Wide-Sense Nonblocking Multicast in WDM Optical Linear Array and
Ring Networks with 2-Length Extension under Index Based Routing
M. SABRIGIRIRAJ1 K. MANOHARAN2 R.KARTHIK3,
1Department of ECE, SVS College of Engineering, Coimbatore, 642 109, Tamilnadu, India
2Department of ECE, SNS College of Technology, Coimbatore, 641 035, Tamilnadu, India.
3Department of ECE, Sri Krishna College of Technology, Coimbatore, 641042, Tamilnadu, India.
Abstract: - Multicast communication is the concurrent transmission of data from one source node to many
destination. It is widely deployed in high performance computing and communication networks.In this paper, a
linear array and ring networks are extended by directly linking all nodes which are separated by one
intermediate node with additional fibers which is referred as linear array and ring network with 2-length
extension and wide sense nonblocking multicast is studied. The wavelength allotment methods are proposed to
realize one-to-many communication over wavelength division multiplexing optical linear array and ring with 2-
length extension under index-based routing and the minimum wavelength number needed is determined. The
minimum wavelength number needed to support for the extended linear array and ring topologies is reduced
approximately by half when compared with that of a linear array network and ring network.
Key-Words: -Wavelength division multiplexing (WDM), Multicast communication, WDM Optical Network,
Wavelength Assignment, Modified Linear Array
Received: August 28, 2021. Revised: April 5, 2022. Accepted: May 7, 2022. Published: June 23, 2022.
1. Introduction
The modern scientific developments in
optical networks based on wavelength division
multiplexing (WDM) are more attractive to satisfy
the high bandwidth necessities of the internet
infrastructure. Additionally, they promise to satisfy
the future bandwidth needs.WDM optical networks
act as the backbone for telecommunication and
high-performance networks. An optical network is a
communication system that uses light signals to
transmit information between two or more points.
Optical networks are based on optical technologies
and components, and are used to route, groom, and
restore wavelength levels and wavelength-based
services.
Multicast communication is the concurrent
transmission of data from one source node to many
destination nodes.It is widely deployed in high
performance computing andcommunication
networks. Multicast assignment [1,2] is a well-
defined type of multicast communication and is
widely used in traffic analysis. Multicast assignment
involves establishing connections between various
nodes of a network in such a way that each
destination node is connected to only one source
node, whereas each source node may be connected
to one or more destination nodes. All forms of
multicast communication can easily be broken down
into multiple multicast assignments. Due to the
absence of optical buffering at the nodes in an
optical network, it is preferred to have a
nonblocking network. Otherwise, data would be
lost with blocked connections. An optical network
can be termed as nonblocking, if it is possible to
establish all the connections of the given multicast
communication without removing or rerouting any
of the existing connections. Such networks are said
to be wide-sense nonblocking [1], if the connections
are established by a definite routing
algorithm.Wide-sense nonblocking multicast is
studied in electronic switching networks and
network topologies namely linear array, ring, torus,
mesh and hypercube [1].Wide-sense nonblocking
multicast and strict-sense nonblocking multicast is
studied for clos networks [3] and elastic optical
switch [4]. Multicast communication is alsostudied
for various optical networks namely clos
network,benes network, elastic optical networks and
for general network topologies [5–10].
Linear array [11-15] and ring networks [16-
20] are the basic network topologies suitable for
interconnection networks and for LAN /WAN and
are well investigated. The linear array and ring
networks are widely adopted for LAN, MAN and
WAN and also used ininterconnection networks due
to its regularity and small node degree. The results
obtained in this paper can be applied for practical
long-haul networks like mesh network, since mesh
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network can be decomposed into multiple linear
array and/or ring networks.
Multicast communication is investigated
well under different scenarios [21-26] in WDM
networks. Wide-sense nonblocking Multicast
communication in linear array with 2-length
extension and unidirectional ring and bidirectional
ring with 2-length extension networks are studied
under longest link first routing is studied already
[24]. In this paper, for the same topologies, index-
based routing strategy is employed to route the
connections based on the index of the source and/or
destination nodes. Particularly, the most important
attention is over the determination of the necessary
and sufficient condition over the minimum
wavelength number needed for the network to be
wide sense nonblocking. Explicit wavelength
allotment techniques are also proposed for each of
the network topologies.
