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Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Rate-Based Stochastic Fusion Calculus and Angelo Troina Continuous Time Markov Chains Fusion Calculus Features Syntax and Semantics Continuous Time

  1. Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Rate-Based Stochastic Fusion Calculus and Angelo Troina Continuous Time Markov Chains Fusion Calculus Features Syntax and Semantics Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Definition Simulating CTMCs Markovian Bisimulation Stochastic Fusion ICTCS’12, September 21, 2012, Varese, Italy Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions

  2. Rate-Based Features of Fusion Calculus Stochastic Fusion Calculus and Continuous Time Markov Chains A global fusion relation modelling global shared state. 1 Gabriel Ciobanu, Angelo Troina Fusion Calculus Features π -Calculus Fusion Calculus Syntax and Semantics Communication Local Local or global Continuous Time Markov Chains Effect Definition Binding Input and Restriction Only one binding operator Simulating CTMCs Markovian Operators Input always binds Neither Input or output Bisimulation Output sometimes binds are binding Stochastic Fusion Calculus Bisimulation Early Only one Syntax and Semantics Process Equivalence Congruence Late congruence: in SFC Open From SFC Processes to CTMCs Hyperequivalence The AM Symbiosis Which one is “best”? Biological System Formal Modelling Simulation Results Conclusions 1 J. Parrow, B. Victor. The Fusion Calculus: Expressiveness and Symmetry in Mobile Processes. Proc. LICS, IEEE Computer Society, 176-185, 1998.

  3. Rate-Based Fusion in Action Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus Features Syntax and Semantics Continuous Time Markov Chains Definition Simulating CTMCs Markovian Bisimulation Stochastic Fusion Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions

  4. Rate-Based Syntax Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus Features Syntax and Semantics Continuous Time Markov Chains Definition Simulating CTMCs Markovian Bisimulation Stochastic Fusion Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions

  5. Rate-Based Semantics Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus Features Syntax and Semantics Continuous Time Markov Chains Definition Simulating CTMCs Markovian Bisimulation Stochastic Fusion Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions

  6. Rate-Based Delayed Input Stochastic Fusion Calculus and Continuous Time Markov Chains In π extended with delayed input, u ( x ) : P stands for a non Gabriel Ciobanu, stopped process P which can perform input on channel u at Angelo Troina any time. Fusion Calculus Features Syntax and Semantics Continuous Time In fusion calculus, delayed input is given by the process: Markov Chains Definition Simulating CTMCs Markovian ( x )( ax | P ) Bisimulation Stochastic Fusion Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions ( x )( c r x | P cell | [ x = s ] P ActPath )

  7. Rate-Based Continuous Time Markov Chains Stochastic Fusion Calculus and Continuous Time Markov Chains A Continuous Time Markov Chain (CTMC) is a triple Gabriel Ciobanu, Angelo Troina � S , T , s 0 � , where: ◮ S is the set of states, Fusion Calculus Features R ≥ 0 is the transition function, ◮ T : S × S �→ I Syntax and Semantics Continuous Time ◮ s 0 ∈ S is the initial state. Markov Chains Definition The system passes from a state s to a state s ′ by consuming Simulating CTMCs Markovian Bisimulation an exponentially distributed quantity of time, in which the Stochastic Fusion parameter of the exponential distribution is T( s , s ′ ). The Calculus cumulative distribution is T( s , C ) = � Syntax and Semantics s ′ ∈ C T( s , s ′ ) Process Equivalence in SFC From SFC Processes to CTMCs Many analysis techniques and tools are available for CTMCs, The AM Symbiosis e.g. the PRISM model checker. 2 Biological System Formal Modelling Simulation Results Conclusions 2 M. Kwiatkowska, G. Norman, D. Parker. Probabilistic Symbolic Model Checking with PRISM: A Hybrid Approach. Int. J. on Soft. Tools for Technology Transfer, vol.6, 128142, 2004.

  8. Rate-Based CTMC: Simulation Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus Features Syntax and Semantics Continuous Time Markov Chains Definition Simulating CTMCs Markovian Bisimulation Stochastic Fusion Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions

  9. Rate-Based CTMC: Simulation Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus Features Syntax and Semantics Continuous Time Markov Chains Definition Simulating CTMCs Markovian Bisimulation Stochastic Fusion Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions

  10. Rate-Based CTMC: Simulation Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus Features Syntax and Semantics Continuous Time Markov Chains Definition Simulating CTMCs Markovian Bisimulation Stochastic Fusion Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions

  11. Rate-Based CTMC: Simulation Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus Features Syntax and Semantics Continuous Time Markov Chains Definition Simulating CTMCs Markovian Bisimulation Stochastic Fusion Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions

  12. Rate-Based CTMCs: Race Condition Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus Features Syntax and Semantics Continuous Time Markov Chains Definition Simulating CTMCs Markovian Bisimulation Stochastic Fusion Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions

  13. Rate-Based CTMCs: Race Condition Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus Features Syntax and Semantics Continuous Time Markov Chains Definition Simulating CTMCs Markovian Bisimulation Stochastic Fusion Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions

  14. Rate-Based CTMCs: Race Condition Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus Features Syntax and Semantics Continuous Time Markov Chains Definition Simulating CTMCs Markovian Bisimulation Stochastic Fusion Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions

  15. Rate-Based CTMCs: Race Condition Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus Features Syntax and Semantics Continuous Time Markov Chains Definition Simulating CTMCs Markovian Bisimulation Stochastic Fusion Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions

  16. Rate-Based CTMCs: Race Condition Stochastic Fusion Calculus and Continuous Time Markov Chains Gabriel Ciobanu, Angelo Troina Fusion Calculus Features Syntax and Semantics Continuous Time Markov Chains Definition Simulating CTMCs Markovian Bisimulation Stochastic Fusion Calculus Syntax and Semantics Process Equivalence in SFC From SFC Processes to CTMCs The AM Symbiosis Biological System Formal Modelling Simulation Results Conclusions with duration ti= min { t1, t2, t3 } .

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