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A Probabilistic Characterization of Fault Rings in Adaptively-Routed Mesh
Interconnection Networks
F. Safaei
1, 2
, A.Khonsari
3, 1
, A. Dadlani
1, 3
, M. Ould-Khaoua
4
1
IPM School of Computer Science, Tehran, Iran
2
Department of Electrical and Computer Engineering, Shahid Beheshti University, Iran
3
Department of Electrical and Computer Engineering, University of Tehran, Iran
4
Department of Computing Science, University of Glasgow, UK
{safaei, ak, a.dadlani}@ipm.ir, mohamed@dcs.gla.ac.uk
Abstract
With increase in concern for reliability in the
current and next generation of Multiprocessors
System-on-Chip (MP-SoCs), multi-computers, cluster
computers, and peer-to-peer communication networks,
fault-tolerance has become an integral part of these
systems. One of the fundamental issues regarding
fault-tolerance is how to efficiently route a faulty
network where each component is associated with
some probability of failure. Adaptive fault-tolerant
routing algorithms have been frequently suggested in
the literature as means of improving communication
performance and fault-tolerant demands in computer
systems. Also, several results have been reported on
usage of fault rings in providing detours to messages
blocked by faults and in routing messages adaptively
around the rectangular faulty regions. In order to
analyze the performance of such routing schemes, one
must investigate the characteristics of fault rings. In
this paper, we derive mathematical expressions to
compute the probability of message facing the fault
rings in the well-known mesh interconnection network.
We also conduct extensive simulation experiments
using a variety of faults, the results of which are used
to confirm the accuracy of the proposed models.
1. Introduction
The requirements of applications necessitate
extremely high processing power, using tens of
thousands of computing elements working in a
coordinated way [1]. Under these circumstances, as
these number of elements increases, the probability of
occurrence of faults grows dramatically. Furthermore,
due to the distributed nature of these machines, the
reliability of the underlying interconnection network
design plays a critical role in keeping these systems
running under faulty conditions. The failure of
components in such systems not only reduces the
machine computational power, but also deforms the
interconnection network.
In recent years, efforts have been made to integrate
performance and reliability of adaptive routing
algorithms to overcome the drawback in the traditional
evaluation methods proposed for interconnection
networks. In designing a fault-tolerant routing
algorithm, choosing a suitable fault model, which in
turn, reflects the fault situations in a real system, is one
of the most important issues. Rectangular fault regions
(also known as block faults) are amongst the most
common approaches for modeling node failures at a
board level in networks [2-4].
For each rectangular fault region embedded in a
network, it is feasible to connect the fault-free
components around the region to form a ring, known
as the fault ring (f-ring for short) for that fault. Many
researchers [3-8] have used the concept of f-rings to
make the existing routing algorithms tolerate multiple
block fault regions without disabling any fault-free
nodes. Messages are routed under a regular adaptive
routing until they face an f-ring. Since the performance
of fault-tolerant routings depends on the availability of
computational resources, and more specifically, the
shapes of fault patterns, the impact of failures should
be taken into account during system performance and
availability evaluation. This paper introduces a novel
performance measure of network reliability,
probability of facing the fault rings, for finding the
reliability and performance measures of arbitrary
adaptive routing algorithms in mesh-based
interconnection networks with a variety of common
cause rectangular faulty regions.
The remainder of the paper is structured as follows.
Section 2 provides a background, introducing the
adaptive routing algorithms and the concept of fault
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models used in this paper. In Section 3, the
mathematical approach for calculating the probability
of facing f-rings is explained, followed by the
simulation platform and conclusion in Sections 4 and
5, respectively.
2. Background
This section explores the basis on which our
mathematical model is founded, including the structure
of a 2-D mesh network and the routing algorithm used
in this study.
