Modelling Probabilistic Wireless Networks (Extended Abstract)
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Presented at the IFIP International Conference on Formal Techniques for Distributed Systems joint international conference 14th Formal Methods for Open Object-Based Distributed Systems 32nd Formal Techniques for Networked and Distributed Systems.
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Modelling Probabilistic Wireless Networks (Extended Abstract)
- 1. The Calculus Testing Networks Proof Techniques Applications Probabilistic Wireless Networks Andrea Cerone and Matthew Hennessy Foundations and Methods Groups Department of Statistics and Computer Science, Trinity College Dublin FMOODS/FORTE 2012 A. Cerone Modelling Probabilistic Wireless Networks
- 2. The Calculus Testing Networks Proof Techniques Applications Outline 1 The Calculus 2 Testing Networks 3 Proof Techniques 4 Applications A. Cerone Modelling Probabilistic Wireless Networks
- 3. The Calculus Testing Networks Proof Techniques Applications Assumptions Network Topology Stations geographically distributed Each station can communicate with its neighbours The topology is static Broadcast communication: A packet sent from a station can be detected by all its neighbours Broadcast is a non-blocking action Reliable Transmission: Transmission primitives modelled at the Datalink Layer (ISO/OSI Standard) Modulation Techniques assumed for modelling virtual channels A. Cerone Modelling Probabilistic Wireless Networks
- 4. The Calculus Testing Networks Proof Techniques Applications Wireless Networks Wireless Network: M = Γ £ M m o1 Γ: Connectivity Graph M : System Term n o2 M ::= 0 n S W1 | W2 Code at stations is probabilistic M = m Sm | n Sn A. Cerone Modelling Probabilistic Wireless Networks
- 5. The Calculus Testing Networks Proof Techniques Applications Well Formedness Internal nodes (running o1 code) Interface nodes (external environment) w m Internal nodes are aware of their neighbours The topology of the external o2 environment is not known A. Cerone Modelling Probabilistic Wireless Networks
- 6. The Calculus Testing Networks Proof Techniques Applications Well Formedness Internal nodes (running o1 code) Interface nodes (external environment) m Internal nodes are aware of their neighbours The topology of the external o2 environment is not known A. Cerone Modelling Probabilistic Wireless Networks
- 7. The Calculus Testing Networks Proof Techniques Applications Processes P, Q ::= P p ⊕ Q | S S, T ::= 0 Empty Process ω Success c! e .P Broadcast c?(x) .P Receive τ.P Internal Activity S+T Non Deterministic Choice if b then S else T Matching A(˜)x Process Definition Process Definition: A(˜) ⇐ S x A. Cerone Modelling Probabilistic Wireless Networks
- 8. The Calculus Testing Networks Proof Techniques Applications Operational Semantics o1 m o2 M = ΓM £ m τ.(c! v 0.81 ⊕ 0) τ M −→ ∆ − ∆ = 0.81 · ΓM £ m c! v + 0.19 · m 0 broadcast detected by o1 , o2 with probability 0.81 A. Cerone Modelling Probabilistic Wireless Networks
- 9. The Calculus Testing Networks Proof Techniques Applications Operational Semantics (2) m o1 n o2 N = ΓN £ m τ.(c! v 0.9 ⊕ 0) | n c?(x) .(c! x 0.9 ⊕ 0) Broadcast detected by o1 with probability 0.9 Broadcast detected by o1 , o2 with probability 0.81 A. Cerone Modelling Probabilistic Wireless Networks
- 10. The Calculus Testing Networks Proof Techniques Applications Question o1 m o1 m n o2 o2 m τ.(c! v 0.9 ⊕ 0) | m τ.(c! v 0.81 ⊕ 0) | n c?(x) .(c! x 0.9 ⊕ 0) Can you replace M with N in a larger network? Compositional reasoning necessary for an answer A. Cerone Modelling Probabilistic Wireless Networks
- 11. The Calculus Testing Networks Proof Techniques Applications Extending Networks Goal: Test a network M with T Definition: (ΓM £ M ) > (ΓT £ T ) = (ΓM ∪ ΓT ) £ (M | T ) Defined only if ΓT does not affect the nodes of M Properties: Interface preservation Well-formedness preservation Associative, Non commutative A. Cerone Modelling Probabilistic Wireless Networks