Section 2 gives preliminaries needed to
analyze wide-sense nonblocking multicast problem
in linear array, unidirectional ring and bidirectional
ring networks with 2-length extension under index-
based routing is discussed. Section 3, the necessary
and sufficient condition on the minimum
wavelength number required to realize wide-sense
nonblocking multicast under index-based routing is
derived and explicit wavelength allotment
techniques are also given. Section 4, discusses the
result obtained for the 2-length extension networks.
Finally, section 5 completes the paper highlighting
future research avenues.
2. Preliminaries
Figure 1(a) and 1(b) shows a 8-node basic
linear array network and a linear array network
with 2-length extension respectively. A linear array
network with 2-length extension is obtained by
additionally connecting the alternate nodes of the
linear array. Each node in the linear array is
additionally connected to an alternate node. At each
node, data can move from node x to node x+1 and
also directly to node x+2, if such nodes exist and
also vice-versa.
Figure 1 (a) An 8-node basic linear array
Figure 1 (b) An 8-node linear array with 2-length
extension
Figure 2(a) and 2(b) shows an 8-node basic
ring network and an 8-node ring network with 2-
length extension respectively. A ring network with
2-length extension is obtained by additionally
connecting the alternate nodes of the basic ring.
Each node in the ring is additionally connected to an
alternate node. At each node, data can move from
node x to node (x+1) mod N and also directly to
node (x+2) mod N.
Figure 2 (a) An 8-node basic ring
Figure 2 (b) An 8-node ring with 2-length
extension
Definition 1: A link that joins two nodes and
(where denotes mod N) is called a
shorter link. For example, in Figure 2,the link that
joins node 0 with node 1 is a shorter link.
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Definition 2: A link that directly joins two nodes,
both labeled with even index is termed as an ‘even
link’. For example, in Figure 2,the link that directly
joins node 2 with node 4 is aneven link.
Definition 3: A link that directly joins two nodes,
both labeled with odd index is termed as an ‘odd
link’. For example, in Figure.2,the link that joins
node 1 with node 3 is an odd link.
Definition 4: A connection (x, y) is defined as the
set of all links that joins source node x with
destination node y under a prescribed routing
method.
3. Main Results
Wavelengths are scarce resources in an
optical networks and its usage need to be minimized
to reduce the cost and complexity of the network. In
this section, the sufficient and necessity condition
on the minimum wavelength number required for
the network to be wide-sense nonblocking is derived
for linear array and unidirectional ring networks
with 2-length extension under index-based routing
technique.
3. 1 Linear Array with 2-length extension
Figure 1(b) shows an 8-nodelinear array
with 2-length extension. The following assumptions
are made to reduce the wavelength number under
index-based routing. If the index of both source and
destination are even, then such lightpaths are routed
only using even links. If the index of both source
and destination are odd, then such lightpaths are
routed only using odd links. All other lightpaths are
routed using shorter links only. Theorem 1 is proved
based on this assumption.
Theorem 1:The sufficient and necessary condition
on the wavelength number for a node linear array
with 2-length extension to be wide-sense non-
blocking is.
Proof: sufficiency: It is to be noted that all the
lightpaths between various sources and destinations
are routed either in rightward direction or in
leftward direction. For a lightwave network, a
particular wavelength can be allotted for multiple
lightpaths as long as they do not overlap with one
another. A wavelength released by a current
lightpath during its termination can be again used to
allot for a new lightpath establishment. It can be
noted that out of multicast lightpaths, there can be
atmost lightpaths either in rightward direction
or in leftward direction. Out of these
lightpaths, there can be a maximum of
lightpaths (first set) established only on the
shorter links whereas the remaining lightpaths
(second set) are established only on the longer links.
So, these two sets of lightpaths can be on the same
set of wavelengths and so needs
wavelengths.
Reusing any one of these
wavelengths, the
remaining lightpath in the opposite direction can be
established. So,
wavelengths are sufficient to
route all the multicast lightpaths.
Necessity: Consider a worst-case multicast
assignment of the form with 0 as the source and all
other nodes as its destinations along with another
connection with 0 as destination for any arbitrary
source. Among those lightpaths whose source
index is 0, there can be almost
lightpaths
established using only shorter links. These
lightpaths (first set) share the link connecting
nodes 0 with 1 and so each of these lightpaths needs
a unique wavelength. So,
wavelengths are
needed to establish the above lightpaths.All the
remaining lightpaths (second set) whose source
index is 0 are established on the even links only.
