Models for Epidemic RoutingModels for Epidemic Routing

Giovanni Neglia

INRIA – EPI Maestro

20 January 2010

Some slides are based on presentations from

R. Groenevelt (Accenture Technology Labs)

and M. Ibrahim (previous Maestro PhD student)

Outline

Introduction on Intermittently Connected Networks (or Delay/Disruption Tolerant Networks)

Markovian models Message Delay in Mobile Ad Hoc Networks, R.

Groenevelt, G. Koole, and P. Nain, Performance, Juan-les-Pins, October 2005

Impact of Mobility on the Performance of Relaying in Ad Hoc Networks, A. Al-Hanbali, A.A. Kherani, R. Groenevelt, P. Nain, and E. Altman, IEEE Infocom 2006, Barcelona, April 2006

Fluid models Performance Modeling of Epidemic Routing, X.

Zhang, G. Neglia, J. Kurose, D. Towsley, Elsevier Computer Networks, Volume 51, Issue 10, July 2007, Pages 2867-2891

Intermittently Connected Networks

mobile wireless networks no path at a given time instant between two nodes

because of power contraint, fast mobility dynamics maintain capacity, when number of nodes (N) diverges

• Fixed wireless networks: C = Θ(sqrt(1/N)) [Gupta99]• Mobile wireless networks: C = Θ(1), [Grossglauser01]

a really challenging network scenario No traditional protocol works

A

V2 V3V1

B C

Some examples

Inter-planetary backbone

DakNet, Internet to rural area

DieselNet, bus network

ZebraNet, Mobile sensor networks

Network for disaster relief team Military battle-field network …

5

Terminology and Focus Why delay tolerant?

support delay insensitive app: email, web surfing, etc.

Why disruption tolerant? almost no infrastructure, difficult to attack post 9/11 syndrome

In this lecture we only consider case where there is no information about future meetings how to route?

Epidemic Routing

A

V2 V3V1

B C

Message as a disease, carried around and transmitted

Epidemic Routing

A

V2 V3V1

B C

Message as a disease, carried around and transmittedo Store, Carry and Forward paradigm

Epidemic Routing

Message as a disease, carried around and transmitted

How many copies? 1 copy ⇨ max throughput, very high delay More ⇨ reduce delay, increase resource consumption

(energy, buffer, bandwidth) Many heuristics proposed to constrain the infection

K-copies, probabilistic…

A

V2

V3

V1

BC

9

An example:Epidemic routing in ZebraNet

10

An example:Epidemic routing in ZebraNet

11

Standard Epidemic Routing

S

22

33

44

Time

55

DD

t1

TD

t2

12

Epidemic Style Routing

Epidemic Routing [Vahdat&Becker00] Propagation of a pkt -> Disease Spread achieve min. delay, at the cost of transm.

power, storage trade-off delay for resources

K-hop forwarding, probabilistic forwarding, limited-time forwarding, spray and wait…

13

2-Hop Forwarding

S

22

33

4

Time

55

DD

t0

TD

At most 2 hop

14

Limited Time Forwarding

S

22

33

Time

55

D

t0

44

3

15

Epidemic Style Routing

Epidemic Routing [Vahdat&Becker00] Propagation of a pkt -> Disease Spread achieve min. delay, at the cost of transm.

power, storage trade-off delay for resources

K-hop forwarding, probabilistic forwarding, limited-time forwarding, spray and wait…

Recovery: deletion of obsolete copies after delivery to dest., e.g., TIMERS: when time expires all the copies are

erased IMMUNE: dest. cures infected nodes VACCINE: on pkt delivery, dest propagates

anti-pkt through network

16

IMMUNE RecoveryS

22

33

44

Time

55

DD

t0

TD

44788

17

VACCINE RecoveryS

22

33

44

Time

55

DD

t0

TD

44788 7

22

Outline

Introduction on Intermittently Connected Networks (or Delay/Disruption Tolerant Networks)

Markovian models the key to Markov model Markovian analysis of epidemic routing

Fluid models

19

The setting we consider N+1 nodes

moving independently in an finite area A

with a fixed transmission range r and no interference

1 source, 1 destination Performance metrics:

Delivery delay Td

Avg. num. of copies at delivery C Avg. total num. of copies made G Avg. buffer occupancy

S

2

3

D

N

20

Standard random mobility models

X1

X2

V1

V2

Directions (αi) are uniformly distributed (0, 2π)

