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Gossiping with IOIMCs

Gossiping with IOIMCs. Pepijn Crou zen Saarland University. Gossiping models: the basics. Networks consist of simple nodes. Broadcasts are forwarded to a (small) number of neighbors. A node does not have to know the entire network.

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Gossiping with IOIMCs

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  1. Gossiping with IOIMCs Pepijn Crouzen Saarland University

  2. Gossiping models: the basics • Networks consist of simple nodes. • Broadcasts are forwarded to a (small) number of neighbors. • A node does not have to know the entire network. • A node does not have to know who has received which messages.

  3. What did we model? • Constant, but arbitrary, number of nodes, • Constant, but arbitrary, interconnections, • Multiple messages from multiple sources, • Individual message reception, • Delayed, probabilistic message forwarding, • Resulting model: labeled CTMC, • Scalable model generation with CADP. • Goal: stochastic validation on message reception times. Focus: Scalable model generation + Information spread

  4. What did we leave out? • Dynamics: • New nodes appearing, • Nodes dying, • Interconnections changing. • Message buffers, • Message content.

  5. How did we model gossiping? • Using Input/Output Interactive Markov Chains: • Each node is modeled by an I/O-IMC, • Messages are sent through output signals and received through input signals. • New messages are received through system-inputs and message reception is signaled using system-outputs. • Network model is constructed through composition of the node models.

  6. M(A)! p.λ M(B)? REC(A)! M(C)? (1-p).λ Simple node model B A • Waiting rate = λ, • Sending probability = p, • Messages are identified by sending node, • While waiting to send, incoming messages are ignored • Node also waits when not sending! C START(A)?

  7. Scalability: Adding links rename ADD2(A) -> ADD(A) in hide ADD(A) in M(A)! ADD(A)! p.λ M(B)? |[ADD(A)]| M(C)? REC(A)! ADD(A)? ADD2(A)? (1-p).λ

  8. Scalability: Adding links, result M(A)! • Now we can generate any gossiping network using: • Composition • Abstraction • Minimization • Renaming • On: • Node model (0 links) • Add-neighbor model p.λ M(B)? M(C)? REC(A)! ADD(A)? (1-p).λ

  9. Case study • 15 node network, • Each node has 3 neighbors, • Convert each node to an I/O-IMC, • Compute the total network model using compositional aggregation, • Compose the network model with a message generation model and a message reception models, • Compute probability that an incoming message reaches all nodes after some time period using resulting labeled CTMC.

  10. REC(NODE2)? START(NODE1)! Node 2 has receivedthe message Message generation and reception Network START signals REC signals x15 • Hide the START and REC signals, • Weak bisimulation minimization • Labeled CTMC

  11. Case study: results • Generation time:+/- 2 hours • Largest appearing model: 223743 states, 1241054 transitions • CTMC size (anonymous reception):233 states • Analysis time: <1 second • And now the probability that all nodes receive a message with send-rate 0.01 and send-probability 70%

  12. Conclusion • Scalable complete state space generation for gossiping networks is possible using very simple base models, but: • We run into the state space explosion fairly early, • Advanced maximal progress cutting is needed to make it feasible, • No dynamics!

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