IEEE Instrumentation & Measurement - September 2023 - 10
black-box device) are not detected promptly, thus invalidating
the results quite before users may realize it.
◗ Other model-based statistical approaches: The statistical
distribution of some crucial metrological characteristics
of a class of measurement instruments as well as the probability
that such characteristics are no longer compliant
with the intended requirements can be also estimated if
suitable stochastic models are defined. This is for instance
the case of atomic clocks, whose phase noise is modelled
by a Wiener process with drift [16]. Once the stochastic
model is chosen, its parameters can be computed through
numerical fitting. Afterwards, the model can be used to
estimate the time after which a new calibration is recommended
with a known level of confidence. The main
drawback of purely statistical techniques is that large sets
of data of homogeneous instruments are needed to build
trustworthy stochastic models. However, usually this
approach is feasible only for metrological institutes or
large calibration laboratories.
As qualitatively shown in Table 1, no interval review policy
is superior in all respects to the others, as all of them exhibit
some advantages and disadvantages. Ultimately, the preferable
approach depends on the type of instrument as well as on
the application whereby the measuring equipment is used.
Furthermore, it should be noted that any adopted policy is also
affected by management and maintenance aspects.
The Simple Response Method (SRM)
Among the " staircase " methods for calibration interval
review, the most straightforward one to use is probably the socalled
Simple Response Method (SRM) [17]. When this policy
is adopted, the duration of metrological confirmation intervals
changes adaptively depending on the outcome of the last confirmation
only, i.e.,
T =
Ta
n Tb
n
n
−
−
1
1
1
1
( + )
()−
, if confirmation successful
,
if confirmation fails
, n > 1 (3)
where Tn denotes the duration of the n-th interval, a>0 and 0−() −
,
t
(4)
where α, β > 0 are the parameters of the distribution, and that
an " adjust always " policy is adopted anytime a calibration is
performed, it can be shown that the probability that an instrument
meets the metrological confirmation requirements after
the nth calibration (for n > 1) is approximately given by the following
recursive expression [17]:
()−1 b
Rnn
≅ R −1
β
1
1
+
−
a
b
→
n→∞
βRn−1
log
log
()−
−
+
1
1
1
b
b
a
It is worth noting the Weibull distribution is a very flexible
and general reliability model, which includes the classic
exponential distribution as a special case when β = 1. Moreover,
if β > 1 the failure rate grows with time, and for β = 2 this
is exactly linear. The relevance of (5) is twofold. First of all, the
steady-state asymptotic value of Rn
depends on the SRM parameters
only, regardless of the underlying Weibull model (4),
although this definitely affects the duration and the trend in
the transient phase. This result suggests that the asymptotic
limit of (5) can be successfully applied also to other distributions
(it was indeed successfully used also to determine the
calibration intervals of Cesium clocks [16]).
Secondly, for a given target probability of compliance to
metrological requirements and a given value of a or b, the
asymptotic limit in (5) can be used as a design criterion to compute
the value of the other parameter, even if the underlying
reliability model is totally or partially unknown. Fig. 4 shows
the results of the comparison between the values returned
by (5) (solid lines) and the probability of compliance computed
through extensive Monte Carlo simulations (dashed
lines) with values of a and b that were purposely exaggerated
to check the validity of (5) under stressed conditions (i.e., for
a = 0.1 and b = 0.55 or a = 0.3 and b = 0.4, respectively). In both
cases, T0
= 12 months and the parameters of the Weibull distribution
are α = 0.0006 e β = 2. The excellent agreement between
theory and simulation results is rather clear.
Reliability
Effort
Table 1 - Qualitative comparison between different methods to review calibration interval duration [15]
Staircase
methods
Medium
Low
Control charts
High
High
Workload balance
Applicability
Instrument availability
10
Medium
Medium
Medium
Medium
Low
Medium
In-use methods
Medium
Medium
Bad
High
Medium
IEEE Instrumentation & Measurement Magazine
In-service
methods
High
Low
Medium
High
High
Medium
High
Bad
Low
Medium
September 2023
(5)
Statistical model
methods
IEEE Instrumentation & Measurement - September 2023
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