Routing Reliability Analysis of Partially Disjoint Paths

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