Download Dissertation Defense

Models 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-lesPins, 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 28672891
Intermittently Connected
Networks
V1
B
V2
A
V3
C
 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))
• Mobile wireless networks: C = Θ(1),
[Gupta99]
[Grossglauser01]
 a really challenging network scenario

No traditional protocol works
Some examples
DieselNet,
bus network
DakNet, Internet to rural area
ZebraNet, Mobile sensor networks



Network for disaster relief team
Military battle-field network
…
Inter-planetary backbone
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?
5
Epidemic Routing
V1

B
A
V2
V3
C
Message as a disease, carried around and transmitted
Epidemic Routing
V1

B
A
V2
V3
C
Message as a disease, carried around and transmitted
o Store, Carry and Forward paradigm
Epidemic Routing
V1
A
V3
B


V2
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…


C
An example:
Epidemic routing in ZebraNet
9
An example:
Epidemic routing in ZebraNet
10
Standard Epidemic Routing
t1
S
t2
2
3
4
5
D
TD
Time
11
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…
12
2-Hop Forwarding
t0
S
2
3
4
5
D
At most 2 hop
TD
Time
13
Limited Time Forwarding
t0
S
2
3
4
5
3
D
Time
14
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

15
IMMUNE Recovery
t0
S
2
3
4
5
TD
D
4
8
7
Time
16
VACCINE Recovery
t0
S
2
3
4
5
TD
D
4
8
7
Time
2
17
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
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
D
3
N
19
Standard random mobility models
Random Waypoint model
(RWP)
V2
Random Direction model
(RD)
X2
T2, V2
X1
α2
V1
T1, V1
R
 Next
positions (Xi)s are
uniformly distributed
 Speeds (Vi)s are uniformly
distributed (Vmin,Vmax)
R
α1
Directions (αi) are uniformly
distributed (0, 2π)
Speeds (Vi) are uniformly
distributed (Vmin,Vmax)
Travel times (Ti) are exponentially
/generally distributed
20
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:

2 wrV *

A
w: mobility specific constant
21
Intuitive explanation
Exponential distribution finds its roots in the
independence 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.
22
Why “almost”?
 pairwise meeting processes are almost
independent Poisson processes with rate:
2 wrV *

w: mobility specific constant
A
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.
23
Examples
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)
24
Pairwise Inter-meeting time
P(T  t )  e t
log P(T  t )  t
25
Pairwise Inter-meeting time
P(T  t )  e t
log P(T  t )  t
26
Pairwise Inter-meeting time
P(T  t )  e t
log P(T  t )  t
27
The derivation of λ
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:
V*
V*Δt
28
The derivation of λ
Probability that nodes meet given by
p x1, y1     ( x, y )dxdy.
A
For small r the points in π(x,y) in A can be
approximated by π(x1,y1) to give
px1 ,y1  2r  V1   t   (x1, y1 ).
*
Unconditioning on (x1,y1) gives
L L

p

p x1, y1   ( x1, y1 )dx1 y1
0 0
 2r  V   t 
*
1
L
L
0
0
 
 2 (x1, y1 )dx1 y1
29
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
  2 r  E[V*]
L
L

2

 (x, y)dxdy,
0
0
Here E[V*] is the average relative speed between
9

(
x
,
y
)
two nodes and
is the pdf in the point

(x,y).
30
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
2rE[V*]
2rE[V*]
RW 
.
RD 
,
2
2
L
L
Here E[V*] is the average relative speed between
two nodes and ω ≈ 1.3683 is the Waypoint
 constant.

If speeds of the nodes are constant and equal to v,
then
8rv
8rv
RD 
L
2
,
RW 
L
2
.
31
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
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.
33
Epidemic 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
Message delay
Proposition: The Laplace transform of the message
delay under the two-hop multicopy protocol is:
T2* ( )
i
( N  1)!   
 i

,
i 1 ( N  i )!  N   
N
and
i ( N  1)!
P( N 2  i )  i
,
N ( N  i)!
i  1,, N
35
Message delay
Proposition: The Laplace transform of the message
delay under epidemic routing is:
1
j(N  1 j)
T ( )   
,
N i1 j1 j(N  1 j)  
N
i
*
E
and

1
P(N E  i)  ,
N
i 1,K ,N
36
Expected message delay
Corollary: The expected message delay under the
two-hop multicopy protocol is
1
i ( N  1)!
1 
1 


,
E[T2 ] 


O



N i 1 ( N  i)! N i   2 N
 N  
N
2
and under the epidemic routing is
1
1  1  log( N )    O 1  ,
 
E[TE ] 


 N 
N i1 i N 
N
Where γ ≈ 0.57721 is Euler’s constant.
37
Relative performance
The relative performance of the two relay protocols:
E[TE ] log( N) 2

E[T2 ]

N
and

E[N E ]
N

E[N 2 ]
2
Note that these are independent of λ!

