Page 213 - Special Topic Session (STS) - Volume 3

STS540 J.-C. Malela-Majika et al.
Section 3 the zero-state and steady-state and average extra quadratic
loss () performance measures are thoroughly investigated using the
Markov chain approach and the BTXII control schemes are compared
̅
with traditional control schemes under zero-state and steady-state
̅
modes. A discussion is provided in Section 4 along with some concluding
remarks.

2. Design of the proposed schemes under the Burr-type XII distribution
The reader is referred to Burr (1942) for details on the Burr-type XII (BTXII)
distribution; details are omitted here to conserve space. Here we simply
mention the advantages of using the BTXII distribution. Note that the BTXII
distribution is used to describe the non-normal probability density function of
the IC process. Advantages of this distribution include the simplicity of its
cumulative distribution function as well as the option of representing a
number of different unimodal distributions. As a result, calculating Type I and
Type II errors are easy and the closed-form of the run-length distribution, of
control charting techniques designed under the BTXII distribution, are easy to

obtain. This paper considers the SS 2--(ℎ + 1) and 1--1 or 2--(ℎ +
1) schemes to expand the Shewhart-type scheme for non-normal
̅
distributed data using the BTXII distribution under the assumption that the
process parameters are known (Case K).
The zero-state and steady-state performances are investigated using the
Markov chain approach. A control chart is typically evaluated using either the
zero-state or the steady-state run-length properties. The former is used to
characterize short term run-length properties of a monitoring scheme as the
zero-state run-length is the number of plotting statistics at which the chart
first signals given it begins in some specific initial state and it is assumed that
the mean shift always takes place at the beginning of the process (Zhang and
Wu, 2005). The steady-state mode is used to characterize long term run-
length properties of a monitoring scheme as the steady-state run-length is
the number of plotting statistics at which the chart first signals given that the
process begins and stays IC for a long time, then at some random time, a
signal is observed (Zhang and Wu, 2005). Although a Markov chain approach
is used here, the details, such as setting up the transition probability matrices
etc., are omitted here to conserve space. The reader is referred to Fu and Lou
(2003) and Shongwe and Graham (2016) for details on the Markov chain
approach and SPCM.

3. Results
The zero-state and steady-state and performance measures
are thoroughly investigated using the Markov chain approach. Both these
measures have been widely used in the SPCM literature (see Human and

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