Outline

• 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