38
Some remarks
•
•
•
•
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 firstmeeting times.
λ depends on:
- mobility pattern
- surface area
- transmission radius
39
Example: two-hop multicopy
40
Example: two-hop multicopy
41
Example: two-hop multicopy
42
Example: two-hop multicopy
Distribution of the number of copies (R=50,100,250m):
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: unrestricted
multicopy
46
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
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
1
( N  1)

2
2( N  2 ) 
2
3
3( N  3)
( N  3)3
( N  2)
3
D

( N  2)2
( N  1)
( N  1)
N
N-2
N-1
N
Infection rate:
rN ( I )  nI ( N  nI )
Delivery rate:
n I
Transient analysis to derive delay, copies made by
delivery; hard to obtain closed form, specially for more
complex schemes
48
Modeling Works:
Small and Haas
 Mobicom 2003 [small03]
 ODE introduced in a naive way for simple epidemic
scheme I ' (t )  I (t )(N  I (t ))
• 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
49
Our contribution
(1/3)
 A unified ODE framework...

limiting process of Markov processes as N
increases [Kurtz 1970]
50
[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)
 F(x(s)),
s
51
[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 E
and a solution of the differential equation
x(s)
 F(x(s)), x(0)  x0
such that x(s)sis in E for 0<=s<=t, then for each δ>0


1
lim Prsup X N (s)  X(s)    0
N 
N
0st 

52
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.
i ' (t )  i (t )(1  i (t ))

with initial condition
i (0)  lim nI (0) / N
N 

multiplying by N
I ' (t )  I (t )( N  I (t ))
53
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)
What can we do
with the fluid model?
 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:
P' (t )  I (t )(1  P(t ))
prob. that pkt is

delivery
ratedelay
 Average
not delivered yet
E[Td nodes-dst
]   (1  P(t )) dt
at time t
infected
0
meeting rate
 Avg. num. of copies sent at delivery

E[C ] 
'
I
(
t
)
P
(t )dt  1

0
55
What can we do
with the fluid model?
 Consider recovery process, eg IMMUNE
(dest. node cures infected node):
I ' (t )  I ( N  I  R )  I
R(t): num. of recovered nodes
R' (t )  I
Num. of susceptible nodes
 Total num. of copies made:

 Total buffer usage
lim R (t )
t 
I
(
t
)
dt

0
56
More flexible
than Markov models
 to model all the different variants,

e.g. limited-time forwarding
I r ' (t )   ( I r (t )  1)( N  I r (t )  T (t )  1)  I r (t ),
T ' (t )  I r (t )
 or
probabilistic forwarding, K-hop forwarding...
 under different recovery schemes (VACCINE,
IMMUNE,...)
57
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
58
Epidemic Routing Average delay
ODE provides good prediction on average delay
59
Delay distribution
CDF of delay under epidemic routing, N=160
Modeling error mainly due to approx. of ODE
60
Some results
 Extensible to other schemes
E[Td ]
I (t ), P (t )
Epidemic
routing
2-hop
forwarding
Prob.
forwarding
I (t ) 
N
1  ( N  1)e Nt
P (t )  1 
ln N
~  ( N  1)
N
N  1  e Nt
I (t )  N  ( N  1)e t
P(t )  1  eN 1Nt ( N 1) e
I (t ) 
 t
N
1  ( N  1)e pNt
P(t )  1  (
N
)1 / p
pNt
N 1 e
~
1 
 2
1
N 1
ln N
ln N
 E[Td ] 
 ( N  1)
p( N  1)
C, G
C
N 1
, G  N  1( IM )
2
N  3  N 2  2N  5
G
( IM _ TX )
2
C~

2
N,
C
G
N 1
2
p( N  1)
1 p
Matching results from Markov chain model,
obtained easier
61
An application:
Tradeoffs evaluation
Delay vs Power
Delay vs Buffer
src/dest
transmission
epidemic
routing
2-hop
3-hop
2-hop
3-hop
62
Other issues
 Not considered in this presentation
Effect of different buffer management
techniques when the buffer is limited
 ODEs by moment closure technique

63
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
64