Probabilistic Performance Analysis of Fault Diagnosis Schemes

2.3 Reliability Theory
In this section, we present a select set of definitions and results from the vast field of
reliability theory. The purpose is to establish two useful probabilistic models for the failure
time of a system or component. For a thorough treatment of reliability theory, see Rausand
and Høyland [77] or Singpurwalla [85].
Let (Ω,F , P) be a probability space, and let τ: Ω → R + := [0,∞) be a random variable that
represents the time at which some system or component fails. As in the previous section,
let P τ and p τ denote the cumulative distribution function (cdf) and probability density
function (pdf) of τ, respectively.
Definition 2.2. The mean time to failure (mttf) of τ is defined as E(τ).
Definition 2.3. The failure rate is defined as the expected number of failures in some interval
of time, given that no failure has occurred yet. For ∆ > 0, the failure rate of τ at time t ≥ 0 is
P(t < τ ≤ t + ∆ | τ > t)
ρ ∆ (t) :=
∆
= P τ(t + ∆) − P τ (t)
∆ ( 1 − P τ (t) ) .
Definition 2.4. The hazard rate of τ at time t ≥ 0 is defined as
h(t) := lim ρ ∆ (t) =
p τ(t)
∆→0 1 − P τ (t) .
Suppose that, for a given sample time T s > 0, the failure time is modeled as a discretevalued
random variable κ: Ω → Z + := {0,1,...}. That is, for all k ∈ Z + , the event {κ = k}
indicates a failure at time kT s . In this case, the interval ∆ must be a multiple of the sample
time T s , so the hazard rate converges to
h(k) = ρ Ts (k) = P κ(k + 1) − P κ (k)
T s
(
1 − Pκ (k) ) .
However, there are cases where the discrete failure time κ does not have an underlying
sample time. In such cases, the hazard rate is defined as
h(k) = ρ 1 (k) = P κ(k + 1) − P κ (k)
.
1 − P κ (k)
For many physical systems, the graph of the hazard rate takes the shape of a “bathtub
curve”, shown in Figure 2.1 [77,85]. Initially, the system goes through a break-in phase where
failures are more likely. If the system survives the break-in phase, the hazard rate remains
roughly constant until the systems begins to wear out and failures become more likely again.
In modeling physical systems, it is common to assume that the break-in phase has already
11