1. 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)

2. 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

3. Intermittently Connected NetworksV2 V3 B V1 C A
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
ICN are mobile wireless networks, where there is no path at a given time instant between two nodes.
This can happen because of transmission power constraints or because of fast mobility dynamics.
But it can also be intended in order to increase network capacity by limiting interference effects.
In fact Grossglauser and Tse in a famous paper that received the Infocom best paper award,
have proved that a constant throughput can be maintained in a mobile disconnected network,
While this is not true for fixed wireless networks.
In any case ICNs represent a very challenging network scenario because no traditional Internet protocol would work in such conditions.
Before I start describing which alternative solutions have been proposed, let me first show some real examples.

4. Some examples DieselNet, bus network DakNet, Internet to rural area
DakNet is a network that provides Internet connectivity to villages scattered through the Cambodian forest.
Motorcycles, or even ox carts, carrying WiFi devices transfer the information among hot spots in different villages.
Even in developed countries, wireless devices installed on buses can increase network coverage. An example is UMass DieselNet.
In ZebraNet project, wireless sensors are deployed on wild animals to monitor their habits.
The current proposal for the interplanetary backbone Is also an intermittently connected network.
Networks to allow communication in disaster relief teams or in soldier platoons are other possible examples.
ZebraNet, Mobile sensor networks
Network for disaster relief team
Military battle-field network …
Inter-planetary backbone

5. Terminology and Focus Why delay tolerant? Why disruption tolerant?
support delay insensitive app: , 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?
In all of these networks we just seen, there is very little or no infrastructure support,
mobile nodes takes up the job of relaying packets, and furthermore the network is often partitioned.
Such network is nevertheless useful to support applications that is not delay sensitive such as s, web serfing etc.
Notice that these network can have different mobility scenario: while buses and motocycles might run
According to fixed schedule and route;
the way zerbra, whale and human beings migrate/move around shows more randomness.
In my thesis, I will focus on mobile DTN where nodes movement is random.

6. Message as a disease, carried around and transmitted
Epidemic Routing V2 V3 B V1 C A
Message as a disease, carried around and transmitted
Up to now we haven’t explicitly explained how messages can be delivered in such networks.
We denote here as epidemic routing a family of forwarding schemes, where messages are carried around by nodes
as a disease in a population and can eventually be transmitted to other nodes.

7. Epidemic Routing Message as a disease, carried around and transmittedV2 V3 B V1 C A
Message as a disease, carried around and transmitted
Store, Carry and Forward paradigm

8. Epidemic Routing Message as a disease, carried around and transmittedV1 A V3 C 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…
If we allow only one copy of each message to be in the system, then there is a scheme that achieves the maximum theoretical throughput, but with a very high delivery delay.
On the contrary we can reduce the delivery delay using multiple copies, but by doing so
We reduce the throughput, we increase energy consumption and we need more space in the buffers.
To address this intrinsic tradeoff, many heuristics have been proposed to constrain the infection, for example
by limiting the max number of copies, or by copying the information at each encounter according to some probability.

9. An example: Epidemic routing in ZebraNet

10. An example: Epidemic routing in ZebraNet

11. Standard Epidemic Routing2 2 t2 3 3 4 4 5 5 D D TD
Time

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…
First, let me describe the various epidemic style routing considered in this talk, and introduce some terminology.
One can make an analogy of the propagation of a packet in DTN to the spread of disease. The nodes that carry a copy of a packet is called an infected node, those that don’t have a copy, but are potential
Carriers, are called susceptible nodes. When two nodes meet, and the packet is being copied from the
One to another, we call the former node infect the other node.
When a node deletes a copy, we say that it recovers.
With these names chosen, epidemic routing, proposed by Vahdat and Becker, is where whenever an infected node meets a susceptible, a copy is forward. This flooding type of scheme achievs minimum
Delay, but also incurs many copies being made, and storage consumption.
In order to trade-off delay performance for resource consumption, many variants of epidemic routing has
been proposed.
They include K-hop, where a packet can traverse at most K hops; prob. Forwarding where a pkt is copied to relay with certain prob.; limited-time forwarding where relay nodes store and forward within
A limited time; etc.
In the first part of my thesis, I will study the problem of characterizing the performance of these routing schemes, the performance trade-off.

13. 2-Hop Forwarding S t0 2 2 3 3 4 5 5 D D TD
At most 2 hop
Time

14. Limited Time ForwardingS t0 2 2 3 3 4 4 5 5 3 D
Time

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
First, let me describe the various epidemic style routing considered in this talk, and introduce some terminology.
One can make an analogy of the propagation of a packet in DTN to the spread of disease. The nodes that carry a copy of a packet is called an infected node, those that don’t have a copy, but are potential
Carriers, are called susceptible nodes. When two nodes meet, and the packet is being copied from the
One to another, we call the former node infect the other node.
When a node deletes a copy, we say that it recovers.
With these names chosen, epidemic routing, proposed by Vahdat and Becker, is where whenever an infected node meets a susceptible, a copy is forward. This flooding type of scheme achievs minimum
Delay, but also incurs many copies being made, and storage consumption.
In order to trade-off delay performance for resource consumption, many variants of epidemic routing has
been proposed.
They include K-hop, where a packet can traverse at most K hops; prob. Forwarding where a pkt is copied to relay with certain prob.; limited-time forwarding where relay nodes store and forward within
A limited time; etc.
In the first part of my thesis, I will study the problem of characterizing the performance of these routing schemes, the performance trade-off.

