Cloud Computing: "network access to shared pool of configurable computing resources"

Transcription

1 Large-Scale Networks Gossip-based Protocols Consistency for Large-Scale Data Replication Uwe Roehm / Vincent Gramoli School of Information Technologies The University of Sydney Page 1 Cloud Computing: "network access to shared pool of configurable computing resources" Source: Jamie's Cloud Based Web Server on an Amazon EC2 The University of Sydney Page 2

2 Shared-Pool of Computing Resources? Data Centers! The University of Sydney Page 3 And not just one data centre of the Big Three (approx., as of 2012) Links: The University of Sydney Page 4

3 Motivation Core requirements for Cloud Computing platforms Scalability Availability / Fail Safety But CAP Theorem: [Brewer, PODC2000] Consistency Availability Partitioning Tolerance Theorem: You can have at most two of these properties for any shared-data system. The University of Sydney Page 5 Motivation (cont d) Highly scalable, fail safe data management? Scalability: Data Partitioning Fail Safety: Data Replication Standard protocols in databases focus on determinism & consistency Example: 2-Phase Commit Protocol 4n messages blocking => this is neither fail safe, nor scalable How can we do better? Idea: Let s try to see what happens if some information/epidemic spreads with some uncertainty The University of Sydney Page 6

4 Introduction Epidemics Branching process SIR model SIS model Relation between SIS and SIR models Gossip-based protocols Slide deck by Vincent Gramoli The University of Sydney Page 7 Epidemics

5 Epidemics Biology meets sociology In extreme cases, a single disease outbreak can have a significant effect on a whole civilization The spread of diseases is impacted by: Properties of the pathogen carrying it (e.g., contagiousness, infectious period) Network structure within a population (e.g., who knows whom) The University of Sydney Page 9 Epidemics The contact network defines the interactions used by an epidemics There is a node for each person Edge between two persons if they come into contact with each other Travel patterns showed an impact on the spread of fast moving diseases 1. within a city [MPN05] or 2. via worldwide airline network [CBB06] The University of Sydney Page 10

6 Epidemics Ebola 2014 Crisis Air traffic among West African countries and the rest of the world The University of Sydney Page Epidemics There are clear connections between: 1. Epidemic diseases in a population 2. Diffusion of ideas in social networks They both spread from person to person There is however some uncertainty in these models as opposed to the previous model we considered about adopting technologies There is a lack of decision making in the transmission of the disease from one person to another In our new model, we will thus consider the person-to-person interaction as random The University of Sydney Page 12 1

7 Branching Process 1 Branching Process Let s model the spread of contagion in waves, one at each level of a tree The University of Sydney Page 14

8 Branching Process A simple model of contagion 1. First wave: A new person carrying a disease enter the population and transmits it to each person he meets independently with probability p. He meets k people while contagious; this k people are the first wave of epidemics. 2. Second wave: each person in the first wave meets k other people. Each infected person of the first wave passes the disease independently to each of the second-wave independently with probability p. 3. Subsequent waves: each person infected in the current wave passes the disease with probability p to k people she meets The University of Sydney Page 15 Branching Process A simple model of contagion 1. With high probability p the contagion spreads widely 2. With low probability p the contagion is likely to die out quickly The University of Sydney Page 16

9 Branching Process A simple model of contagion There are two possibilities in this branching process model: 1. Contagion dies out: contagion reaches a wave where it infects no one 2. Infinite contagion: continues to infect people in every wave The basic reproduction number R 0 is a simple condition to tell these two possibilities apart R 0 is the expected number of new (secondary) cases of the disease caused by a single (typical) infection In our model, this is R 0 = pk because k persons are infected with proba p Þ the outcome is determined by whether R 0 < 1. The University of Sydney Page 17 Branching Process A simple model of contagion Claim: If R 0 < 1, then with probability 1, the disease dies out after a finite number of waves. If R 0 > 1, then with probability greater than 0 the disease persists by infecting at least one person in each wave. The University of Sydney Page 18

10 SIR Model 1 SIR Model An individual node in the branching process model goes through three potential stages during the course of the epidemic: 1. Susceptible: before the node has caught the disease, it is susceptible to be infected by its neighbors 2. Infectious: once the node has caught the disease, it is infectious and has some probability of infecting each of its susceptible neighbors 3. Removed: after a particular node has experienced the full infectious period, it is removed from consideration, since it no longer poses a threat of future infection. The contact network is represented as a directed graph Edge <v,w> indicates that if v is infected, the disease can spread to w If either has the potential to infect the other, we can represent edges point both ways The University of Sydney Page 20

