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Performance Evaluation of Epidemic Routing: Markovian and Mean Field Models

This lecture by Giovanni Neglia from INRIA explores the performance evaluation of epidemic routing systems using Markovian and mean field models. It covers the limitations of Markovian approaches, introduces mean field models, and discusses various applications, including Bianchi’s model. Key topics include the modeling of inter-meeting times in mobile networks, the behavior of message delays in ad-hoc scenarios, and analytical methods for deriving exact results and approximations. The lecture also references notable works and research pertaining to these models across communication systems.

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Performance Evaluation of Epidemic Routing: Markovian and Mean Field Models

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  1. Performance Evaluation Lecture 2: Epidemics Giovanni Neglia INRIA – EPI Maestro 9 January 2014

  2. Outline • Limit of Markovian models • Mean Field (or Fluid) models • exact results • extensions • Applications • Bianchi’s model • Epidemic routing

  3. Mean fluid for Epidemic routing (and similar) • Approximation: pairwise intermeeting times modeled as independent exponential random variables • Markov models for epidemic routing • Mean Fluid Models

  4. Inter-meeting times under random mobility (from Lucile Sassatelli’s course) • Inter-meeting times mobile/mobile have been shown to follow an exponential distribution • [Groenevelt et al.: The message delay in mobile ad hoc networks. Performance Evalation, 2005] • Pr{ X = x } = μ exp(- μx) • CDF: Pr{ X ≤ x } =1 -exp(- μx), CCDF: Pr{ X > x } =exp(- μx) MAESTRO M. Ibrahim, Routing and Performance Evaluation of Disruption Tolerant Networks, PhD defense, UNS 2008

  5. Pairwise Inter-meeting time

  6. Pairwise Inter-meeting time

  7. Pairwise Inter-meeting time

  8. 2-hop routing Model the number of occurrences of the message as an absorbing Continuous Time Markov Chain (C-MC): • 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.

  9. Epidemic routing Model the number of occurrences of the message as an absorbing C-MC: • 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.

  10. (from Lucile Sassatelli’s course) T. G. Kurtz, Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes, Journal of Applied Probability, vol. 7, no. 1, pp. 49–58, 1970. M. Benaïm and J.-Y. Le Boudec, A class of mean field interaction models for computer and communication systems, Performance Evaluation, vol. 65, no. 11-12, pp. 823–838, 2008.

  11. (from Lucile Sassatelli’s course) T. G. Kurtz, Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes, Journal of Applied Probability, vol. 7, no. 1, pp. 49–58, 1970. M. Benaïm and J.-Y. Le Boudec, A class of mean field interaction models for computer and communication systems, Performance Evaluation, vol. 65, no. 11-12, pp. 823–838, 2008.

  12. (from Lucile Sassatelli’s course) T. G. Kurtz, Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes, Journal of Applied Probability, vol. 7, no. 1, pp. 49–58, 1970. M. Benaïm and J.-Y. Le Boudec, A class of mean field interaction models for computer and communication systems, Performance Evaluation, vol. 65, no. 11-12, pp. 823–838, 2008.

  13. (from Lucile Sassatelli’s course) T. G. Kurtz, Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes, Journal of Applied Probability, vol. 7, no. 1, pp. 49–58, 1970. M. Benaïm and J.-Y. Le Boudec, A class of mean field interaction models for computer and communication systems, Performance Evaluation, vol. 65, no. 11-12, pp. 823–838, 2008.

  14. (from Lucile Sassatelli’s course) Epidemic Two-hop MAESTRO Zhang, X., Neglia, G., Kurose, J., Towsley, D.: Performance Modeling of Epidemic Routing. Computer Networks 51, 2867–2891 (2007)

  15. A further issue Model the number of occurrences of the message as an absorbing Continuous Time Markov Chain (C-MC): • We need a different convergence result [Kurtz70] Solution of ordinarydifferentialequations as limits of pure jump markovprocesses, T. G. Kurtz, Journal of AppliedProbabilities, pages 49-58, 1970

  16. [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 RmxZm 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:

  17. The formal result [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

  18. Application to epidemic routing • rN(nI)=λnI (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

  19. Application to epidemic routing , The solution is And for the Markov system we expect

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