1
Jian Pu, Eric Manning, Gholamali C. Shoja
PANDA group, Department of Computer Science
University of Victoria
August, 2001
Routing Reliability Analysis
of Partially Disjoint Paths
2
Outline
Background
Cases of edge sharing
Deriving the formulas
Results
3
Background
OSPF (open shortest path first)
a widely used link

state routing protocol
ROSPF (Reliable OSPF)
our proposed reliability extension to OSPF
two alternate paths are pre

calculated as backups
faster path switches may reduce routing convergence time
Multi

path approach in ROSPF
the calculated three paths can be partially disjoint
however, the shared edges should be as few as possible
and, the overall connection reliability should be acceptable
4
Our goals
To derive mathematical formulas for computing the
overall connection reliability
R
overall
of the three partially
disjoint paths between two nodes
To use these formulas to compute
R
overall
in our
proposed alternate path

finding algorithms.
5
Cases of edge sharing
Edge sharing among the three paths (P1, P2 and P3)
6
Assumptions
assume the number of edges on a path (hop number) represent
the length of the path
omit all other metrics, e.g. distance, delay or bandwidth
assume every edge has equal failure rate
f
e
therefore, the reliability of every link,
r
e
, is identical
assume we can detect the failed path and shift to an available
alternative with 100% probability
only consider edge failures, omit node failures in calculation
node failures can be treated as multiple edge failures
7
P1, P2 and P3 are three paths between source
s
and sink
t
path
P
1
length
(m
1
)
:
m
1
=
L
1
+
L
12
+
L
13
+
L
123
path
P
2
length
(m
2
)
:
m
2
=
L
2
+
L
12
+
L
23
+
L
123
path
P
3
length
(m
3
)
:
m
3
=
L
3
+
L
13
+
L
23
+
L
123
L
i
,
the
number
of
unshared
edges
used
only
by
path
P
i
;
i
=
{
1
,
2
,
3
}
L
ij
,
the number of double

shared edges used by path P
i
and path P
j;
i, j = {1, 2, 3}, i != j
L
123
,
the number of triple

shared edges used by all three paths
Notations
8
Methods
The shared edges make the three paths
dependent with each other
If a shared edge fails, two or three paths may
be affected
Therefore, we apply the 3

D Venn Diagram
model to deal with the dependent events
9
3

D Venn Diagram
To Calculate occurrence probability of
three dependent events
E
1
,
E
2
and
E
3
3

D Venn Diagram model
S = S
0

S
1
+ S
2
S

occurrence
probability
of non

null event
from E
1
, E
2
and E
3
S
0
=
probability
of all 3 independent events
= Pr(E
1
) + Pr(E
2
) + Pr(E
3
)
S
1
=
probability
of double overlap components
= Pr(E
12
) + Pr(E
13
) + Pr(E
23
)
S
2
=
probability
of triple overlap components
= Pr(E
123
)
E
1
E
2
E
3
E
12
E
13
E
23
E
123
10
Deriving the formulas (1)
Three dependent events in our 3

path routing
Event
1
:
the
connection
success
caused
by
success
of
path
P
1
P
r
(
1
)
=
probability
of
path
P
1
success
=
r
e
m
1
Event 2
: the connection success caused by success of path P
2
P
r
(2) = probability of path
P
2
success = r
e
m2
Event 3
: the connection success caused by success of path P
3
P
r
(3) = probability of path
P
3
success = r
e
m3
11
Deriving the formulas (2)
S
0
=
probability of
connection success
using one path
independently
= P
r
(1) + P
r
(2) + P
r
(3) = r
e
m1
+ r
e
m2
+ r
e
m3
S
1
=
probability of
connection success
using two paths
simultaneously
= P
r
(12) + P
r
(13) + P
r
(23)
= r
e
(m1+m2)
–
(L123+L12)
+ r
e
(m1+m3)
–
(L123+L13)
+ r
e
(m2+m3)
–
(L123+L23)
S
2
=
probability of
connection success
using all three paths
simultaneously
= P
r
(123)
= r
e
(m1+m2+m3)
–
(2*L123+L12+L13+L23)
Thus, according to 3

D Venn Diagram model:
R
overall
=
successful probability of overall network (reliability)
= S
=
S
0

S
1
+ S
2
12
Some special cases
(1). The three paths (P
1
, P
2
and P
3
) are entirely disjoint (case 1)
R
overall
= 1
–
(1
–
r
e
m1
) ∙ (1
–
r
e
m2
) ∙ (1
–
r
e
m3
)
(2). Only triple

shared edges exist among P
1
, P
2
and P
3
(case 2)
R
overall
= r
e
L123
∙ [1
–
(1
–
r
e
m1

L123
) ∙ (1
–
r
e
m2

L123
) ∙ (1
–
r
e
m3

L123
)]
(3). Only double

shared edges between P
1
and P
2
exist (case 3)
R
overall
= 1
–
(1
–
r
e
m3
) ∙ {1
–
r
e
L12
∙ [1
–
(1
–
r
e
L1
) ∙ (1
–
r
e
L2
)]}
13
Calculation for
R
overall
(1)
Connection reliability decreases with the increase of the number of triple

or double

shared
edge, e.g.
L
12
or
L
123
. Here, the length of each path is
40
; reliability of every edge
r
e
= 0.99
14
Calculation for
R
overall
(2)
The relation of connection reliability
R
overall
, and the edge reliability
r
e
when only triple

shared edges exist. Here, length of all paths is
20
15
Calculation for
R
overall
(3)
The comparisons of connection reliability
R
when different types of double

shared edges
exist. Here, length of all three path is
40
; reliability of every edge
r
e
= 0.99
16
References
J. Moy,
OSPF Version 2
, RFC2328, April 1998
J. Moy,
OSPF, Anatomy of an Internet Routing Protocol
, Chapter 7,
pp. 137

150, Addison

Wesley, 1998
J. W. Suurballe,
Disjoint Paths in a Network
, Networks, 4: 125

145,
1974
J. E. Freund and R. E. Walpole,
Mathematical Statistics
, 3
rd
edition,
chapter 2, 1980, pp. 24

59
Jian Pu, Eric Manning, G. C. Shoja,
A New Algorithm to Compute
Alternate Paths in Reliable OSPF (ROSPF),
PDPTA'2001 , June 2001,
Las Vegas, Nevada, USA
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