A 2-D
RC×
mesh topology, denoted by
RC
M
×
,
comprises of
2 ERCRC=⋅−−channels and
NRC=⋅
nodes. The address of each node
u
is
denoted by
(
)
,
xy
uu
, where
{}
0, 1, , 1
x
uC∈−…
and
{}
0, 1, , 1
y
uR∈−…
. Two nodes
(
)
:,
xy
uuu
and
(
)
:,
xy
vvv
are connected if their addresses differ
in one and only one dimension, say dimension
X
,
and
1
xx
uv−=
. Similarly, along dimensionY ,
1
yy
uv−=
. The mesh topology is inherently
asymmetric due to the absence of wrap-around
connections along each dimension. As a result, nodes
at the corners and edges in the network have two and
three neighbors, respectively.
2.1. Routing algorithms
While a topology determines the connectivity
between nodes, a routing algorithm describes which
nodes and links are visited by messages on their
journey from source to destination. This section
presents the terminology of routing algorithms.
A routing algorithm
ℜ
can formally be described
as a function
: Eℜ×→
N
N
, where
N
and
E
denote the set of nodes and channels, respectively
[9]. Given the current node
c
x
and the destination
node
d
x
,
(
)
,
cd
xxℜ
returns a channel
,
c
xy〈〉
, where
y
is an immediate neighbor of
c
x
.
ℜ
can be divided
into a routing restriction function,
R
ℜ
, and a selection
function,
(
)
:
S
EEℜ→
P
, where
(
)
E
P
is the
power set of
E
. A routing restriction function
(
)
:
R
Eℜ×→
N
N
P
of a routing algorithm
ℜ
takes the original input of
ℜ
and returns
{}
12
,,,,,, , 0
cc cm
xy xy xy m〈〉〈〉〈 〉≥… as the set of
channels, with
i
y
(
)
1 im≤≤
as the neighboring
nodes of
c
x
[9].
If
R
ℜ
always returns a channel set of size lesser
than or equal to 1, then
ℜ
is deterministic, otherwise it
is adaptive [9]. Deterministic routing has the advantage
of being simple; however, if any channel along the
message path is heavily loaded, the message
experiences large delays and if any node or channel
along the path is faulty, the message is not delivered at
all. Alternatively, adaptive routing allows paths to be
chosen dynamically, resulting in improved network
performance.
2.2. Fault models and fault patterns
Beyond improving network utilization, adaptive
routing can also support fault-tolerance by exploiting
redundant paths in the network. The fault-tolerant
computing literature is extensive and thorough in the
definition of fault models for the treatment of faulty
digital systems. Various forms of faults such hardware
faults, where a node or a link fails to function,
software bugs, and malicious sniffing or removal of
packets can be experienced in a network. In this paper,
we merely focus on hardware faults. Failures not only
reduce the computational power, but also deform the
structure of interconnect network, which may
consequently lead to a disconnected network.
Definition 1 A topology is said to be connected if
there exists a path between all source-destination pairs
(
)
,sd
for allsd≠ . Otherwise, it is disconnected.
A fault is considered as the failure of one singular
physical network component. If a link fails, all other
network components resume functionality, while if a
node fails, it renders all links attached to it useless. An
attached link can also fail without disrupting the
validity of our methods, and thus both faults can be
regarded as one singular node fault. However, the
network might experience problems with identifying
this situation as a node fault. Multiple faults can either
be scattered all around the network, or be closely
located to each other as two or more neighboring
nodes. Hardware faults are relatively infrequent in
modern interconnection networks, so under normal
operation, having two neighboring nodes fail is
unlikely. However, closely located nodes are likely to
go down due to external reasons, such as a power
outage. Adjacent faulty nodes are coalesced into fault
regions, which may lead to different patterns of failed
components. Faulty regions extended by faulty
components, may form convex or concave shaped fault
patterns [2-5, 7, 8, 10-13]. Some faulty patterns in
convex and concave regions are
|-shape, ||-shape,
-shape and L-shape, U-shape, T-shape, H-shape, +-
shape, respectively [9, 12-14].
3. The mathematical model
This section provides a derivation for the
mathematical model. The model uses the assumptions
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commonly used in the literature for the construction of
similar mathematical models [9, 13-15].
