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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