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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

[1] V. Puente, and J.-Ángel Gregorio, “Immucube: Scalable

Fault-Tolerant Routing for k-ary n-cube Networks”, IEEE

Trans. on Parallel and Distributed Systems, Vol. 18, No. 6,

June 2007, pp. 776-788.

[2] R. V. Boppana, and S. Chalasani, “Fault-Tolerant

Wormhole Routing Algorithms for Mesh Networks”, IEEE

Trans. Computers, Vol. 44, No. 7, 1995, pp. 848-864.

[3] Y. M. Boura, and C. R. Das, “Fault-tolerant routing in

mesh networks”, International Conference on Parallel

Processing, 1995, pp. I-106-I-109.

[4] J. D. Shih, “Fault-tolerant wormhole routing in torus

networks with overlapped block faults”, IEEE Proc. Comp.

Digital Tech., Vol. 150, No. 1, Jan. 2003, pp. 29-37.

[5] S. Park, J.-H. Youn, and B. Bose, “Fault-Tolerant

Wormhole Routing Algorithms in Meshes in the Presence of

Concave Faults”, Proceedings of the 14

th

International

Symposium on Parallel and Distributed Processing, 2000.

[6] C. L. Chen, and G. M. Chiu, “A Fault-tolerant routing

scheme for meshes with non-convex faults”, IEEE Trans. on

Parallel and Distributed Systems, Vol. 12, No. 5, May 2001,

pp. 467-475.

[7] J. Zhou, and F. C. M. Lau, “Adaptive fault-tolerant

wormhole routing with two virtual channels in 2D meshes”,

Proc. of 7

th

Int. Symposium Parallel Architectures,

Algorithms and Networks, 2004, pp. 142-148.

[8] H. Gu, et al., “A new routing method to tolerate both

convex and concave faulty regions in mesh/torus networks”,

The International Conference on Parallel and Distributed

Computing, Applications and Technologies (PDCAT05),

Dec. 2005, pp.714-719.

[9] I. Theiss, “Modularity, Routing and Fault Tolerance in

Interconnection Networks”, PhD thesis, Faculty of

Mathematics and Natural Sciences, University of Oslo, Feb.

2004.

[10] J. Wu, and Z. Jiang, “On Constructing the Minimum

Orthogonal Convex Polygon in 2-D Faulty Meshes”, IPDPS

2004.

[11] Y.J. Suh, et al., “Software-based rerouting for fault-

tolerant pipelined communication”, IEEE Trans. On Parallel

and Distributed Systems, Vol. 11, No. 3, Mar. 2000.

[12] J. Duato, and S. Yalamanchili, L.M. Ni,

“Interconnection networks: An engineering approach”,

Morgan Kaufmann Publishers, 2003.

[13] M. Hoseiny Farahabady, F. Safaei, A. Khonsari, and M.

Fathy, “Characterization of Spatial Fault Patterns in

Interconnection Networks”, Journal of Parallel Computing,

Vol. 32, No. 11-12, 2006, pp. 886-901.

[14] J. Xu, “Topological structure and analysis of

interconnection networks”, Kluwer Academic Publishers,

2001.

[15] F. Safaei, M. Fathy, A. Khonsari, M. Gilak, and M.

Ould-Khaoua, “A New Performance Measure for

Characterizing Fault-Rings in Interconnection Networks”,

Submitted to Journal of Information Sciences, 2007.

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