Stochastic Equivalence of CPDP-Automata and Piecewise Deterministic Markov Processes

Communicating Piecewise Deterministic Markov Processes

“…A CPDP automaton can be seen as a PDP type process enhanced with interaction/communication possibilities (see [14] for the relation between PDPs and CPDPs). Also, CPDPs can be seen as a generalization of Interactive Markov Chains (IMCs, see [8]).…”

Stochastic semantics for Communicating Piecewise Deterministic Markov Processes

“…It is known that the superposition of two (or more) Poisson processes is again a Poisson process (see, in the context of CPDP, [8] for a proof of this result). This means that if we combine all spontaneous transitions ofX n with origin location l to one spontaneous transition (l, λ l , L,R tot,l ), with λ l (x) = ∑ and if we replace all spontaneous transitions by these combined spontaneous transitions, then the stochastic behavior (concerning the evolution of the state) will not change.…”

“…Proof: From the text above and from the results of [8], it is clear that if R(E, x) = 1 for all x, then the PDP suggested by the theorem models the state evolution of X. If for some x ∈ E, R(E, x) < 1, then it can be seen that this must mean that there exists a hybrid jump with multiplicity infinity such that the probability of this hybrid jump at x is greater than zero.…”

“…Corollary 4.3: If for some n ∈ N we have that T n u = / 0, then the multiplicity of the hybrid jumps of X is bounded by n and the state of X exhibits a PDP behavior, with the same PDP as the corresponding PDP of X n (which can be constructed according to [8] because all hybrid jumps of X n have multiplicity one).…”

“…Also, the theorem says that if the state evolution can be modelled by a stochastic process, then it can be modelled by a stochastic process from the class of PDPs. The proof of the theorem makes use of the results from [8].…”