• Dynamic Source Routing( DSR)1. Multipath extension to DSR
– Disjoint Multipath(protocol 1)– Multipath protocol 2
2. Analytic Modeling• Modeling of Protocol 1 • Modeling of Protocol 2• Modeling of Braided path• Modeling of Advanced Braid
Outline
Dynamic Source Routing( DSR)
1
4
3
2
5
6
7
8
<1>
<1>
<1,2>
<1,3>
<1>
<1,4>
<1,3,5>
<1,3,5,7>
<1,4,6>
<1,4,6>
<1,4,6>
<1,4,6>
S
D
Disjoint Multipath DSR( 1/3)• Many flooded query message arrive at the destination
via different routes.• Primary source route is the route taken by the first query
reaching the destination.• The destination “remembers” the primary source, in
order to figure out disjoint routes.• The destination controls the total number of replies ,thus
preventing a reply flood• The source keeps all routes received on reply packets in
its route cache.• When the primary route breaks ,the shortest remaining
alternate route is used.
Disjoint Multipath DSR( 2/3)
DS
P1
P2
PN
Disjoint Multipath DSR( 3/3)• Shortcoming An intermediate link failure on the primary route still
sends an error packet back to the source ,which will then use an alternate route
This cases a temporary loss of route for the data packets that are already in transit upstream from the failed link
Let us call this technique as protocol 1
Multipath protocol 2 (1/3)
• The destination replies to each intermediate node in the primary route with an alternate disjoint route to the destination.
• The reply is targeted to the intermediate node instead of the source
Multipath protocol 2 (2/3)
S D
L1 L2 L3 L4 Lk
P1
P2
P3
P4
n1
n2
n3
n4
nk+1
Multipath protocol 2 (3/3)
• When the link Li is broken ,the node ni replace the unused portion of the route, Li to Lk ,in the data packet header by the alternate route Pi.
• The node ni is responsible for modifying the source route on all later data packets to use its own alternate route.
• This will continue until a link on Pi breaks.• It will cause an error packet transmitted backward up to
node ni-1,which will quench the error packet and switch all later data packets to its own alternate route Pi-1 .
• This process continues until the source gets an error packet and has no alternate route
Analytic Modeling(1/2)
• The lifetime of Li is denoted by XLi
• We represent the lifetime of a wireless link between a pair of nodes by a random variable
• Assume that XLi ,i=1,2,….k, are independent and identically distributed exponential random variables ,each with mean ι
• EX:
1
tLiX
Analytic Modeling(2/2)
• Since a route fails when any one of the wireless links in its path breaks , the lifetime of a route P ,consisting of k wireless links, is a random variable XP that can be expressed asXP =min(XL1, XL2,….., XLk)
• It is well known that XP is also an exponentially distributed random variable with a mean of
k
Modeling of Protocol 1 (1/4)
• Assume a source S has N routes to a destination D .
• The primary route is denoted by P1 and the N-1 alternate routes are denoted by P2, P3,…. PN
• The length of route Pi is ki • The time after which none of the routes are usef
ul is a random variable T, whereT =max(XP1, XP2,….., XPN)
• T represents the time between successive route discoveries
Modeling of Protocol 1 (2/4)
• XP1, XP2,….., XPN are exponential random variables ,where
the pdf of XPi is
Note that XPi ‘s are independent• the cdf of XPi is
Nitf tiX
i
iP,.....,2,1,)(
tt
XXi
iPiPdttftF 1)()(
0
Modeling of Protocol 1 (3/4)
• The cdf of T ,FT(t) is obtained as
N
iX
PPP
PPP
T
tF
tXtXtXP
tXXXP
tTPtF
iP
N
N
1
)(
)](.....)()[(
]),...,,[max(
][)(
21
21
Modeling of Protocol 1 (4/4)
• The pdf of T ,the time between successive route discoveries ,is given by
• Where λi =ki /ι=1/lifetime of the i-th route
)1()1)(1(
)1()1)(1(
)1()1)(1()(
)(')(
121
312
321
2
1
ttttN
tttt
ttttT
TT
NN
N
Ntf
tFtf
Modeling of Protocol 2 (1/5)
• The time until the next route discovery T is the time until the event E is true ,where E is described by the following logical expression:
• Then T can be expressed as