Since the first set of lightpaths and second set of
lightpaths don’t share any link in the same direction,
and they can be established using the same set of
wavelengths. Also, the lightpath whose
destination index is 0 is routed in opposite direction
to all the above lightpaths and can be established on
any one of the already used
wavelengths. Hence,
the wavelength number requirement is
.
Therefore,
wavelengths are the sufficient and
necessary condition on the wavelength number.
Wavelength allotment technique
Let the tablesand be used for managing
lightpaths whose source and destination are even
indexed and routed in rightward direction and
leftward direction respectively.
Let the tables and be used for managing
lightpaths whose source and destination are odd
indexed and routed in rightward direction and
leftward direction respectively.
Let the tables and be used for managing
lightpaths of all other connections routed in
rightward direction and leftward direction
respectively.
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Let all the above tables be initially empty.
Let be used for storing the available
wavelengths namely
Let x and y be the index of source and destination
node respectively.
The step-by-step procedure is described as follows,
Step 1: If and are both even and allot a
wavelength that is not available in but available
in Then include this wavelength in
Step 2: Else If and are both even and allot
a wavelength that is not available in but
available in Then include this wavelength in
Step 3: Else If and are both odd and allot a
wavelength that is not available in but available
in Then include this wavelength in
Step 4: Else If and are both odd and allot a
wavelength that is not available in but available
in Then include this wavelength in
Step 5: Else If allot a wavelength that is not
available in but available in Then include this
wavelength in
Step 6: Else allot a wavelength that is not available
in but available in Then include this
wavelength in
3. 2 Unidirectional ring with2-length extension
Figure 2(b) shows an 8-node ring with 2-
length extension. The routing of lightpaths is
assumed to be in clockwise direction without loss of
generality. The following assumptions are made to
reduce the wavelength number when is even. If
the index of both source and destination are even,
then such lightpaths are routed only using even
links. If the index of both source and destination are
odd, then such lightpaths are routed only using odd
links. All other lightpaths are routed using shorter
links only. Theorem 2 is proved based on this
assumption.
Theorem 2:When N is even, the sufficient and
necessary condition on the wavelength number for a
node unidirectional ring with 2-length extension
to be wide-sense non-blocking is.
Proof: Sufficiency: It is to be noted that all the
lightpaths between various sources and destinations
are routed only in clockwise direction without loss
of generality. For a lightwave network, a particular
wavelength can be allotted for multiple lightpaths as
long as they do not overlap with one another. A
wavelength released by a current lightpath during its
termination can be again used to allot for a new
lightpath establishment. It can be noted that out of N
multicast lightpaths, there can be atmost
lightpaths originating from the same sourcex. Out
of these lightpaths originating from the
source x, there can be almost
lightpaths (first set)
sharing the link joining the source node x with
.These lightpaths use only shorter links from the
source to the corresponding destination. The other
lightpaths (second set) with the same source index
uses only longer links and do not share any link in
the same direction with the first set of lightpaths.
The remaining lightpath whose destination is xdo
not share any link with the lightpath already
established between the nodesand , and they
can be on the same wavelength. So,
wavelengths
are sufficient to route all connections.
Necessity: Consider a worst-case multicast
assignment of the form with 0 as the source and all
other nodes as its destinations along with another
lightpath with 0 as the destination for an arbitrary
source. Among those lightpaths (first set) whose
source index is 0, there can be almost
lightpaths
sharing the link joining the nodes 0 with 1. So, each
of these
lightpaths needs a unique wavelength.
The other
lightpaths (second set) whose source
index is 0 (second set) are all established only on the
even links. Since the first set of lightpaths and
second set of lightpaths do not share any link in the
same direction, they can be established on the same
set of wavelengths. The remaining lightpath whose
destination index is 0 for an arbitrary source may be
allotted the same wavelength that is already allotted
for the lightpath (0, 1). Hence, the wavelength
number requirement is
. Therefore,
wavelengths are the sufficient and necessary
condition on the wavelength number.
Wavelength allotment technique
Let the table be used for managing lightpaths
whose source and destination are even indexed and
routed in clockwise direction.
Let the table be used for managing lightpaths
whose source and destination are odd indexed and
routed in clockwise direction.
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Let the tables be used for managing lightpaths of
all other connections routed through shorter links
only.
Let all the above tables be initially empty.
Let be used for storing the available
wavelengths namely
Let x and y be the index of source and destination
node respectively.
