Abstract
In this paper, we study the model of stochastic timed automata and we target the definition of adequate composition operators that will allow a compositional approach to the design of stochastic systems with hard real-time constraints. This paper achieves the first step towards that goal. Firstly, we define a parallel composition operator that (we prove) corresponds to the interleaving semantics for that model; we give conditions over probability distributions, which ensure that the operator is well-defined; and we exhibit problematic behaviours when this condition is not satisfied. We furthermore identify a large and natural subclass which is closed under parallel composition. Secondly, we define a bisimulation notion which naturally extends that for continuous-time Markov chains. Finally, we importantly show that the defined bisimulation is a congruence w.r.t. the parallel composition, which is an expected property for a proper modular approach to system design.
The first and the third authors are supported by ERC project EQualIS. The second author is partly supported by FP7-EU project Cassting. The fourth author was a postdoctoral researcher at the Belgian National Fund for Scientific Research (FNRS).
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Notes
- 1.
- 2.
Somehow, the clock behaviour in GSMPs and in
-STA is that of countdown timers (which can be seen as event-predicting clocks of [3]), which is not as rich as general clocks in standard timed automata. - 3.
We restrict to open guards for technical reasons due to stochastic aspects.
- 4.
Two measures \(\mu \) and \(\nu \) on the same measurable space are equivalent whenever for every measurable set A, \(\mu (A)>0\) iff \(\nu (A)>0\).
- 5.
We recall that a run
is Zeno if \(\sum _{i\ge 1}\tau _i<+\infty \). - 6.
A STA is said almost-surely fair whenever \(\mathbb {P} _{\mathcal {A}}(\textsf {fair})=1\), where a run is fair if and only if (roughly speaking) any edge enabled infinitely often is taken infinitely often.
-STA is that of countdown timers (which can be seen as event-predicting clocks of [
is Zeno if References
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Bouyer, P., Brihaye, T., Carlier, P., Menet, Q. (2016). Compositional Design of Stochastic Timed Automata . In: Kulikov, A., Woeginger, G. (eds) Computer Science – Theory and Applications. CSR 2016. Lecture Notes in Computer Science(), vol 9691. Springer, Cham. https://doi.org/10.1007/978-3-319-34171-2_9
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