- 12. The Calculus Testing Networks Proof Techniques Applications Example o1 o1 o1 m > = m o2 o2 o2 The converse extension is not defined! A. Cerone Modelling Probabilistic Wireless Networks
- 13. The Calculus Testing Networks Proof Techniques Applications Testing Networks Idea: Use interface nodes to test the behaviour of a network ω used to denote success Computation step: internal or broadcast action A. Cerone Modelling Probabilistic Wireless Networks
- 14. The Calculus Testing Networks Proof Techniques Applications Behavioural Preorders M, N share the same interface M may N : M > T leads to success with probability p implies N > T leads to success with probability q ≥ p M must N : N > T leads to success with probability q implies M > T leads to success with probability p ≤ q Compositionality: M ∗ N implies (M > L) ∗ (N > L) A. Cerone Modelling Probabilistic Wireless Networks
- 15. The Calculus Testing Networks Proof Techniques Applications Broadcast Vs. Multicast o1 m o1 m n o2 o2 N = ΓN £ m c! v | n c! v M = ΓM £ m c! v Distinguishing M from N : M may N, N mayM o1 receives, then broadcasts w N must M, M must N o2 receives two values, then compares the latter with v A. Cerone Modelling Probabilistic Wireless Networks
- 16. The Calculus Testing Networks Proof Techniques Applications Extensional semantics Which activities can be observed by the external environment? τ M − → ∆ - internal activity − n.c?v M − − − ∆ - input performed by node n − −→ c!v£η M − − − ∆ - output detected by nodes in η − −→ α Lifting to distributions: ∆ − → Θ − A. Cerone Modelling Probabilistic Wireless Networks
- 17. The Calculus Testing Networks Proof Techniques Applications Weak Transitions τ -transitions not observed by the external environment Impossibility to distinguish broadcast from multicast τ ∆ |== Θ - Internal behaviour on the long run (infinite ⇒ sequences allowed) n.c?v τ τ n.c?v ∆ |== = ⇒ Θ if ∆ |== − − − |== Θ == ⇒ − −→ ⇒ c!v£η τ τ c!v£η ∆ |== = ⇒ Θ if ∆ |== − − − |== Θ == ⇒ − −→ ⇒ c!v£η1 c!v£η2 c!v£(η1 ∪η2 ) ∆ |== = = |== = = Θ implies ∆ |== = = = ⇒ Θ, if = =⇒ = =⇒ ==== η1 ∩ η2 = ∅ A. Cerone Modelling Probabilistic Wireless Networks
- 18. The Calculus Testing Networks Proof Techniques Applications Simulation/Deadlock Simulation Simulation: ∆ ¡sim Θ α α ∆ |==⇒ i∈I pi · ∆i implies Θ |==⇒ i∈I pi · Θi and ∆i ¡sim Θi Deadlock Simulation:1 ∆ DS Θ τ τ ∆ |== ∆ and ∆ deadlocked implies Θ |== Θ and Θ ⇒ ⇒ deadlocked 1 Apologies for wrong definition in the paper A. Cerone Modelling Probabilistic Wireless Networks
- 19. The Calculus Testing Networks Proof Techniques Applications Results Assumption: M, N finite state, finite branching and not using ω Theorem 1: M ¡sim N implies M may N Theorem 2: M DS N implies M must N Remark: Simulation/Deadlock Simulation not complete (Broadcasts with probability < 1 cannot be matched by multicasts) A. Cerone Modelling Probabilistic Wireless Networks
- 20. The Calculus Testing Networks Proof Techniques Applications Applications In the paper: Probabilistic Sequential Routing ( may ) Other applications: Probabilistic Connectionless Routing Probabilistic Connection-oriented Routing Implementation at both Network and Transport layers Multicast Routing Virtual Shared Memory A. Cerone Modelling Probabilistic Wireless Networks
- 21. The Calculus Testing Networks Proof Techniques Applications Conclusions Contribution: Compositional theory for wireless networks Definition of behavioural preorders Development of sound proof techniques Applications to real world scenarios Future directions: Full abstraction for probabilistic networks More applications Introducing mobility A. Cerone Modelling Probabilistic Wireless Networks
- 22. The Calculus Testing Networks Proof Techniques Applications Thanks Thank you!!! A. Cerone Modelling Probabilistic Wireless Networks