Performance Analysis of Fault Tolerant Irregular MINs

Performance Analysis of Fault Tolerant
Irregular MINs
Deepti Malhotra
Department of Computer Science &Engineering
Thapar Institute of Engineering and Technology
Patiala, 147004, INDIA
E-mail [email protected]
Abstract: The effectiveness of parallel or distributed
systems is often determined by its communication
network. Multistage Interconnection Network plays a
vital role on the performance of the multiprocessor
system. This paper introduces a class of fault tolerant
Multistage Interconnection Network (MINs) named
as Modified Double Tree Network (MDOT), Four
Tree Network (FT), Modified Four Tree Network
(MFT), Fault Tolerant Double Tree Network (FDOT)
and an attempt has been made to analyze these
different irregular Multistage Interconnection
Networks in terms of no of stages, no of switches,
fault-tolerance, cost-effectiveness, path length,
reliability and performance. A MFT and FDOT
network can provide a “full access” capacity in
presence of multiple faults. FDOT network supports
on-line repairability and is cost-effective than other
fault-tolerant Multistage Interconnection Networks
with a similar fault-tolerant capability. FT network is
a single switch fault-tolerant.If both switches in a loop
are simultaneously faulty then clearly some sources
are disconnected from some destinations.
Keywords: Multistage Interconnection Network, Double
Tree Network, Four Tree Network, Modified Four Tree
Network, Modified FT, Fault Tolerant Double Tree
Network.
I. INTRODUCTION
With the present state of technology, building
multiprocessor systems with hundreds of multistage
processors is feasible. The Multistage Interconnection
Networks (MINs) are an important part of these systems
that
enable the processors to communicate with themselves
and with the memory modules. In MINs, the fixed interstage connections exist between the adjacent stages with
a number of switches at each stage that are dynamically
set to establish desired connections to route the requests
from the source to the destination. Multistage
interconnection networks (MINs) achieve the full access
property with far fewer connections and hence have
become very popular.
Rinkle Aggarwal
Department of Computer Science &Engineering
Thapar Institute of Engineering and Technology
Patiala, 147004, INDIA
E-mail [email protected]
Many ways of providing fault-tolerance to the multistage
interconnection networks (MINs) have been proposed in
the literature. The basic idea for fault-tolerance is to
provide multiple paths for a source-destination pair, so
that alternate paths can be used in case of faults in a path.
Fault-tolerance is the ability of the network to operate
even in the presence of faults. A network may be
provided with fault-tolerance by providing rnultiple paths
between an input and output pair. This can be done by
increasing the number of stages, by providing multiple
links between stages, increasing the size of the switch,
portioning a unique path network into several
subnetworks and incorporating multiple copies of a basic
network.
The organization of the paper is as follows: Section II
comprises of DOT network. Section III provides details
of MDOT network. Section IV provides description of
FT network. Section V comprises of MFT network.
Details and design of FDOT network are proposed in
Section VI. Comparison of these networks on the basis of
different parameters is proposed in Section VII. Section
VIII comprises of some important terms used in the paper.
Finally conclusions are given in Section IX.
II. DOT NETWORK
As the name indicates, the DOT network consists of two
full binary trees with a common root. Though the DOT
network is irregular and has fewer switching elements
than a conventional Multistage Network, it maintains the
full access property.
Unlike regular MINs, the DOT network has paths of
varying lengths between pairs of input and output
terminals. Certain pairs of input and output terminals,
called the favourite pairs, have a shorter path length
between them than others. Hence, the DOT network is
expected to perform better than a conventional MIN. The
DOT network of size 8X8 is shown in figure1. The DOT
network is a 2(log N) - 1 stage interconnection network
having 2n+1 - 3 switching elements. It is an irregular
interconnection network in the sense that the number of
switching elements (SE) differs from stage to stage.
The SEs in stage j are numbered from 0 to N/2j+1 – 1
starting from the topmost SE. The upper and lower input
(output) terminals of an SE i are numbered as, 2i and
2i+1, respectively. As shown in Fig. l, the SEs of the
DOT network have an interconnection pattern which is a
combination of two trees (in opposite directions) having
a common root node in the centre. The interconnection
between the SEs is described below.
in each stage j, 0 <= j < n - 1
for each SE i, 0 <= i < 2n-j-1 , with i0 = 0
output 2i Æ input 2i of stage 2n - 2 - j
output 2i+1 Æ1 input [2i + 1)/2] of stage j+1
for each SE i, 0 <= i < 2n-j-1 , with i0 = 1
output 2i Æ input 2i/2 = I of stage j + 1
output 2i +1Æinput 2i +1 of stage 2n - 2 - j
in each stage j, n - 1 <= j < 2n - 2
for each SE i, 0 < i < 2j-n+1
output 2i Æ input 4i + 1 of stage j + 1
output 2i Æ1 + input 4i + 2 of stage j + 1
It can be seen that, when the central switch (CS) of an N
x N DOT network is in the through or straight state, the
network is split into two disjoint N/2 x N/2 networks of
2n - 2 stages. Each of the two N/2 x N/2 subnetworks
will have two central switching elements, namely the left
central switch (LCS) and the right central switch
(RCS).The central switches of the jth N/2 x N/2
subnetwork are labelled LCSi, j and RCSi, j, respectively.
