DES - Definition of 'Stochastic Timed Automata'

Stochastic Timed Automaton:

A stochastic timed automaton is a 6-tuple $\left(\mathcal{E},\mathcal{X},\Gamma ,p,p_{x_0},F\right)$,where:

• $\mathcal{E}$ is a discrete set of events.
• $\mathcal{X}$ is a discrete set of states.
• For each $x\in \mathcal{X}$, $\Gamma (x)\subseteq \mathcal{E}$ is the set of events that are possible in state $x$.
• For each $x^{\prime },x\in \mathcal{X}$ and $e\in \Gamma$, $p(x^{\prime }\mid x,e)$ is the probability that the next state is $x^{\prime }$, given the current state $x$ and the next event $e$.

NOTE: When we talk about next event immagine always that at state 0: $x_0$ the next event to finish is event $e_1$, also that event $e_0$ does not exist.

Then to give a general rule: at state $x_3$ event $e_3$ has already occurred, and the next event to occur will be $e_4$.

• $F=\left\{F_e:e\in \mathcal{E}\right\}$ is a stochastic clock structure.

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~Example:

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Observation $\mathrm{I}$:

Timed Automata are a special case of Stochastic Timed Automata, where all probability $p(x^{\prime }\mid x,e)$ = 1, which means that all conditional probability are certain. And $p_{x_0}(x)=1⟺x=x_0$.

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Observation $\mathrm{II}$:

There could also exist mixed timed automata, where only the clock structure is stochastic ($f$, $x_0$, $F$), one where the transition function and the initial state are stochastic: ($p$, $p_{x_0}$, $V$), and all other possible combinations.

TERMINOLOGY $p$ is defined as probability mass function (pmf) > $p_{x_0}$ is defined as pmf of the initial state. > $p_{E_1}$ is defined as pmf of the first event. > $p_{X_5}$ is defined as pmf of the fifth state.