Speeds (Vi) are uniformly distributed (Vmin,Vmax)

Travel times (Ti) are exponentially /generally distributed

Directions (αi) are uniformly distributed (0, 2π)

Speeds (Vi) are uniformly distributed (Vmin,Vmax)

Travel times (Ti) are exponentially /generally distributed

R

T1, V1

T2, V2

R

Next positions (Xi)s are uniformly distributed

Speeds (Vi)s are uniformly distributed (Vmin,Vmax)

Next positions (Xi)s are uniformly distributed

Speeds (Vi)s are uniformly distributed (Vmin,Vmax)

α1

α2

Random Waypoint model (RWP)

Random Waypoint model (RWP)

Random Direction model (RD)

Random Direction model (RD)

21

The key to Markov model

[Groenevelt05] if nodes move according to standard random

mobility model (random waypoint, random direction) with average relative speed E[V*],

and if Nr2 is small in comparison to A pairwise meeting processes are almost

independent Poisson processes with rate:

A

wrV *2 w: mobility specific constant

22

Exponential distribution finds its roots in theindependence assumptions of each mobility

model:• Nodes move independently of each other• Random waypoint: future locations of a

node are independent of past locations of that node.

• Random direction: future speeds and directions of a node are independent of past speeds and directions of that node.

There is some probability q that two nodes will meet before the next change of direction. At the next change of direction the process repeats itself, almost independently.

Intuitive explanation

23

Why “almost”?

pairwise meeting processes are almost independent Poisson processes with rate:

1. inter-meeting times are not exponential if N1 and N2 have met in the near past they are

more likely to meet (they are close to each other)• the more the bigger it is r2 in comparison to A

2. meeting processes are not independent if in [t,t+τ] N1 meets N2 and N2 meets N3, it is

more likely that N1 meets N3 in the same interval• the more the bigger it is r2 in comparison to A

moreover if Nr2 is comparable with A (dense network) a lot of meeting happen at the same time.

A

wrV *2 w: mobility specific constant

24

Nodes move on a square of size 4x4 km2 (L=4 km)

Different transmission radii (R=50,100,250 m)

Random waypoint and random direction:no pause time[vmin,vmax]=[4,10] km/hour

Random direction: travel time ~ exp(4)

Examples

25

Pairwise Inter-meeting time

ttTP

etTP t

)(log

)(

26

Pairwise Inter-meeting time

ttTP

etTP t

)(log

)(

27

Pairwise Inter-meeting time

ttTP

etTP t

)(log

)(

28

Assume a node in position (x1,y1) moves in a straight line with speed V1.

Position of the other node comes from steady-state distribution with pdf π(x,y).

Look at the area A covered in Δt time:

The derivation of λ

V*

V*Δt

29

The derivation of λ

Probability that nodes meet given by

A

yx dxdyyxp .),(1,1

For small r the points in π(x,y) in A can be approximated by π(x1,y1) to give

€

px1 ,y1≈ 2r ⋅V1

* ⋅Δ t ⋅π (x1,y1).

Unconditioning on (x1,y1) gives

11111,100

),( ydxyxpp yx

LL

€

≈2r ⋅V1* ⋅Δ t ⋅

0

L

∫0

L

∫ π 2(x1,y1)dx1y1

30

Proposition: Let r<<L. The inter-meeting time for the random direction and the random waypoint mobility models is approximately exponentially distributed with parameter

Here E[V*] is the average relative speed between two nodes and is the pdf in the point (x,y).

The derivation of λ

€

≈2 r ⋅E[V*]⋅0

L

∫ π 2(x,y)dxdy0

L

∫ ,

),( yx 9

31

The derivation of λ

Proposition: Let r<<L. The inter-meeting time for the random direction and the random waypoint mobility models is approximately exponentially distributed with parameter

Here E[V*] is the average relative speed between two nodes and ω ≈ 1.3683 is the Waypoint constant.€

RD ≈2rE[V*]

L2,

€

RW ≈2rωE[V*]

L2.

,

82L

rvRD

.

82L

rvRW

If speeds of the nodes are constant and equal to v,

then

Summary up to now

First steps of this research a good intuition some simulations validating the intuition for

a reasonable range of parameters What could have been done more

prove that the results is asymptotically (r->0) true “in some sense”

What can be built on top of this? Markovian models for routing in DTNs

33

2-hop routing

Model the number of occurrences of the message as an absorbing Markov chain:

• State i{1,…,N} represents the number of occurrences of the message in the network.