In general, the central switches of the N/2i x N/2i, (0 <= i
<= n - 1) subnetwork will have the labels, LCSi, j and
RCSi, j respectively, where j, (0 <= j <=2i - 1) represents
the subnetwork number (with the numbering starting
from the top most subnetwork and proceeding
downwards). LCS0,0 is the same as RCS0,0 and is
referred to as the CS of the network. Because of the
central switches, the DOT network is blocking with
limited permutation capability. The structure of the DOT
network allows multiple paths of varying lengths (in
terms of the number of SEs passed through) between a
pair of source and destination terminals.
III. MDOT NETWORK
MDOT
network
is
an
Irregular Multistage
Interconnection Network .It consists of a right and a left
half. Each half of the network resembles a binary tree.
The left and right trees are mirror images of each other.
MDOT network of size 2n X 2n has 2n source and 2n
destination and 2n-1 stages. Further, it has (2n+1 -3)
switches. Each of the stage i and 2n-i have exactly 2n-1
switches of size 2X2 for i= 1, 2…n. The MDOT network
of size 8X8 is shown in figure2.
IV. FT NETWORK
The FT network of size 2n X2n contains two identical
groups G0 and G1 each consisting of a MDOT network of
size 2n-1 X 2n -1 ,which are organized one above the other.
The two groups are formed based on the most significant
bit (MSB) of the source-destination terminals. Thus, half
of the source destination terminals with MSB 0 fall into
group G0 and others having MSB 1 fall into the group G1 .
The FT network of size 2n X 2n has (2m-1) stages and
total of (2m+2 -6) switches with 2n-1. of size 2X2 and rest
of size 3X3. There are 2n multiplexers and an equal no
of demultiplexers of 2X1 and 1X2 sizes respectively.
Both the stages i and 2m-i in the group have 2n-i-1
switches and are numbered as 1,2…..2n-i-1 (where i =
1,2…,m , m = log2 N/2 and N = 2n ). Every 3X3 SE in a
stage forms a loop with the corresponding numbered 3X3
SE of other sub –network in the same stage. Every source
and destination is connected to both the subgroups by
means of multiplexers and demultiplexers. In case the
primary path is busy or faulty, requests will be routed
through secondary path in the sub-network. FT network
is single switch fault-tolerant. If both switches in a loop
are simultaneously faulty then clearly some sources are
disconnected from some destinations.FT network of size
16X16 is shown in figure3.
V. MFT NETWORK
Modified FT network of size 16X16 has one stage and
two switch less compared to the FT network. It contains
total of (2m+2-8) switches and (2m-2) stages .It contains
the multiplexers and demultiplexers equal to FT network.
The 2X1 multiplexers are connected in such a manner
that each multiplexers between pair of sources have all
the bits (sn-2…..s2s0) equal except the bit s1 . The 2X1
demultiplexers are so connected that each multiplexer is
connected to destinations having all bits (dn-1…d1) the
same. MFT network of size 16X16 is shown in figure4.
VI. FDOT NETWORK
An FDOT-k network of size NXN is designed by
dividing N inputs and N outputs into k disjoint partition
of N/k sources and N/k destinations where k (>=2) and N
(>k) are the powers of 2.There are k independent
subnetworks, and an extra one, such that a connection
path between each source-destination pair can be
established via any one of the subnetwork. All the (k+1)
subnetworks are of identical type. The extra subnetwork
helps to enhance fault-tolerant capability and to keep a
desired level of performance even in the presence of
faults. Each source and destination are linked to all the
(k+1) subnetworks via kX1 multiplexers and 1Xk
demultiplexers.An FDOT-k consists of (2n-1) number of
stages and (k+1) (2n+1-3) number of switching elements,
where n = log2 N/k. FDOT network of size 8X8 is shown
in figure 5.
Figure1: DOT Network of size 8X8
Figure2: MODIFIED DOT NETWORK
Figure3: FOUR TREE NETWORK OF SIZE 16X16
Figure4: MODIFIED FOUR TREE NETWORK OF SIZE 16X16
Figure5: FDOT NETWORK OF SIZE 8X8
VII. COMPARISON OF IRREGULAR MINS
In this paper MINs are compared as:
A. On the basis of the Hardware Structure:
such as no of stages, no of switches, no of switches
per stage and no of multiplexers/demultiplexers.
Comparisons are shown in table I.
B. On the basis of Performance Parameters.
Comparisons are shown in table II.
TABLE I: COMPARISON ON THE BASIS OF HARDWARE
Sr No.