11 SIR Model Each node loops in a Susceptible (S), Infectious (I), Removed (R) cycle Initially, some nodes are in the I state and all others in the S state Each node v that enters the I state remains infectious for a fixed number of steps t I During each of these t I steps, v has probability p of passing the disease to each of its susceptible neighbors After t I steps, node v is in state R: it is no longer in I or S and can no longer catch or transmit the disease The University of Sydney Page 21 SIR Model Example Example of a SIR epidemic where shaded nodes with dark border are in I, shaded nodes with thin border are in R and the rest are in S. Each node remains in I for a number of steps t I =1 Initially, nodes y and z are infected Step 0: Step 1: The epidemic spreads to some but not all of the remaining nodes The University of Sydney Page 22

12 SIR Model Example (cont d) Example of a SIR epidemic where shaded nodes with dark border are in I, shaded nodes with thin border are in R and the rest are in S. Step 2: Step 3: The University of Sydney Page 23 SIR Model We can derive a more realistic model Some nodes are more likely to catch the disease This can be represented by a different probability p v,w depending on each edge <v,w> Each node may stay in the I state a random period This can be modeled by a probability of leaving the state I with probability q in each step spent in this state We could have disease with highly contagious incubation periods followed by a less contagious periods where symptoms appear This can be modeled with substate of state I (early, middle and later periods of I) The University of Sydney Page 24

13 SIS Model SIS Model We have considered a model where nodes get infected at most once However, a simple variation of this model allows us to reason about epidemics where nodes can be re-infected To this end, we have nodes that alternate between Susceptible (S) and Infectious (I) states, these models are SIS models. The University of Sydney Page 26

14 SIS Model Let us describe the SIS model as follows 1. Initially, some nodes are in the I state and all others are in the S state 2. Each node v that enters the I state remains infectious for a fixed number of steps t I 3. During each of these t I steps, v has a probability p of passing the disease to each of its susceptible neighbors 4. After the t I steps, node v is no longer infectious, and it returns to the S state The University of Sydney Page 27 SIS Model Example of an execution in the SIS model In each step, the nodes in the Infectious (I) state are shaded In the SIS model, nodes can be infected, recover and then be infected again The University of Sydney Page 28

15 Relationship between the SIS and the SIR Models SIS Model vs. SIR Model Differences of SIS and SIR models: In the SIR model, the epidemic spreads on a finite graph Burns through a bounded supply of nodes, since nodes can never be re-infected Comes to an end after a finite number of steps In the SIS model, the epidemic may die if all nodes are free from the disease In a finite graph, there will eventually (with probability 1) come a point in time where all contagion attempts simultaneously fail for t I steps in a row, and at this point the epidemics will be over A key question in the SIS model, is to know how long the epidemics will last and how many individuals will be affected at different points in time The University of Sydney Page 30

16 SIS model vs. SIR model For contact networks where the structure is mathematically tractable, some work investigated the time it takes for an epidemic in the SIS model to die out [Berger et al. SODA 05, Liggett Stochastic Interacting Sys. 99] For a particular value of the contagion probability p, a SIS epidemic on the network will undergo a rapid shift: from one that dies out quickly to one that persists for a very long time This type of analysis is quite complex and involves a probability p that depends on the network structure The University of Sydney Page 31 SIS model vs. SIR model We can represent some of the basic variants of the SIS model as special cases of the SIR model This shows the flexibility of epidemic models Consider the SIS model in which t I = 1 where each node is infectious during one step If we think about each node v as a different node in each time step, then we represent the model where nodes are never re-infected We create a separate copy of each node for each time step t = 1, 2 The University of Sydney Page 32

17 SIS model vs. SIR model We create a time-expanded contact network where For each edge <v,w> in the original contact network, we create edges in the time-expanded contact network from the copy of v at time t to the copy of w at time t+1 This encode the idea that w can catch the disease at time t+1 if v is infectious at time t The University of Sydney Page 33 SIS model vs. SIR model Applying the time-expanded contact network to this contact network: Leads to the following time-expanded contact network: The University of Sydney Page 34

18 SIS model vs. SIR model The idea is that the same SIS disease dynamics that previously circulated around the original contact network can now flow forward in time through the time-expanded contact network, with copies of nodes that are in the I state at time t producing new infections in copies of nodes at time t+1 But on this time-expanded graph we have a SIR process, since any copy of a node can be treated as removed (R) once its one time step of infection is over With this view of the process we have the same distribution of outcomes as the original SIS model The University of Sydney Page 35 SIS model vs. SIR model Hence the course of the SIS epidemic seen before: Can be represented as the following course of the SIR epidemic: The University of Sydney Page 36