3.1. Calculating the probability of a message
facing f-rings in mesh networks
Consider an
RC×
mesh network with some faulty
nodes located in a rectangular region of size
hl×
(1 , 1 )lC hR≤< ≤ <
.
Also, assume that the
faulty nodes do not disconnect the network. We name
such a network as a connected
RC×
mesh with
rectangular faulty pattern.
Definition 2 Consider a connected
RC× mesh with
a rectangular faulty pattern of
hl× . The f-ring is a
set of all fault-free adjacent nodes and links formed
around each rectangular fault pattern such that the
interior of the f-ring contains only faulty components.
Definition 3 When a faulty region touches one or
more boundaries of the mesh, the formation of an f-
ring around the faulty region is not possible. In this
case, a fault chain, or f-chain, is formed rather than an
f-ring.
Definition 4 (The minimum path from point
= (, )
aa
axy
to point
= (, )
bb
bxy
) We denote by
S
,
the set of all sequences containing forward,
backward, upward and downward directions, so that
all
S
elements start from (, )
aa
axy= and terminate
at
(, )
bb
bxy=
. We let
0
s
be the length of the shortest
sequences of
S
. Therefore, one path from
a
to
b
is
an element of the set
S
with length
0
s
.
The distance between the first components of points
(, )
aa
axy= and (, )
bb
bxy= in an RC× mesh is
equal to
ba
xx−
which is denoted by
symbol
(
)
,
x
abΔ . Similarly, the distance between the
second components of points
(, )
aa
axy= and
(, )
bb
bxy= is equal to
ba
yy− and demonstrated
by
(
)
,
y
abΔ
. The set of the
RC×
mesh network
vertices, denoted by
()
RC
vM
×
,
is defined as:
(
){}
() ,:1 ,1
RC
vM xy x C y R
×
=≤≤≤≤
(1)
The number of paths (minimal) between two non-
faulty points
,( )
RC
ab vM
×
∈
is denoted by
(,)LT a b
and its value is given by:
ba ba
ba
xx yy
xx
−+−
⎛⎞
⎟
⎜
⎟
⎜
⎟
⎜
⎟
−
⎟
⎜
⎝⎠
(2)
In this section, we intend to calculate the probability
of a path facing the f-ring, which we denote by
hit
P
.
In
order to calculate
hit
P
, we should enumerate the
number of all existing paths facing the f-ring and
divide them by the number of all existing paths in the
connected mesh network. This can be expressed as:
hit
The number of all minimal paths crossing the f ring
P
The number of minimal paths existing in the network
−
(3)
A path facing the f-ring means that there exist one
or more points from the set of points residing on the f-
ring along the given path. Since the position of the
faulty points in the mesh network is important.
Therefore, to indicate the exact location of the faulty
points, we should know one of the four points placed
on the corners of the rectangular faulty pattern. For
compatibility, the point in the bottom-left corner of the
rectangular faulty pattern is considered as the
characteristic point and is denoted by
α
. Thus, to
indicate the exact location of the block fault region, in
addition to knowing the length and width of the
rectangular
(, )lh , its characteristic point should be
determined. The set of
(, )lh
rectangular faulty points
with the characteristic point
α
is shown as (, , )Flhα ,
and the set of f-ring points around this faulty pattern is
illustrated by
(, , )Rlh α . It is apparent that (, , )xFlhα∈ ,
but
(, , )xRlhα∉
. Also, the set
(, , ) (, , )Flh Rlhαα∪
is
denoted as
(, , )FR l h α .
Theorem 1: In a connected
RC× mesh network
with the
hl×
rectangular fault region and
characteristic point
α
, the number of all existing paths
between any pair of non-faulty nodes is given by:
,( )\(,,)
(,)
ab v M F lh
RC
LT a b
α∈
×
∑
(4)
where
the symbol "\" signifies the difference between
two sets.
Proof: Consider a connected
RC×
mesh with
hl×
rectangular fault pattern surrounded by an f-ring.