The step-by-step procedure is described as follows,
Step 1: If and are both even, allot a wavelength
that is not available in but available in Then
include this wavelength in
Step 2: Else If and are both odd, allot a
wavelength that is not available in but available
in Then include this wavelength in
Step 3: Else allot a wavelength that is not available
in but available in Then include this
wavelength in
Now the following assumptions are made to reduce
the wavelength number when N is odd.
Let a flag be established for each node and let it
be zero initially.
Let x and y be the index of the source and
destination respectively.
If no intermediate node is present in between x
and y along the primary ring (primary ring is
the basic ring network without modification) in
the clockwise direction, then the lightpath
between x and yis established using the shorter
link only.
If only one intermediate node is present in
between x and y along the primary ring in the
clockwise direction, then the lightpath between
x and y is established using the longer link
only.
If two or more intermediate nodes are present
in between x and y along the primary ring in
the clockwise direction, and if then the
lightpath is routed first from source xto next
node . From the node , the lightpath
is routed using the longest link first routing.
If two or more intermediate nodes are present
in between x and y along the primary ring in
the clockwise direction, and if then the
lightpath is routed using the longest link first
routing.
Theorem 3 is proved based on this assumption.
Theorem 3:When is odd, the sufficient and
necessary condition on the wavelength number for a
unidirectional 2-length extension ring with nodes
to be wide-sense non-blocking is
.
Proof: Sufficiency: It is to be noted that all the
lightpaths between various sources and destinations
are routed only in clockwise direction without loss
of generality. For a lightwave network, a particular
wavelength can be allotted for multiple lightpaths as
long as they do not overlap with one another. A
wavelength released by a current lightpath during its
termination can be again used to allot for a new
lightpath establishment. It can be noted that out of
multicast lightpaths, there can be atmost
lightpaths originating from the same sourcex. Out of
these lightpaths, there can be
almost
lightpaths (first set) sharing the link
joining the source node x with. So, each of
these
lightpaths needs a unique wavelength and
wavelengths are needed to route all the
lightpaths in the first set. The other
lightpaths
(second set) from the same source x are established
using the longer link joining the nodes x with .
So, each of these
lightpaths needs a unique
wavelength and
wavelengths are needed to
route all the lightpaths in the second set. However,
all the lightpaths of the first set do not share any link
with any of the lightpaths of the second set. So,
these two set of lightpaths can be established on the
same set of wavelengths. The remaining lightpath
whose destination is xdo not share any link with the
lightpath already established between the nodes x
and and they can be on the same wavelength.
So,
wavelengths are sufficient to route all
connections.
Necessity: Consider a worst-case multicast
assignment of the form with 0 as the source and all
other nodes as its destinations along with another
lightpath with 0 as the destination for an arbitrary
source. Among those lightpaths (first set) whose
source index is 0, there can be atmost
lightpaths
sharing the link joining the nodes 0 with 1 (
So, each of these
lightpaths needs a unique
wavelength. The other lightpaths (second set) with
the same source index 0 (second set) are all
established mostly on the even links
Since the first set of lightpaths and second set of
lightpaths do not share any link in the same
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direction, they can be established on the same set of
wavelengths. The remaining lightpath whose
destination index is 0 may be allotted the same
wavelength that is already allotted for the lightpath
(0, 1). Hence, the wavelength number requirement
is
. Therefore,
wavelengths are the sufficient
and necessary condition on the wavelength number.
Wavelength allotment technique
In this technique, two tables namely for
rightward direction and for leftward direction are
used for managing wavelength allotment for the
lightpaths.
Let all the tables be initially empty
Let be used for storing the available
wavelengths namely
.
Let
Let x and y be the index of the source and
destination respectively.
The step by step procedure is described as follows,
Step 1: If then the lightpath is
established using the wavelength
Step 2: Else if then the
lightpath is established using the wavelength
Step 3: Else if x is even and assign a
wavelength that is in but not in Add this
wavelength to and make
Step 4: Else if x is odd and , assign a
wavelength that is in but not in Add this
wavelength to and make
Step 5: Else if x is even and assign a
wavelength that is in but not in Add this
wavelength to and make
Step 6: Else assign a wavelength that is in but
not in Add this wavelength to and make
3. 3 Bidirectional ring with2-length extension
Figure 4 shows an 8-node ring with 2-length
extension. The routing of lightpaths is assumed to
be in shortest path direction. If the length of the
lightpath is same in both clockwise and
anticlockwise direction, then clockwise direction is
always followed. The following assumptions are
made in the routing procedure: If the index of
destination is odd, then the lightpath is established
using longest link first routing. If the index of
destination is even, then the lightpath is established
using only shorter links. Theorem 4 is proved based
on this assumption.
