Gossiping with IOIMCs

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.

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Simple node model
B
A

• Waiting rate ?,
• 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)?
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Scalability Adding links
rename ADD2(A) -gt ADD(A) in
hide ADD(A) in
M(A)!
ADD(A)!
p.?
M(B)?
ADD(A)
M(C)?
REC(A)!
ADD(A)?
ADD2(A)?
(1-p).?
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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.?
M(B)?
M(C)?
REC(A)!
ADD(A)?
(1-p).?
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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.

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Message generation and reception
Network
START signals
REC signals
x15

• Hide the START and REC signals,
• Weak bisimulation minimization
• Labeled CTMC

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Case study results

• Generation time /- 2 hours
• Largest appearing model
• 223743 states, 1241054 transitions
• CTMC size (anonymous reception) 233 states
• Analysis time lt1 second
• And now the probability that all nodes receive a
  message with send-rate 0.01 and send-probability
  70

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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!