Also, consider two non-faulty points
a
and
b
in the
network mentioned above. The number of paths from
a
to
b
is given by:
(,)
ba ba
ba
xx yy
LT a b
xx
−+−
⎛⎞
⎟
⎜
⎟
⎜
=
⎟
⎜
⎟
−
⎟
⎜
⎝⎠
Thus, the total number of paths in the aforementioned
network can be calculated as the aggregate of the total
number of paths existing between any two of non-
faulty points in the network, which completes the
proof. ■
To carry on, it is required to determine the direction
of each path in the network with respect to the
coordinate axes. Consider two points
(, )
aa
axy= and
(, )
bb
bxy=
in the
RC×
mesh. We wish to move
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from a to b . Table 1 summarizes the information for
Table 1 Description of the directions to be used for adaptive
routing of messages in the mesh network
.
Message direction
Condition
satisfied
The message is routed from
(, )
aa
axy=
to
(, )
bb
bxy= alongX
+
.
0
ba
xx−≥
The message is routed from (, )
aa
axy= to
(, )
bb
bxy= along
X
−
.
0
ba
xx−<
The message is routed from (, )
aa
axy= to
(, )
bb
bxy=
alongY
+
.
0
ba
yy−≥
The message is routed from (, )
aa
axy= to
(, )
bb
bxy=
along
Y
−
.
0
ba
yy−<
the directions of the messages along
X
and
Y
, where
−
and
+
reveal the orientation in the negative or
positive side of the coordinate axes, respectively.
Now, if we consider an arbitrary path from
a
to
b
and multiply the set of the first components of the
existing nodes with the set of the second components
of the existing nodes in a Cartesian way, we obtain a
mesh subnetwork
,
ba ba
xx yy−×−
,
from the
RC×
mesh in which nodes
a
and
b
are placed on
opposite corners of
the subnetwork. It is
straightforward to show that all the existing paths from
a
to
b
lie in this subnetwork, which we denote
by
(,)Mab .
Example. Consider a
67× mesh network with a
34×
block faulty region and the characteristic point
(
)
2, 3α =
as illustrated in
Figure 1. We wish to
route messages from
(
)
1, 2a =
to
(
)
6, 4b =
. In this
network, a minimal path from
a to b is:
(
)
(
)
(
)
(
)
()()()()
1,2 2,2 3,2 4,2
5, 2 6, 2 6, 3 6, 4
a
b
=→→ → →
→→→=
Therefore, the set of first and second components of
the existing nodes along this path
are
{}
1, 2, 3, 4, 5, 6
and
{}
2, 3, 4
, respectively. So, the
product of
(,)Mab
is:
{}{}
( , ) 1,2,3,4,5,6 2,3,4Mab =×
Figure 1 (a) A 67× mesh network with two arbitrary points
a
and
b
in the presence of block fault region along with the
corresponding f-ring; (b) illustration of
(,)Mab subnetwork.
with the existing faulty points in (,)Mab as:
( , ) ( , , ) {2, 3, 4, 5} {3, 4}Mab Fhlα =×∩
Definition 5 The restriction function of
RC
M
×
network
, given by
(
)
:( )
RC RC
MvM
××
→P
F
, in
which
()
RC
M
×
P
is the set of all mesh subnetworks of
RC
M
×
, has the norm of
(
)
2
2
RC⋅
and criterion as:
(1) If the set of faulty points resembles a
hl×
rectangular fault pattern with characteristic
point
α
and an f-ring around it.
