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-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

Intermittently Connected Networks V2 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.

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

Terminology and Focus Why delay tolerant? Why disruption 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? 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 emails, 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.

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.

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

Epidemic Routing Message as a disease, carried around and transmitted 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… 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.

An example: Epidemic routing in ZebraNet

An example: Epidemic routing in ZebraNet

Standard Epidemic Routing 2 2 t2 3 3 4 4 5 5 D D TD Time

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.

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

Limited Time Forwarding S t0 2 2 3 3 4 4 5 5 3 D Time

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.

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

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

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 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.

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)

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

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.

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.

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)

Pairwise Inter-meeting time

Pairwise Inter-meeting time

Pairwise Inter-meeting time

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

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

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

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 ≈ 1.3683 is the Waypoint constant. If speeds of the nodes are constant and equal to v, then

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.

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.

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

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

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.

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!

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

Example: two-hop multicopy

Example: two-hop multicopy

Example: two-hop multicopy

Example: two-hop multicopy Distribution of the number of copies (R=50,100,250m):

Example: two-hop multicopy Distribution of the number of copies (R=50,100,250m):

Example: two-hop multicopy Distribution of the number of copies (R=50,100,250m):

Example: unrestricted multicopy

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 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.

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.

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

[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:

[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

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

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: 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.

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) ….

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,...)

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

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

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

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

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

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.

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 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