16. IMMUNE Recovery S t0 2 2 3 3 4 4 5 5 D D TD 4 4 7 8 8
Time

17. VACCINE Recovery S t0 2 2 3 3 4 4 5 5 D D TD 4 4 7 7 8 8
Time 2 2

18. 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 D 3 N
I will now talk about the network model we studied.
We assume there are N+1 mobile nodes moving according to common random waypoint/direction mobility
Model with a terrain of area A.
The mobile nodes are equipped with wireless device with fixed transmission range, r, that is small compared
to the area A.
Furthermore, we assume that when two nodes meet, they can exchange all packets that they need to instantaneously.
We also assume nodes have unlimited buffer first, and discuss how the model is be extended to limited buffer case later.
Given the network model I just described, the delay of a packet is then just the summation of
Times that it spends in each hop relays waiting for next contact.

20. Standard random mobility models
Random Waypoint model (RWP)
Random Direction model (RD) X2 V2
T2, V2 X1 α2 V1
T1, V1 R α1 R
Directions (αi) are uniformly distributed (0, 2π)
Speeds (Vi) are uniformly distributed (Vmin,Vmax)
Travel times (Ti) are exponentially /generally distributed
Next positions (Xi)s are uniformly distributed
Speeds (Vi)s are uniformly distributed (Vmin,Vmax)

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 recent work by Groevelt etc. partly address the problem of modeling the random contacts between nodes, and give support for Markovian model.
They shows the inter-meeting time between any pair of nodes is close to a Poisson process. Actually,
If the transmission range is small compared to the area, and the speed is relatively high,
Then such approximation is good. Also the following formula has been derived for calculating
of the meeting rate from system parameters.
The intuition behind this is: for a trip of a node, it meets with another node with certain prob. => a geometric distr. => exponential distr.
w: mobility specific constant

22. 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.

23. Why “almost”?
pairwise meeting processes are almost independent Poisson processes with rate:
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
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
moreover if Nr2 is comparable with A (dense network) a lot of meeting happen at the same time.
w: mobility specific constant
A recent work by Groevelt etc. partly address the problem of modeling the random contacts between nodes, and give support for Markovian model.
They shows the inter-meeting time between any pair of nodes is close to a Poisson process. Actually,
If the transmission range is small compared to the area, and the speed is relatively high,
Then such approximation is good. Also the following formula has been derived for calculating
of the meeting rate from system parameters.
The intuition behind this is: for a trip of a node, it meets with another node with certain prob. => a geometric distr. => exponential distr.

24. 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)

25. Pairwise Inter-meeting time

26. Pairwise Inter-meeting time

27. Pairwise Inter-meeting time

28. 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

29. The derivation of λ Probability that nodes meet given by
For small r the points in π(x,y) in A can be
approximated by π(x1,y1) to give
Unconditioning on (x1,y1) gives

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
Here E[V*] is the average relative speed between two nodes and is the pdf in the point (x,y). 9

31. If speeds of the nodes are constant and equal to v,
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 ≈ is the Waypoint constant.
If speeds of the nodes are constant and equal to v,
then

32. 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. 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.

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

36. Message delay
Proposition: The Laplace transform of the message delay under epidemic routing is:
and

37. Expected message delay
Corollary: The expected message delay under the
two-hop multicopy protocol is
and under the epidemic routing is
Where γ ≈ is Euler’s constant.

38. Relative performance and Note that these are independent of λ!
The relative performance of the two relay protocols:
and
Note that these are independent of λ!

39. 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 first-meeting times.
λ depends on: - mobility pattern - surface area - transmission radius

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

47. 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 1 2 3
N-2 N
N-1 D
Infection rate:
Delivery rate:
Now, with this result, in this same work, Markov Chain models are used to study epidemic routing and
2-hop routing. Here I show the model for epidemic case.
Given a network of N+1 nodes, where pair-wide meeting process follows i.i.d. Poisson process with
Rate lamba, the MC model has N states each corresponding to the state where there is I infected node and
The receiver has not received the pkt yet; the state “A” denotes the state where the packet has been delivered.
The initial state is 1. Starting from this state, the new infection rate is given by (n-1)\lambda; and then at two infected
Node state, the infection rate is 2(N-2)\lambda, etc. At any state, the rate of entering “A” state is just the rate
That any infected node meets the destination, given by Ni*lambda.
We know that the delivery delay is then the time to reach state A, and the copies made
At the time of delivery is then the state from which A is reached.
Actually, a Laplace transform of delay is derived in the paper, from which an asymptotic average delay is then found.
The drawback is the derivation is difficult (2-hop case deriviation is already quite complex), even so for extended models.

