performance evaluation
DESCRIPTION
Performance Evaluation. Lecture 2: Epidemics Giovanni Neglia INRIA – EPI Maestro 9 January 2014. Outline. Limit of Markovian models Mean Field (or Fluid) models exact results extensions Applications Bianchi ’ s model Epidemic routing. Mean fluid for Epidemic routing (and similar). - PowerPoint PPT PresentationTRANSCRIPT
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Performance EvaluationPerformance Evaluation
Lecture 2: Epidemics
Giovanni Neglia
INRIA – EPI Maestro
9 January 2014
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Outline
Limit of Markovian modelsMean Field (or Fluid) models
• exact results• extensions• Applications
- Bianchi’s model- Epidemic routing
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Mean fluid for Epidemic routing (and similar)1. Approximation: pairwise intermeeting
times modeled as independent exponential random variables
2. Markov models for epidemic routing3. Mean Fluid Models
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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)
M. Ibrahim, Routing and Performance Evaluation of Disruption Tolerant Networks, PhD defense, UNS 2008
MA
ESTR
O
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Pairwise Inter-meeting time
ttTP
etTP t
)(log
)(
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Pairwise Inter-meeting time
ttTP
etTP t
)(log
)(
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Pairwise Inter-meeting time
ttTP
etTP t
)(log
)(
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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.
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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.
Epidemic routing
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10T. 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.
(from Lucile Sassatelli’s course)
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11T. 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.
(from Lucile Sassatelli’s course)
![Page 12: Performance Evaluation](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813879550346895da02875/html5/thumbnails/12.jpg)
12T. 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.
(from Lucile Sassatelli’s course)
![Page 13: Performance Evaluation](https://reader036.vdocuments.net/reader036/viewer/2022062517/56813879550346895da02875/html5/thumbnails/13.jpg)
13T. 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.
(from Lucile Sassatelli’s course)
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14Zhang, X., Neglia, G., Kurose, J., Towsley, D.: Performance Modeling of Epidemic Routing. Computer Networks 51, 2867–2891 (2007)
Two-hopEpidemic
MAESTRO
(from Lucile Sassatelli’s course)
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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 ordinary differential equations as limits of pure jump markov processes, T. G. Kurtz, Journal of Applied Probabilities, pages 49-58, 1970
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[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 thatrN(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:
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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 Eand a solution of the differential equation
such that x(s) is in E for 0<=s<=t, then for each δ>0
)),(()(
sxFs
sx
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Application to epidemic routingrN(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
))(1)(()(' tititi
Nni IN
/)0(lim)0(
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Application to epidemic routing
,
The solution is
And for the Markov system we expect
))(1)(()(' tititi Nni IN
/)0(lim)0(