• State A represents the destination node receiving (a copy of) the message.

34

Model the number of occurrences of the message as an absorbing Markov chain:

• State i{1,…,N} represents the number of occurrences of the message in the network.

• State A represents the destination node receiving (a copy of) the message.

Epidemic routing

35

Proposition: The Laplace transform of the message delay under the two-hop multicopy protocol is:

Message delay

, )!(

)!1()(

1

*2

iN

i NiN

NiT

NiiN

N

N

iiNP

i,,1 ,

)!(

)!1()( 2

and

36

Proposition: The Laplace transform of the message delay under epidemic routing is:

Message delay

€

TE*(θ) =

1

N

jλ (N +1− j)

jλ (N +1− j) +θj=1

i

∏i=1

N

∑ ,

€

P(NE = i) =1

N, i =1,K ,N

and

37

Corollary: The expected message delay under the

two-hop multicopy protocol is

Expected message delay

N

iiNiN

Ni

NTE

1

2

2)!(

)!1(1 ][

and under the epidemic routing is

€

E[TE ] = 1

λN

1

ii=1

N

∑

Where γ ≈ 0.57721 is Euler’s constant.

,1

2

1

N

ON

,1

)log(1

N

ONN

38

Relative performance

and

€

E[TE ]

E[T2]≈

log(N)

N

2

π

€

E[NE ]

E[N2]≈

N

2π

The relative performance of the two relay protocols:

Note that these are independent of λ!

39

• These expressions hold for any mobility model which has exponential meeting times.

• Two mobility models which give the same λ also have the same message delay for both relay protocols! (mobility pattern is “hidden” in λ)

• Mean message delay scales with mean first-meeting times.

• λ depends on:- mobility pattern- surface area- transmission radius

Some remarks

40

Example: two-hop multicopy

41

Example: two-hop multicopy

42

Example: two-hop multicopy

43

Example: two-hop multicopy

Distribution of the number of copies (R=50,100,250m):

44

Example: two-hop multicopy

Distribution of the number of copies (R=50,100,250m):

45

Example: two-hop multicopy

Distribution of the number of copies (R=50,100,250m):

46

Example: unrestricted multicopy

Outline

Introduction on Intermittently Connected Networks (or Delay/Disruption Tolerant Networks)

Markovian models the key to Markov model Markovian analysis of epidemic routing

Fluid models

48

Why a fluid approach?

[Groenevelt05] Markov models can be developed States: nI =1,…, N: num. of infected nodes, different

from destination; D: packet delivered to the destination

Transient analysis to derive delay, copies made by delivery; hard to obtain closed form, specially for more complex schemes

Infection rate:

Delivery rate:

)()( IIN nNnIr

In

)1( N )2(2 N )3(3 N

1 2 3 N-2 NN-12)2( N3)3( N )1( N

D

2 3 )2( N )1( N N

49

Modeling Works: Small and Haas Mobicom 2003 [small03]

ODE introduced in a naive way for simple epidemic scheme

• N is the total number of nodes,• I the total number of infected nodes• λ is the average pairwise meeting rate

Average pair-wise meeting rate obtained from simulations

TON 2006 [haas06] consider a Markov Chain with N-1 different

meeting rates depending on the number of infected nodes (obtained from simulations)

Numerical solution complexity increases with N

))()(()(' tINtItI

50

Our contribution (1/3)

A unified ODE framework... limiting process of Markov processes as N

increases [Kurtz 1970]

51

[Kurtz1970]

{XN(t), N natural} a family of Markov process in Zm

with rates rN(k,k+h), k,h in Zm

It is called density dependent if it exists a continuous function f() in Rm such that rN(k,k+h) = N f(1/N k, h), h<>0

Define F(x)=Σh h f(x,h)

Kurtz’s theorem determines when {XN(t)} are close to the solution of the differential equation:

€

∂x(s)

∂s= F(x(s)),

52

[Kurtz1970]

Theorem. Suppose there is an open set E in Rm and a constant M such that |F(x)-F(y)|<M|x-y|, x,y in E supx in EΣh|h| f(x,h) <∞,

limd->∞supx in EΣ|h|>d|h| f(x,h) =0

Consider the set of processes in {XN(t)} such that limN->∞ 1/N XN(0) = x0 in Eand a solution of the differential equation

such that x(s) is in E for 0<=s<=t, then for each δ>0

€

∂x(s)

