Performance Comparison of Simple Regular Meshes and Their k-ary n-cube Variants in Optical Networks

Abstract

The need for supporting the ever-growing number of multi-computers in the contemporary Internet has persuaded researches worldwide to search for suitable underlying network topologies with desirable properties. Even in the presence of speedy optical technologies, the arrangement of nodes in a network can be highly influential. Among all the topologies investigated so far, k-ary n-cubes have been reported to be widely adopted in the literature. But in hybrid networks, where the network is composed of a mixture of different types of topologies, the type and size of k-ary n-cubes can greatly affect the per-formance factor of the network. In this paper, we study and compare the per-formance behaviour of simple regular meshes with their wrap-around variants (2-D and 3-D torus) for Optical Packet Switching (OPS) systems with various sizes under a common traffic condition through simulation results.

Full text

Performance Comparison of Simple Regular Meshes
and their k-ary n-cube variants in Optical Networks
Ahmad Kianrad1, Aresh Dadlani2, 1, Ali Rajabi2, 1, Mohammadreza Aghajani3,
Ahmad Khonsari1, 2, Seyed Hasan Seyed Razi1
1Department of Electrical and Computer Engineering, University of Tehran, Iran
2IPM School of Computer Science, Tehran, Iran
3Sharif University of Technology, Tehran, Iran
a.kianrad@ece.ut.ac.ir, {a.dadlani, alirajabi, aghajani, ak}@ipm.ir, seyedraz@ece.ut.ac.ir
Abstract. The need for supporting the ever-growing number of multi-
computers in the contemporary Internet has persuaded researches worldwide to
search for suitable underlying network topologies with desirable properties.
Even in the presence of speedy optical technologies, the arrangement of nodes
in a network can be highly influential. Among all the topologies investigated so
far, k-ary n-cubes have been reported to be widely adopted in the literature. But
in hybrid networks, where the network is composed of a mixture of different
types of topologies, the type and size of k-ary n-cubes can greatly affect the per-
formance factor of the network. In this paper, we study and compare the per-
formance behaviour of simple regular meshes with their wrap-around variants
(2-D and 3-D torus) for Optical Packet Switching (OPS) systems with various
sizes under a common traffic condition through simulation results.
Keywords: Optical Packet Switching (OPS), simple regular mesh, k-ary n-
cubes, performance measures
1 Introduction
In the past few years, networking has been experiencing a migration towards optical-
based technologies. The reason for such diversion is the high flexibility, increased
scalability, Quality of Service (QoS) management, and unlimited bandwidth provi-
sioning introduced by such optical networks. Among all the proposed paradigms [1-
3], OPS has been the subject of several research projects [4, 5]. Its potential to inter-
face with the WDM transport layer and bridge the gap between the electrical (IP)
layer and the optical (WDM) layer has added to its popularity [2, 6].
Despite the implementation of OPS as the underlying technology [7, 8], there are yet
other prominent factors that influence the performance of networks such as network
topology, switching method, routing algorithm and traffic load [9]. The performance
of OPS networks with regular and irregular mesh topologies has been studied in the
literature [10]. But, due to the high adoption of k-ary n-cubes in OPS networks, in this
paper, we compare the performance factors of the basic instances of such a topology
with simple regular meshes for systems with various sizes and under a given traffic
load using results obtained through simulations.
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The rest of this paper is structured as follows. In Section 2, we provide an introduc-
tion on the types of topologies available and emphasize mainly on k-ary n-cubes.
Next, we delineate the performance factors under investigation in Section 3, followed
by the simulation scenarios with relevant explanations on the results obtained through
simulations in Section 4. Towards the end, in Section 5, we summarize our work and
enlighten directions for possible future works
2 Network Topologies
Network topology is the physical or logical arrangement of nodes interconnected
through links. Generally, existing networks are broadly classified into indirect and di-
rect networks. Examples of indirect networks are crossbar and bus, as their nodes are
connected to other nodes by means of multiple intermediate stages of switches. On
the contrary, in direct networks, each node is connected directly to some other nodes,
such as in a k-ary n-cube, mesh and tree. Due to high scalability, direct networks, in
particular k-ary n-cubes, have been extensively employed in several networks.
(a) (b) (c)
Fig. 1. The various k-ary n-cubes under consideration (a) a simple 3x3 regular mesh (b) a 3-ary
2-cube (2-D torus), and (c) a 3-ary 3-cube (3-D torus).
The k-ary n-cube, where k denotes the radix and n symbolizes the dimension, is an n-
dimensional grid structure with k nodes accommodated in each dimension. Each of
the nodes can be identified by an n-digit radix k address [11]. Fig. 1 illustrates a
simple 3 x 3 regular mesh, followed by instances of k-ary n-cubes. If we denote the
number of nodes in topology i as
n
k
i
N
and the number of edges as i
L
, then the follow-
ing equations hold for regular meshes (SRM) similar to that shown in Fig. 1-a:
=
n
SRM
N
k ,
(
)
=
−21
SRM
Lkk
.