Theorem 4: The sufficient and necessary condition
on the wavelength number for a node
bidirectional ring with 2-length extension to be
wide-sense non-blocking is
.
Proof: Sufficiency: All the lightpaths between
various sources and destinations are routed only in
the shortest path direction. If the length of lightpath
is same in both clockwise and anticlockwise
direction, then clockwise direction is always
followed. For a lightwave network, a particular
wavelength can be allotted for multiple lightpaths as
long as they do not overlap with one another. A
wavelength released by a current lightpath during its
termination can be again used to allot for a new
lightpath establishment. It can be noted that a
multicast lightpath spans up to a maximum of
longer links. Since, there are Nlonger links present
in a ring network, two multicast lightpaths which do
not share any link in the same direction can be on
the same wavelength. So,
wavelengths are
sufficient to route all multicast lightpaths.
Necessity:
Consider a worst-case multicast assignment of the
form
for If N is even,
then for every integer i such that 0 ≤ i ≤
lightpaths of the form
can
be allotted a unique wavelength as they do not share
any link. This necessitates
wavelengths. If N is
odd, then for every integer i such that 0 ≤ i ≤
lightpaths of the form
can be allotted a unique wavelength as they
do not share any link. This necessitates
wavelengths. The remaining connection of the form
may be allotted another
unique wavelength. Hence,
wavelengths are
sufficient to route all multicast lightpaths
irrespective of whether is even or odd.
Wavelength allotment technique
Let x and y be the index of source and destination
node respectively.
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Step 1: When is even, and if
, allot the
wavelength Wy for this lightpath. Else if
,
allot the wavelength
for this lightpath.
Step 2: When is odd, and if
, allot the
wavelength Wyfor this lightpath. Else if
,
allot the wavelength
for this lightpath.
4. Results and Discussion
Table1 shows the comparison between basic
network topologies and network topologies with 2-
length extension under longest link first routing and
index-based routing in terms of the number of
wavelengths required to support wide-sense
nonblocking multicast communication. The results
were obtained in the previous section for network
topologies with 2-length extension under index-
based routing. From Table 1, it can be observed that
the wavelength number needed to support wide-
sense nonblocking multicast is reduced
approximately by half for linear array and
unidirectional ring with 2-length extensions under
index-based routing when compared with
corresponding network topologies under longest
link first routing. However, it is observed that there
is no change in wavelength requirement for a bi-
directional ring with 2-length extension. In addition
to LAN, WAN and MAN, the results obtained in
this work can be used to analyze the practical long-
haul networks like mesh network (both regular and
irregular), since mesh network can be decomposed
into a combination of basic networks and networks
with 2-length extension.
5. Conclusion and future work
In this work, wide-sense nonblocking
multicast communication is studied for WDM
optical linear array and unidirectional and
bidirectional ring with 2-length extensions under
index-based routing. The sufficient and necessary
condition on the minimum wavelength number
needed for the network to be wide-sense
nonblocking is derived for each of the above
network topologies. The results found in this work
are then compared with their corresponding network
topologies under longest link first routing. The
wavelength number needed to support wide-sense
nonblocking multicast in linear array and
unidirectional ring is reduced approximately by half
when compared with corresponding network
topologies under longest link first routing. However,
it is observed that there is no change in wavelength
requirement for a bi-directional ring with 2-length
extension.Future work includes extending this study
of wide-sense nonblocking multicast
communication for WDM optical linear array and
unidirectional and bidirectional ring with 3-length
and higher length extensions.
Table 1. Comparison of wavelength requirement
between basic network topologies and network
topologies with 2-length extension
Network
Topology
Conventional[
1]
2-length
extension
with
longest
link first
routing[2
4]
2-length
extensio
n with
index-
based
routing
N-node
Linear
Array
N-1
N-2
2
N
N-node
Unidirection
al ring
N
N-2
2
N
N-node
Bidirectiona
l ring
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Creative Commons Attribution License 4.0
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This article is published under the terms of the Creative
Commons Attribution License 4.0
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WSEAS TRANSACTIONS on COMMUNICATIONS
DOI: 10.37394/23204.2022.21.22
M. Sabrigiriraj, K. Manoharan, R. Karthik
E-ISSN: 2224-2864
188
Volume 21, 2022