19 Gossip-based Protocols Gossip-based Protocols: 1. Information Dissemination Also called epidemic protocols [DGH87]: "Epidemic Algorithms for Replicated Database Maintenance" History: Based on the theory of epidemics, studying the spreading of infectious diseases Opposite goal: trying to spread (information) as fast as possible Terminology: An infected node holds data that is willing to spread to others A susceptible node is a node that has not yet seen this data A removed node is no willing/able to spread its data An update is a version of a data (data can be updated and timestamped) Advantages: Propagates in O(log n) where n is the number of peers if peers are picked u.a.r. Tolerates peer failures and message losses The University of Sydney Page 38

20 Gossip-based Protocols: Anti-Entropy Anti-Entropy: p i picks p j at random, and exchanges updates with p j Three approaches of exchanging updates: 1. p i only pushes its own updates to p j 2. p i only pulls in new updates from p j 3. p i and p j send updates to each other (i.e., push-pull approach) Remarks: Push-based approaches are bad at spreading rapidly updates With pull-based, a susceptible will likely get the update from many infected nodes Even more efficient is the combination of the two: push-pull. The University of Sydney Page 39 Gossip-based Protocols: Rumour Mongering Rumor Mongering: send only important messages or abstain More specifically: 1. If node p i has just been updated for data item x, it contacts an arbitrary other node p j and tries to push the update to p j 2. If p j was already updated by another node, then p i may become removed with some probability 1/k Efficient at spreading news/updates Some nodes may not receive the news/updates and remain susceptible In large scale network, the fraction s of nodes that remain susceptible is: s = e -(k+1)(1-s) Typically, if k=4, ln(s) = -4.97, s < The University of Sydney Page 40

21 Gossip-based Protocols: Mixing with Anti-Entropy Rumor mongering spreads updates fast with low traffic however, non-zero probability of nodes remaining susceptible after the apidemic Antri-entropy can be run (infrequently) in the background to ensure all nodes eventually get the update with probability 1 since a single rumor is already known by most nodes, dies out quickly The University of Sydney Page 41 Gossip-based Protocols Principle General principle Each peer executes periodically: 1. p = selectrandompeer(); 2. peerstate = p.getstate(); 3. mystate = me.getstate(); 4. newstate = merge(mystate, peerstate); 5. p.putstate(newstate); 6. me.putstate(newstate); The University of Sydney Page 42

22 Gossip-based Protocols: 2. Overlay Networks Example: self-stabilizing protocols [JMB09] T-Man: gossip protocol that can build overlay networks from scratch: "what are my preferred neighbours?" Eventually nodes organize themselves into a torus overlay The University of Sydney Page 43 Gossip-based Protocols: 3. Membership Management Multi-layer Gossip [JVG07] Peers gossip to construct a communication overlay resistant to crashes They maintain a small view of neighbors to communicate with They update this view periodically with the one from their neighbors Peers may use this overlay to disseminate information by gossiping They can use this view to push and/or pull updates They may piggyback the view information into these messages The University of Sydney Page 44

23 Other Gossip-Based Protocol Usages Gossip-based Computation of Aggregate Information [KDG2003] Gossip-based failure detection and membership protocol [Gupta et al, ACM PoDC2001] used eg. Amazon Dynamo [DeCandia SOSP2007] and also Apache Cassandra to build the DHT The University of Sydney Page 45 References [MPN05] Meyers, Pourbohloul, Newman, Skowronski, Brunham. Network Theory and SARS: Predicting outbreak diversity. J. Theoretical Biology. 232:71-81, [CBB06] Colizza, Barrat, Barthelemy, Vespignani. The role of Airline transportation network in the prediction and predictability of global epidemics. Proc. Natl. Acad. Sci, USA. 103(7): , [DGH87] Demers, Greene, Hauser, Irish, Larson, Shenker, Sturgis, Swinehart, Terry. Epidemic algorithms for replicated database maintenance. PODC [JMB09] Jelasity, Montresor, Babaoglu. T-Man: Gossip-based fast overlay topology construction. J. Computer Networks, 53(13): , [JVG07] Mark Jelasity, Spyros Voulgaris, Rachid Guerraoui, Anne-Marie Kermarrec, Martin van Steen. Gossip-based peer sampling. ACM TOCS [KDG03] D. Kempe, A. Dobra and Johannes Gehrke. Gossip-based computation of aggregate information. In: 44 th IEEE Symposium on Foundations of Computer Science, The University of Sydney Page 46