(2) If the set of faulty points resembles two
overlapping rectangular faulty patterns with
characteristic points
1
α
and
2
α
together with
their associated f-rings, then,,
(,) (, , )
((,))
(,)
overlap
Mab FRlh
Mab
Mab FR
α⎧
⎪
⎪
=
⎨
⎪
⎪
⎩
∩
∩
F
(5)
Definition 6 Let
a
and
b
be two non-faulty points
in
()
RC
vM
×
. For any
two arbitrary points
,((,))
ij
CC Mab∈
F
, the number of possible paths
from
j
C
to
i
C
is indicated by
,
(,)
ab j i
LM C C
,
such
that the direction of each path along dimension
(
)
XorY
is collinear with the direction of a path from
a
to
b
along dimension
(
)
XorY
and is defined as:
,,
,
(,) (,)
(,)
ab ab
xy
ji ji
ab
x
ji
CC CC
CC
⎛⎞
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎝⎠
Δ+Δ
Δ
(6)
where
,
(,)
ab
xji
CCΔ denotes the number of orientations
along a path from
j
C to
i
C in dimension X
(or
Y
)
collinear with the orientations along a path from
a
to
b having the following criterion:
(
)
()
,
00 or
0 0
(,)
||
ij ij
ij
ij
CC ba CC
ba C C
ab
xji
CC
xx xx andxx
xx andx x
CC
x x otherwise
⎧
⎪
−−≥−≥
⎪
⎪
⎪
⎪
⎪
−< − ≥
⎪
Δ=
⎨
⎪
⎪
⎪
⎪
⎪
−−
⎪
⎪
⎩
(7)
Given that
a and b are two non-faulty points of a
mesh network and
(
)
(
)
12
,{,,,}
k
Mab CC C= …
F
,
the number of paths from
a
to
b
not traversing points
12
,,,
k
CC C…
can be calculated as follows [15]:
0,
det ( , )
ij
ij k
dab
≤≤
(8)
where,
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0,1
,
(,) ( , ) 0,1, ,
( , ) ( , ) 1, 2, , , 0, 1, ,
jabjk
ij a b j i
dab LM CC j k
dab LM CC i k j k
+
==
===
…
……
(9)
Theorem 2: Given that
a
and
b
are
two non-faulty
points of a mesh network with
hl×
block fault region
and characteristic point α together with an f-ring
around it, the number of paths from
a
to
b
that
do
not cross the f-ring is given by:
,
0,
,( )\(,,)
det ( , )
ab
RC
ij
ij C
ab v M FR lh
dab
α
×
≤≤
∈
∑
(10)
in which,
(,,) (,,) (,,)FR h l F h l R h lααα= ∪
(11)
where
,ab
C
is
the number of elements of
()
(,)Mab
F
.
Proof: Considering the fact that each path
confronting
(,, )FR h l α
will also be incident with the f-
ring, the number of paths not traversing the points on f-
ring is equal to the number of paths in
RC
M
×
not
crossing the
(,, )FR h l α
points. Consider two arbitrary
points
a
and
b
from the set
(
)
\(,,)
RC
vM FRhlα
×
.
According to equation (8), the number of minimal
paths not traversing the points
()
(,)Mab
F
is equal to
,
0,
det ( , )
ab
ij
ij C
dab
≤≤
. Therefore, the number of all
existing paths in
RC
M
×
not facing the f-ring will be
equal to the total number of paths between any two
non-faulty points in
RC
M
×
not traversing the points of
(,, )FR h l α
.
That is, the same as equation (10). ■
It follows from Theorem 3 that probability of a path in
RC
M
×
not facing the fault ring,
miss
P
, is:
,
0,
,( )\(,,)
,( )\(,,)
det ( , )
(,)
ab
RC
RC
ij
ij C
ab v M FR hl
miss
ab v M FR hl
dab
P
LT a b
α
α
×
×
≤≤
∈
∈
=
∑
∑
(12)
Thus, it is trivial that
1
hit miss
PP=−
.
3.2. Extending the model to overlapping f-rings
There can be several f-rings in a network with
multiple block fault regions. So far, we have assumed
that faults are such that the f-rings do not overlap. We
will now show that the proposed mathematical model
can be extended to the case of two overlapping f-rings.
Two f-rings are said to overlap if they share one or
more links [2, 4, 5]. In order to calculate
hit
P
, we
assume that there are two block faults named
1
11 2
(,, )Fhlα
and
2
22 2
(,, )Fhlα
in the network. The set of
all faulty points is indicated by five main
characteristics:
121212
,,,,,hhllαα , and the common
length (width). The set of all faulty points in the case
of overlapping f-rings is denoted by
overlap
F .