(1)
Similarly, for the 2-D and 3-D tori instances of k-ary n-cubes, as illustrated in Figs. 1-
b and 1-c, the number of nodes is the same as that of , while the number of links
are as follows: SRM
N
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−=2
22
D
L
k ,
−=3
33
D
L
k .
(2)
In the following section, we introduce the factors chosen to study the performance of
the different instances of each type of topology.
3 Performance Metrics
The four metrics commonly used to evaluate the performance of a network in terms of
network utilization, reliability and latency are network throughput, packet-loss rate,
average end-to-end delay and mean hop distance [11]. If we denote by and ,
the total number of packets lost during transmission and the total number of packets
generated, respectively, then the packet-loss rate is defined as:
L
PT
P
−=
L
T
P
Packet loss Rate
P
. (3)
Network throughput is the fraction of network resource that successfully delivers
data. Because packets are dropped, a part of the network capacity is wasted in trans-
porting the bits that are dropped. An ideal situation in which the network throughput
tends to reach unity is when no packets are dropped and no link remains idle. Thus,
the network throughput can be defined as:
=⎛⎞
×
⎜⎟
⎝⎠
bits
network simulation
ideal
N
Network Throughput CT
H
. (4)
where bits
N
, , and denote respectively, the total number of bits
successfully delivered, the transmission capacity of the network, simulation time and
the ideal average hop distance. In (4), is, in turn, calculated as follows:
network
Csimulation
Tideal
H
network
C
=
××
network link wavelength data
CNN R
. (5)
in which link
N
, and
wavelength
Ndata
R
symbolize respectively, the total number of links,
total number of wavelengths and data rate. Mean hop distance is the number of hops
taken by a packet to traverse a path, averaged over all possible source-destination
pairs in the network.
Since our main interest is to study the topologies feasible for the optical domain, there
exist other metrics that should be regarded in a network. The primary key metric is
the cost of fibers required to implement a network with higher performance. Due to
the high cost of optical fibers, it has always been feasible to minimize the amount of
fiber used in optical networks, unless the addition of a few links would efficiently im-
prove the performance. This metric is in trade-off with the average number of hops
mentioned earlier and depends on the structure of the topology under study. It is upon
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the routing algorithms to decide whether to utilize an extra fiber link or increase the
hop distance of a packet traveling through the network. Based on the cost and mean
hop distance, we account another metric called the routed capacity. This factor de-
termines the percentage of packets routed through the network by the routing algo-
rithm in use. Another metric dependent on the trade-off between fiber cost and mean
hop distance is link utilization which indicates the percentage of available links in use.
In order to study the performance behavior of different k-ary n-cubes with different
sizes under the optical domain, it is mandatory to analyze such systems through pre-
defined or meaningfully derived measures.
4 Simulation Scenarios and Results
For purpose of illustration, the network topologies under study are size variants of
those shown in Fig. 1. For sake of clarity, we use the notation to denote a topol-
ogy of type k,
()
k
Ti
{}()
DDSRMk
−
−∈ 3,2, , and size , i
{
}
(
)
16,14,1012,8,6,4
∈
i. For in-
stance, a simple 6x6 regular mesh is denoted as , a 10-ary 2-cube by
, and a 4-ary 3-cube by .
(6)
SRM
T
−2(10)
D
T−3(4)
D
T
We have created and compared our scenarios in the OPNET WDM Guru environ-
ment. In each of the following scenarios, each bi-directional link i is i
L
long. The
number of fiber pairs in each link is f
N
and the user-defined cost of each pair is .
Every fiber contains W wavelengths, each carrying a data stream at rate
f
C
R
. In addi-
tion, the traffic matrix is generated randomly, and all packets are chosen to be routed
through the shortest path. The values for the parameters used in the simulation are
=
i
L
20 km, Nf = 50, and Cf = 50.
In our first scenario, we compare the three aforementioned topologies on the basis of
their routed capacity. As shown in Fig. 2, it is obvious that as k-ary 3-cubes are more
effective in routing packets than k-ary 2-cube and simple regular mesh for smaller
sized topologies, this percentage decreases with increase in topology size. This proves
the fact that in spite of the being very expensive, k-ary 3-cubes provide better per-
formance than the other topologies for smaller-sized networks, but as the size in-
creases, it is not feasible to adopt such k-ary 3-cubes as their routing capacity ap-
proaches that of simple regular meshes.
