DEPENDABLE SYSTEMS AND SOFTWARE
Prof. Dr.-Ing. Holger Hermanns
Dr. Lei Song
M.Sc. Hassan Hatefi
4. Exercise for Quantitative Model Checking (WS 13/14)
http://depend.cs.uni-sb.de/index.php?570
Exercise 4.1
Consider the DTMC below:
Let A = {s3 } and B = {s2 }.
a)
Compute the probability, from each state of the Markov chain, of reaching a state in A
within 4 steps.
b)
Compute the probability, from each state of the Markov chain, of reaching a state in A.
c)
Compute the probability, from the initial state, of reaching the set of states A ∪ B.
d)
Compute the probability, from the initial state, that a state from A ∪ B is visited
infinitely often.
Exercise 4.2
Consider a finite DTMC (S, P, π(0), L) and subsets of states A, B ⊆ S. Show that the
following two sets of paths are measurable, i.e. contained in the σ-algebra F:
a)
the set of paths starting in state s ∈ S and remaining forever in states from A;
b)
the set of paths starting in state s ∈ S, remaining forever in states from A and passing
through a state in B after exactly 5 time-steps.
Hint: For the first part, show the complement of the event is in the σ-algebra. For the second
part, express the event as an intersection of two measurable events.
Exercise 4.3
Consider the DTMC below. Illustrate the execution of the PCTL model checking algorithms
to determine which states of the Markov chain satisfy:
a)
P≥ 17 ( b U c )
19
b)
P≥ 1 [ X P> 1 [ (b ∨ c) U ≤2 (b ∧ c) ] ]
2
3
Exercise 4.4
Show that ∼ := ∪R is bisimulation R is an equivalence relation.