Models for Epidemic Routing

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 2867-2891

3. 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 V2 V3 B V1 C A

4. Some examples Inter-planetary backbone DieselNet, bus network DakNet, Internet to rural area 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?

6. Epidemic Routing V2 V3 B V1 C A • Message as a disease, carried around and transmitted

7. Epidemic Routing V2 V3 B V1 C A • Message as a disease, carried around and transmitted • Store, Carry and Forward paradigm

8. Epidemic Routing V1 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…

9. An example:Epidemic routing in ZebraNet

10. An example:Epidemic routing in ZebraNet

11. Standard Epidemic Routing t1 S 2 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…

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

14. Limited Time Forwarding t0 S 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

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

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

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: 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 • 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. w: mobility specific constant

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. 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. 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 ≈ 0.57721 is Euler’s constant.

38. Relative performance 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. 1 2 3 N-2 N N-1 D 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:

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

50. Our contribution (1/3) • A unified ODE framework... • limiting process of Markov processes as N increases [Kurtz 1970]