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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 2rE[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 8rv 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 i1 j1 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 i1 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 Prsup X N (s) X(s) 0 N N 0st 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 1Nt ( N 1) e I (t ) t N 1 ( N 1)e pNt P(t ) 1 ( N )1 / p pNt 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