Moreover, the f-rings around the first and the second
block fault regions are demonstrated by
111 1
(,, )Rhl α
and
222 2
(,, )Rhl α , respectively. The total number of
existing paths in a connected
RC
M
×
mesh network
with
overlap
FR
is
,( )\
(,)
RC overlap
ab v M F
LT a b
×
∈
∑
(13)
If
RC
M
×
is a connected mesh network with two
overlapping f-regions with points of the first and
second faulty regions, respectively, denoted by
1
11 2
(,, )Fhlα and
2
22 2
(,, )Fhlα , and the set of points lying
on their corresponding f-rings be denoted by
111 1 222 2
(,, ) (,, )
overlap
RRhlRhlαα= ∪ ,then the paths
not intersecting the overlapping f-rings is given by:
,
0,
,( )\
det ( , )
ab
RC overlap
ij
ij C
ab v M FR
dab
×
≤≤
∈
∑
(14)
As a result, the number of all existing paths in
RC
M
×
not crossing
overlap
FR
is:
()
,
0,
,( )\
,\
det ( , )
(,)
ab
R C overlap
R C overlap
ij
ij C
ab v M FR
miss
ab v M F
dab
P
LT a b
×
×
≤≤
∈
∈
=
∑
∑
(15)
So, we get,
() ()
1
hit overlap miss overlap
PP=−
(16)
4. Experimental results
In Section 3, we have derived mathematical
expressions to calculate the probability of message
facing the f-rings (f-chains) with and without
overlapping. A program for calculating the probability
of message facing the fault rings was developed in C
and executed to simulate the failure of nodes and the
subsequent constructing of the corresponding f-rings
(f-chains). The simulator generates faults in the
network so that the resulting faulty regions are convex
(block). It also checks that all nodes in the network are
still connected using adaptive routing algorithm. The
goal of the simulation is to calculate the values of the
probability of message facing the f-rings (f-chains) for
different number of faulty nodes in the mesh topology.
Table 2 gives the simulation results and the
mathematical expressions in a 2-D mesh for various
network sizes and fault patterns characteristics.
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5. Conclusions
In order to evaluate the adaptive routing algorithm,
its reliability and performance measures should be
quantified. This paper introduces a novel performance
measure of network reliability, probability of message
facing fault rings and presents the mathematical
approach for its evaluation of arbitrary rectangular
shapes in a given network with mesh architecture.
Capturing the network performance measures, such as
message latency and link waiting times throughout a
faulty network are straightforward applications of the
results obtained in this paper. Further research can be
devoted to incorporation of common cause faulty
patterns and extension of our work to propose
mathematical expressions for calculating the
probability that a message faces fault patterns when
failures occur in the network.
6. References
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Table 2 Experimental results of the probability of message facing fault rings (chains) in 2-D mesh architecture with different
shapes of fault rings (chains), and various sizes of the network which agree with the mathematical expressions.
Mesh network
(
)
RC
M
×
Faulty pattern characteristics
3×3 6×5 6×6 8×8 9×7 9×6 10×10
f-ring:
()()
1, 2,2lh α== =
1.0 0.63 0.51 0.42 0.47 0.46
0.38
f-ring:
(
)
3, 2, 2, 2lhα===
1.0 0.88 0.76 0.69 0.79 0.78
0.55
f-chain:
(
)
3, 2, 1, 1lhα===
0.95 0.47 0.37 0.31 0.34 0.33
0.29
Overlapped f-rings:
() ()
12 1 2 1 2
1, 2, 2, 2 , 4, 3llh h αα== = = = =
0.0 0.86 0.74 0.69 0.79 0.75
0.56
Overlapped f-chains:
() ()
12 1 2 1 2
1, 1, 3 , 2, 1llhh αα== = = = =
1.0 0.64 0.52 0.4 0.46 0.45
0.35
Pre-print
