Lecture Notes in Computer Science 3472

242 Verena Wolf
(a, 1
4 ) (a, 1
4 )
u1 u2 ⊥ u3
(stop, 1)
sP
(b, 2
9 )
(stop, 1
6 )
(b, 1
9 )
(stop, 1) (stop, 1)
Fig. 9.3. Removing nondeterminism in a probabilistic process.
9.5 Action-Labeled Continuous-Time Markov Chains
Action-labeled continuous-time Markov chains (aCTMCs) are the underlying
model of stochastic process algebras like TIPP, EMPA and PEPA [GHR93,
BG96, Hil96] and the action-labeled extension of continuous-time Markov chains.
The difference between aCTMCs and fully probabilistic processes (as presented
in Section 9.3) is that the process acts in continuous time and an exponentially
distributed residence time X is associated with each state. Transition probabilities
are replaced by rates and the process remains X time units in the corresponding
state and chooses a successor state with regard to the probabilities
implicitly given by the rates. Thus, aCTMCs model real-world systems with
time-dependence and randomness.
Definition 9.5. An action-labeled continuous-time Markov chain M is
given by a tuple (SM , −→M , sM )where
• SM is a countable set of states,
•−→M⊆SM × (Actτ × R>0) × SM is a transition relation and
• sM ∈ SM is an initial state.
Furthermore, for all s ∈ SM the exit rate
E(s) =
�
r
is finite.
s ′ ,a,r:
(s,a,r,s ′ )∈→ M
Let ACTMC denote the set of all action-labeled continuous-time Markov chains.
We write s (a,r)
−−−→M s ′ for (s, a, r, s ′ ) ∈→M . We interpret an aCTMC as follows:
Every element s (a,r)
−−−→M s ′ corresponds to an a-transition in the chain from
state s to state s ′ . Every transition has an associated stochastic delay with a
duration determined by r. Ifs has only one successor s ′ ,ana-transition to s ′ can
take place after a delay which lasts an exponentially distributed time period. We
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