∂s= F(x(s)),

€

x(0) = x0

€

limN→∞

Pr sup0≤s≤ t

1

NXN (s) − X(s) > δ

⎧ ⎨ ⎩

⎫ ⎬ ⎭= 0

53

Application to epidemic routing rN(nI)=λ nI (N-nI) = N (λN) (nI/N) (1-nI/N) assuming β = λ N keeps constant (e.g.

node density is constant) f(x,h)=f(x)=x(1-x), F(x)=f(x) as N→∞, nI/N → i(t), s.t.

with initial condition

multiplying by N

))(1)(()(' tititi

Nni IN

/)0(lim)0(

))()(()(' tINtItI

What can we do with the fluid model? Derive an estimation of the number of

infected nodes at time t e.g. if I(0)=1 -> I(t)=N/(1+(N-1)e-Nλt)

55

Delivery delay Td: time from pkt generation at the src until the dst receives the pkt CDF of Td, P(t) := Pr(Td<t) given by:

Average delay

Avg. num. of copies sent at delivery

infected nodes-dst meeting rate

What can we dowith the fluid model?

))(1)(()(' tPtItP

0

))(1(][ dttPTE d

1)()(][0

'

dttPtICE

delivery rate at time t

prob. that pkt is not delivered yet

56

What can we dowith the fluid model? Consider recovery process, eg IMMUNE

(dest. node cures infected node):

Total num. of copies made:

Total buffer usage

ItR

IRINItI

)('

)()('

)(lim tRt

0

)( dttI

R(t): num. of recovered nodesNum. of susceptible

nodes

57

More flexible than Markov models to model all the different variants,

e.g. limited-time forwarding

or probabilistic forwarding, K-hop forwarding...

under different recovery schemes (VACCINE, IMMUNE,...)

)()('),()1)()()(1)(()('

tItTtItTtINtItI

r

rrrr

58

Our contribution

Closed formulas for average delay, number of copies and CDF in many cases

Asymptotic results Numerical evaluation always possible

without scaling problems Study of delay vs buffer occupancy or

delay vs power consumption for different forwarding schemes

59

Epidemic Routing Average delay

ODE provides good prediction on average delay

60

Delay distribution CDF of delay under epidemic routing,

N=160

Modeling error mainly due to approx. of ODE

61

Some results Extensible to other schemes

teNNtI )1()(

NteN

NtP

1

1)(

NteN

NtI

)1(1)(

teNNtNetP )1(11)(

NtpeN

NtI

)1(1)(

pNtpeN

NtP /1)

1(1)(

)1(

ln

N

N

1

1

2

1

N

)1(

ln][

)1(

ln

Np

NTE

N

Nd

)(1,2

1IMNG

NC

)_(2

523 2

TXIMNNN

G

2

1,

2

N

GNC

p

NpC

1

)1(

Epidemicrouting

2-hopforwarding

Prob.forwarding

)(),( tPtI ][ dTE GC,

Matching results from Markov chain model, obtained easier

~

~ ~

62

An application: Tradeoffs evaluation

Delay vs Power Delay vs Buffer

2-hop3-hop

2-hop3-hop

src/dest transmission

epidemicrouting

63

Other issues

Not considered in this presentation Effect of different buffer management

techniques when the buffer is limited ODEs by moment closure technique

64

References

Papers discussed Markovian models

• Message Delay in Mobile Ad Hoc Networks, R. Groenevelt, G. Koole, and P. Nain, Performance, Juan-les-Pins, October 2005

• Impact of Mobility on the Performance of Relaying in Ad Hoc Networks, A. Al-Hanbali, A.A. Kherani, R. Groenevelt, P. Nain, and E. Altman, IEEE Infocom 2006, Barcelona, April 2006

Fluid models• Performance Modeling of Epidemic Routing, X. Zhang, G. Neglia, J.

Kurose, D. Towsley, Elsevier Computer Networks, Volume 51, Issue 10, July 2007, Pages 2867-2891

Other [Grossglauser01] Mobility Increases the Capacity of Ad Hoc

Wireless Networks, M. Grossglauser and D. Tse, IEEE Infocom 2001

[Gupta99] The capacity of Wireless Networks, P. Gupta, P.R. Kumar, IEEE Conference on Decision and Control 1999

[Kurtz70] Solution of ordinary differential equations as limits of pure jump markov processes, T. G. Kurtz, Journal of Applied Probabilities, pages 49-58, 1970