)( 11123312211 PPPLPPPLPPLPLE kkk
),,,,max(,,
),,max(),,max(min
11
12211
PPPL
PPLPL
XXXX
XXXXXT
kkk
Multipath protocol 2 (2/3)
S D
L1 L2 L3 L4 Lk
P1
P2
P3
P4
n1
n2
n3
n4
nk+1
Modeling of Protocol 2 (2/5)
• Let us denote the random variable by Zi ,
• the pdf of Zi ,fZi(t) is given by
• Where
),...,,max(21 NPPP XXX
1
1
1
,1
)1()(i
j
i
jkk
ttjZ
kj
itf
11
,,2,1
ijfor
ijfork j
j
Modeling of Protocol 2 (3/5)
• the cdf of Zi , FZi(t) is given by
• where
11
,,2,1
)1()()(1
10
ikfor
ikfork
dttftF
k
k
i
k
tt
ZZk
ii
Modeling of Protocol 2 (4/5)
• The pdf of T =min(Z1 ,Z2 ,…, Zk )• min(Z1 ,Z2 ,…, Zk )= max((1-Z1 ),(1-Z2 ),…,(1- Zk
))• the cdf of T , FT (t) is given by
N
iX
ZZZ
ZZZT
tF
FFFP
FFFPtF
iP
k
1
)(1
])1(),...,1(),1([max
)],...,,[min()(
111
21
Modeling of Protocol 2 (5/5)
• the pdf of T , fT (t) is given by
k
i
k
ijjZZ
PPPL
PPLPL
T
TT
tFtf
XXXX
XXXXXtf
tFtf
ii
kkk
1 ,1
k21
)(1)(
) Z,, Z, min(Z
),,,,max(,,
),,max(),,max(min)(
)(')(
11
12211
• The expected value of T ,E[T] is given by
• No. of disjoint paths=N• Mean lifetime of a single link, ι=5
0
)(][ dttftTE T
)1(2.2,:
1,1,:
:
1Pr
21
21
21
NkkkkkkCCASE
NkkkkkkBCASE
kkkkACASE
otocol
N
N
N
3:
2:
1:
2Pr
ikkCCASE
ikkBCASE
ikkACASE
otocol
i
i
i
Protocol 1
00.5
11.5
22.5
33.5
4
2 3 4 5 6 7Length of the primary route (k)
Inter
val b
etwee
n ro
ute
disc
over
ies,E
[T]
Case A (ki=k),N=2
Case B (ki=k+i-1),N=2
Case C (ki=k+2(i-1)),N=2N=1
protocol 1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
1 2 3 4 5 6 7No. of disjoint paths(N)
Inte
rval
bet
wee
n ro
ute
disc
over
ies,E
[T]
Case A,N=3
Case B,N=3
Case C,N=3
Case A,N=6
Case B,N=6
Case C,N=6
Protocol 2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
2 3 4 5 6 7
Length of the primary route (k)
Inte
rval
bet
wee
n ro
ute
disc
over
ies,E
[T]
Case A,ki=k-i+1
Case B,ki=k-i+2
Case C,ki=k-i+3
Single path
Node fail
• Link fail
• Node fail
)),,,,max(,
),,,max(),,min(max(
)(
121
12312
121123412312
PPPn
PPnPn
kkk
XXXX
XXXXXT
PPPnPPPnPPnPnE
kkk
),,,,max(,,
),,max(),,max(min
)(
11
12211
11123312211
PPPL
PPLPL
kkk
XXXX
XXXXXT
PPPLPPPLPPLPLE
kkk
Multipath protocol 2
S D
L1 L2 L3 L4 Lk
P1
P2
P3
P4
n1
n2
n3
n4
nk+1
Protocol 2 (node fail)
00.5
11.5
22.5
33.5
44.5
55.5
66.5
77.5
8
2 3 4 5 6 7Length of the primary route (k)
Inte
rval
bet
wee
n ro
ute
disc
over
ies,E
[T]
Case A,ki=k-i+1
Case B,ki=k-i+2
Case C,ki=k-i+3
Single path
Modeling of Braided path
)),max(,),,max(),,max(,
),max(,),,max(),,min(max(
13221
14332
1322114332
kk
kk
npnpnp
nnnnnn
kkkk
XXXXXX
XXXXXXT
npnpnpnnnnnnE
1 2 3 K4 K+1K-1
P1
P2
K-2
Pk-2
Pk-1
Braided path
0
0.5
1
1.5
2
2.5
3
3.5
4
3 4 5 6 7
Length of the primary route (k)
Inte
rval
bet
wee
n ro
ute
disc
over
ies,E
[T]
Case A,alternate path=1
Case B,alternate path=2
Case C,alternate path=3
Single path
Modeling of Advanced Braid(1/2)
• Assume length of a= length of Pi
2 3 K4 K+1K-1
P1
P2
K-2
Pk-2
Pk-1
5 6
P3
P4
a
a
K-3
Pk-3 a
1
Modeling of Advanced Braid(2/2)
)),max(),,max(,
),,max(),,min(max(
112
3221
1123221
kkkk npanp
anpnp
kkkk
XXXXX
XXXXXT
npanpanpnpE
Advanced Braid
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
4 5 6 7
Length of the primary route (k)
Inte
rval
bet
wee
n ro
ute
disc
over
ies,E
[T]
Case A,alternate path=1
Case B,alternate path=2
Case C,alternate path=3
Single path
Comparison of CASE A
k 4 5 6 7
Advanced path
3.881 3.4623 3.1741 2.9591
Braided path
2.5317 2.0621 1.7755 1.5786
Protocol 2
2.9497 2.3402 1.9602 1.6979
Comparison of CASE B
k 4 5 6 7
Advanced path
2.7621 2.4068 2.1747 2.0073
Braided path
2.055 1.6803 1.4497 1.2905
Protocol 2
2.4906 2.027 1.7313 1.5227
Comparison of CASE C
k 4 5 6 7
Advanced path
2.3521 2.0055 1.788 1.6355
Braided path
1.8355 1.494 1.2853 1.1418
Protocol 2
2.2478 1.8374 1.5795 1.3986
Comparison
0
1
2
3
4
5
4 5 6 7
Length of the primary route (k)
Inter
val b
etwee
n ro
ute
disc
over
ies,E
[T]
Case A,mybraid
Case B,mybraid
Case C,mybraid
Case A,braid
Case B,braid
Case C,braid
Case A,protocol 2
Case B,protocol 2
Case C,protocol 2
single path