehensive approach for
traffic grooming and wavelength assignment in SONET/WDM rings.
IEEE/ACM Transactions on Networking, pages 608–617, 2000.
[9] R. Berry and E. Modiano. Reducing electronic multiplexing costs in
SONET/WDM rings. IEEE Journal on Selected Areas in Communications, October 2000.
[10] T. Chow and P. J. Lin. The ring grooming problem. SIAM J. Discrete
Math., 2001.
[11] O. Gerstel, P. Lin, and G. Sasaki. Wavelength assignment in a WDM
ring to minimize the cost of embedded SONET rings. Proceedings of
INFOCOM, San Francisco, pages 94–101, 1998.
[12] R. Ramaswami O. Gerstel and H. Sasaki. Cost effective grooming in
WDM rings. IEEE/ACM Transactions On Networking, 8(5), October
2000.
[13] R. Dutta and G. N. Rouskas. On optimal traffic grooming in WDM rings.
IEEE Journal on Selected Areas in Communications, pages 110–121, January 2002.
[14] O. Gerstel and R. Ramaswami. Cost effective grooming in WDM rings.
Proceedings of INFOCOM, San Francisco, April 1998.
[15] X. Zhang and C. Qiao. An effective and comprehensive approach to traffic grooming and wavelength assignment in SONET/WDM rings. SPIE
Proceedings of Conf. All-optical Networking, 3531:221–232, September
1998.
[16] W. Cho, J. Wang, and B. Mukherjee. Improved approaches for costeffective traffic grooming in WDM ring networks: Uniform- traffic case.
Photonic Network Communications, pages 245–254, 2001.