In Fig. 3, the total hop count in k-ary 3-cubes is far greater than that in case of simple
regular meshes and k-ary 2-cubes. In fact, the total hop counts of the latter two nearly
overlap with each other. This implies that the cost of taking more number of hops is
far lesser than that of increasing the fiber cost value. Thus, routing algorithms that
consider fiber and hop counts as their routing cost parameter tend to increase the hop
distance rather than add to the fiber cost.
Our next comparison involves the mean hop count for topologies with various sizes.
As illustrated in Fig. 4, it can be easily deduced that the expected mean hop count
would have a trend similar to that of the total hop count depicted in Fig. 3. As mean
hop is the overall average of the total hop count taken to travel all possible source-
destination pairs in the network, it can be inferred that in k-ary 3-cubes, the fiber cost
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is minimized at the expense of the mean hop value which, in turn, depends on the to-
tal number of hops.
0
5
10
15
20
25
30
35
40
45
0 2 4 6 8 10 12 14 16 18
Network Size
Routed Capacity (in %)
SRM Topology
k-ary 2-cube Topology
k-ary 3-cube Topology
Fig. 2. Performance comparison of simple regular meshes, k-ary 2-cubes, and k-ary 3-cubes in
terms of routed capacity and under random traffic condition. The sizes of all three topologies
are even numbers ranging from 4 to 16.
0
5000
10000
15000
20000
25000
30000
35000
0 2 4 6 8 1012141618
Network Size
Total Hops
SRM Topology
k-ary 2-cube Topology
k-ary 3-cube Topology
Fig. 3. Performance comparison of simple regular meshes, k-ary 2-cubes, and k-ary 3-cubes in
terms of total number of hops and under random traffic condition. The sizes of all three topolo-
gies are even numbers ranging from 4 to 16.
In order to study the relation between routed capacity and link utilization, we defined
the maximum number of hops to be some constant value. As can be inferred from Ta-
ble 1, with increase in network complexity, the routing cost decreases. That is, it is
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not quite feasible to adopt topologies of large sizes in order to enhance the network
performance. Also, for topologies with smaller sizes, routing algorithms tend to per-
form well in k-ary 3-cubes than their simple mesh and k-ary 2-cube variants.
0
2
4
6
8
10
12
14
024681012141618
Network Size
Mean Hop Distance
SRM Topology
k-ary 2-cube Topology
k-ary 3-cube Topology
Fig. 4. Performance comparison of simple regular meshes, k-ary 2-cubes, and k-ary 3-cubes in
terms of mean hop distance traveled by a packet and under random traffic condition. The sizes
of all three topologies are even numbers ranging from 4 to 16.
5 Conclusions
Performance measures adopted to study network behavior with various underlying to-
pologies have been plenty. In this paper, we studied and analyzed the performance
behavior of the three most efficient topologies namely, simple regular meshes, k-ary
2-cubes and k-ary 3-cubes in terms of routed capacity, mean hop distance and net-
work link utilization. We learnt that routing algorithms with routing cost defined as
fiber cost and hop count, tend to boost network performance in case of smaller sized
topologies. But with increase in network size, the performance gradually reduces in-
dependent of the type of topology chosen. Thus, it is not feasible to adopt large sized
k-ary n-cubes in any hybrid network as such topologies would not only increase the
cost of implementation, but also perhaps increase performance insignificantly. The
work done in this paper can be further extended to involve other feasible topologies
and k-ary n-cube variants studied under real-world traffics existing in the optical net-
work.
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Table 1. Comparison results obtained through simulations for all performance metrics. The
four performance measures are listed as column labels and the size and types of topologies un-
der consideration are listed as row labels. SRM, 2-D and 3-D denote, respectively, a simple
regular mesh, a k-ary 2-cube, and a k-ary 3-cube.
Routed
Capacity (%) Total
Hops Mean
Hop Link
Utilization (%)
SRM 25.45 384 2.27 100
2-D 34.64 511 2.22 99.8
4
3-D 39.45 5708 2.95 99.1
SRM 17.09 2270 3.87 94.58
2-D 21.78 2490 3.29 97.27
6
3-D 24.18 11357 4.25 98.59
SRM 8.11 4198 4.71 93.71
2-D 8.94 4415 4.5 95.15
8
3-D 9.1 14871 5.9 96.42
SRM 4.44 7102 5.88 98.64
2-D 4.63 7246 5.75 98.45 10
3-D 4.69 18324 6.55 99.5
SRM 3.4 10243 5.3 97
2-D 3.18 10539 5.84 98.31
12
3-D 3.1 21359 7.01 99.1
SRM 2.26 14252 6.06 97.88
2-D 2.23 14408 6.12 97.62
14
3-D 2.18 25812 8.31 98.52
SRM 1.7 18861 6.19 98.44
2-D 1.5 19048 6.98 98.39 16
3-D 1.2 29903 9.57 99.3
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