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
To model epidemic routing, one need to first model the random contacts between nodes, which is a
Hard problem.
The following two previous works adopted a hybrid approach, using models with parameters obtained through simulations.
Small and her colleages in their 2003 paper applied ODE model, commonly used in epidemiology studies to study
Delivery delay under epidemic routing. In particular, for I(t), the num. of infected nodes in the network at time t, the following ODE is given
With the intuition that the increasing rate of infected node equals to the meeting rate between infected nodes, and susceptible nodes. Here lambda, the average pair-wise meeting rate between nodes is obtained from simulations.
Attributing modeling errors observed from the above model to the usage of single parameter to capture meeting rate, they later adopt a discrete markov chain model. They use N-1 parameters obtained from simulation to capture the mobility models.
They use numerical solution to solve the model, which has a complexity increasing with N. Also it’s hard to obtain insights
Into the schemes from numerical results.

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:

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 E
and a solution of the differential equation
such that x(s) is in E for 0<=s<=t, then for each 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

54. 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. 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:
Average delay
Avg. num. of copies sent at delivery
prob. that pkt is not delivered yet
delivery rate at time t
infected nodes-dst meeting rate
The ODE can be solved explicitly to get I(t). Let’s now see how to derive delivery delay from this result.
Delivery delay Td is defined as the time from …. For the CDF of Td, we can derive the following ODE:
The intuitive explanation is that the infestiminal rate of delivery at time t is given by the rate that one infected node meets destination, \lambda I(t) , times the prob. that the packet has not been delivered yet.
P(t) can also be explicitly solved.
With P(t), the average delay can be obtained by integrating (1-P(t)) from zero to infinity.
The num. of copies sent at delivery is the number of infected node before delivery time. The following equation follows.

56. What can we do with the fluid model?
Consider recovery process, eg IMMUNE (dest. node cures infected node):
Total num. of copies made:
Total buffer usage
R(t): num. of recovered nodes
Num. of susceptible nodes
To calculate the total number of copies sent and average storage consumption, we need to consider
the recovery process that starts after the first copy is delivered.
For example, for IMMUNE recovery where infected node delete the copy and store anti-packet on
Meeting destination, the following ODEs can be derived. We introduced R(t), which is the average num. of
Recovered nodes at time t.
From the ODE, the total num. of copies made is then given by ; and the total amount of buffer
taken by a packet during its life time is the intergral of I(t) ….

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,...)

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
We have perform model validation through simulation. Here we show the model predicted
delay versus simulation result, for the basic epidemic routing.
From the smaller figure, which plots for varying node number, the average delay in log scale. We can see
That the ODE model provides good estimation of the delay (trend), with slight under estimation.
Now let’s look at the predicted delay distribution for N=160 case. We observe that the model overpredict the distribution.
A Poisson simulation result is also obtained, and is very close to the simulation result. This suggests that the Poisson meeting
Process is a good approximition. The modeling error comes from the small number of infected node. We have also used a moment
Closure technique to obtain ODE of higher moments, and solved it numeraically. From the figure, we see that the extended ODE
Predicts the delay much better.
ODE provides good prediction on average delay

60. Modeling error mainly due to approx. of ODE
Delay distribution
CDF of delay under epidemic routing, N=160
We have perform model validation through simulation. Here we show the model predicted
delay versus simulation result, for the basic epidemic routing.
From the smaller figure, which plots for varying node number, the average delay in log scale. We can see
That the ODE model provides good estimation of the delay (trend), with slight under estimation.
Now let’s look at the predicted delay distribution for N=160 case. We observe that the model overpredict the distribution.
A Poisson simulation result is also obtained, and is very close to the simulation result. This suggests that the Poisson meeting
Process is a good approximition. The modeling error comes from the small number of infected node. We have also used a moment
Closure technique to obtain ODE of higher moments, and solved it numeraically. From the figure, we see that the extended ODE
Predicts the delay much better.
Modeling error mainly due to approx. of ODE

61. Matching results from Markov chain model, obtained easier
Some results
Extensible to other schemes
Epidemic
routing ~
2-hop
forwarding ~ ~
ODEs can be derived to model various other schemes. For some of them, we are able to derive close-form
results. We have also couplted the ODE model for packet propagation with a nodal buffer model to
consider the case where nodal buffer is constrained.
Prob.
forwarding
Matching results from Markov chain model, obtained easier

62. An application: Tradeoffs evaluation
Delay vs Power Delay vs Buffer
src/dest transmission
epidemic
routing
2-hop
2-hop
3-hop
3-hop

63. Other issues Not considered in this presentation
Effect of different buffer management techniques when the buffer is limited
ODEs by moment closure technique
I will now talk about the network model we studied.
We assume there are N+1 mobile nodes moving according to common random waypoint/direction mobility
Model with a terrain of area A.
The mobile nodes are equipped with wireless device with fixed transmission range, r, that is small compared
to the area A.
Furthermore, we assume that when two nodes meet, they can exchange all packets that they need to instantenously.
We also assume nodes have unlimited buffer first, and discuss how the model is be extended to limited buffer case later.
Given the network model I just described, the delay of a packet is then just the summation of
Times that it spends in each hop relays waiting for next contact.

64. References Papers discussed Other Markovian models Fluid 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
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