Probabilistic Performance Analysis of Fault Diagnosis Schemes

Main Results
The following theorems relate the tractability of Markov chains to easily-verifiable properties
of directed graphs.
Theorem 4.11. Given a non-degenerate, time-homogeneous Markov chain θ ∼ (Θ,Π,π 0 ),
define the matrix A as follows:
⎧
⎨1 if i ≠ j, Π(i , j ) ≠ 0,
A(i , j ) :=
⎩0 otherwise,
for all i , j ∈ Θ. Then, the Markov chain θ is tractable if and only if the directed graph with
vertices Θ and adjacency matrix A is acyclic.
(4.4)
Theorem 4.12. Given a non-degenerate Markov chain θ ∼ ( Θ,{Π k },π 0
)
with time-varying
transition probabilities, define the matrix A as follows:
⎧
⎨1 if i ≠ j, Π k (i , j ) ≠ 0 for some k ≥ 0
A(i , j ) :=
⎩0 otherwise,
for all i , j ∈ Θ. Then, the Markov chain θ is tractable if the directed graph with vertices Θ and
adjacency matrix A is acyclic.
(4.5)
Remark 4.13. Note that Theorem 4.11 gives a necessary and sufficient condition for tractability,
while Theorem 4.12 only gives a sufficient condition. Indeed, Example 4.18 (below) shows
that the graph-theoretic condition stated in Theorem 4.12 is not necessary for tractability.
Remark 4.14. The presence of cycles in a directed graph G = (V,E) can be determined
using the Depth-First Search (dfs) algorithm in O(|V | + |E|) time, where V is the set of
vertices and E is the set of edges [19, 21]. For the graphs considered in Theorems 4.11
and 4.12, the number of vertices is |Θ| = m + 1, and the number of edges is no more than
(m + 1) 2 − (m + 1) = m 2 +m, since the diagonal entries of A must be 0. Hence, the tractability
of a given Markov chain can be verified using dfs in O(m 2 ) time.
Example 4.15. Suppose that Θ = {0,1} and
[ ]
1 0
Π =
,
1 − p p
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