Characteristics
1
Full Form
2
MDOT
FT
MFT
FDOT
modified double
tree
Four tree
modified four tree
Fault-tolerant
double tree
Number of stages
For 2nX2n
network no. of
stages are (2n-1)
where N= 2n
For 2nX2n network
no. of stages are
(2m-1) where
m=log2N/2 & N= 2n
For 2nX2n
network no. of
stages are (2m-2)
where m=log2N/2
& N=2n
For 2nX2n
network no. of
stages are (2n-1)
where n=
log2N/k ,k(>=2)
are independent
subnetworks
and N(>k)
3
Number of switches
(2n+1-3) switches
(2m+2-6) switches
(2m+2-8) switches
(k+1)(2n+1-3)
switches
4
Number of switch per
stage
Stage i &(2n-i)
have exactly 2n-1
switches where
i=1,2,…..,n
Stage i &(2m-i) in
the group have 2n-i-1
switches where
i=1,2,…..,m
Stage i &(2m-i-1)
in the group have
2n-i switches
where
i=1,2,..,n+1
Stage i & 2(n1)-i have
exactly
(k+1). 2n-i-1
switching
element for
i=0,1…,(n-1)
5
Number of multiplexers
& demultiplexers
multiplexers &
demultiplexers
are not used
2n multiplexers & 2n
demultiplexers are
used
2n multiplexers
& 2n
demultiplexers
are used
number of
multiplexers &
demultiplexers
are (N+2n)
TABLE II: COMPARISON ON THE BASIS OF PERFORMANCE PARAMETERS
Sr No.
Characteristics
MDOT
FT
MFT
FDOT
1
Fault –Tolerant
Not
faulttolerant,
the
central switch is
critical
and
presence of a
single
fault
breaks down the
system
completely.
It is single switch
fault-tolerant. If both
switches in a loop are
simultaneously faulty
then clearly some
sources
are
disconnected
form
some destinations.
If both switches
in a loop are
simultaneously
faulty then even
some sources are
connected to the
destinations.
K switch fault –
tolerant means it
can
provide
a
connection for any
source-destination
pair in the presence
of any instance of
upto k faults in the
network.
2
Cost-Effectiveness
It is also costeffective
FT is more costeffective
than
MFT.Proposed cost is
(9.75 2n+1-54)
Proposed cost of
MFT is
(9.75 2n+1-72)
More cost-effective
than other faulttolerant MINs. Its
cost is
(k+1)(2n+3+2N-12)
3
Path Length
It is a unique
path
network
where only a
single path exists
between
any
input-output pair.
The network has
multiple path of
different
lengths
between an inputoutput terminal pair
Multiple paths of
varying lengths
are
available
between
each
sourcedestination pair.
If
[(sn-1
d n-1) +
(sn-2
d n-2)
d2)]
+…+ (s2
is zero. Then
Minimum path
is available along
with the other
possible
pathlength
Multiple paths of
varying lengths are
available between
each
sourcedestination pair.
If
[(sn-1
d n-1) +(sn-2
d n-2) +…+(sj
dj)] is zero
(where 1<=j<=(n1))
then
all paths of length
equal to or greater
than 2j are possible
else
paths of length 2n1(i.e. longest path
is possible only.
4
Reliability &
Performance
It is less reliable
because single
switching
element
is
present
and
breakdown
of
switching
element leads to
the breakdown of
the network that
has derogatory
effect on the
performance of
the system.
The network has
improved reliability.
The
improved
reliability obtained
from the multiple
paths depends upon
how effectively the
alternate paths are
used by the routing
algorithm.
MFT
has
improved
reliability
&
performance
than FT.
FDOT-k is k faulttolerant, so it is
quite reliable as
compared to the
other
networks
containing nearly
same amount of
hardware.
It
possesses
the
property of on-line
repairability.
VIII. IMPORTANT TERMS USED IN THE PAPER
Fault–Tolerant:
A
fault
tolerance
multistage
interconnection network provides service routing even
under the faults. Fault-tolerance is the ability of the
network to operate even in the presence of faults. The
basic idea for fault-tolerance is to provide multiple paths
for a source-destination pair, so that alternate paths can
be used in case of faults in a path.
Path-Length: The path length p, of a path between an
input terminal and an output terminal in a MIN is the
number of SEs encountered by a message sent along that
path. It gives the information about the different pathlengths possible between the source-destination pair.
On-line Repairable: A network is said to be On-line
repairable that if some faults develop in any component
in the network, then the faulty component can be simply
replaced by the new one without causing a break-down in
the system.
Cost-Effectiveness: To estimate the cost of a network the
assumption is that the cost of a switch used in a network
is proportional to the number of cross-points within a
switch.
[3]
[4]
[5]
[6]
[7]
[8]
[9]
Reliability: It is defined to be the probability of existence
of at least one fault-free path between a given pair of
input and output.
[10]
IX. CONCLUSION
[11]
In this paper irregular fault-tolerant Multi-Stage
Interconnection Networks named Modified double tree
network, four tree network, Modified four tree network,
and Fault-tolerant double tree network have been
proposed and analyzed. Further, these different irregular
multistage interconnection networks are compared in
terms of different parameters like no of stages, no of
switches, fault-tolerance, cost-effectiveness, path length,
reliability.
In MDOT single switching element is present and
breakdown of switching element leads to the breakdown
of the network as a result the performance of the network.
MFT and FDOT can achieve significant tolerance to
faults and good performance with relatively low costs
and simple control schemes.
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