PFDavg generalized formulas for SIS subject to partial and full periodic tests based on multi-phase Markov models

• PFDavg formulas that include partial testing capability, relying on the IEC 61508 formulas scheme, are formally wrong.
• Their consistency was investigated based on multi-phase Markov models related to 1oo1 and 1oo2 architectures.
• New PFDavg generalized formulas have been developed.
• The new formulation considers the different repair times related to the various failures modes (detected failure (DD), detected by partial tests (PT), detected by proof tests (FT)).

IEC 61508 is a standard on design and operation of safety-instrumented systems (SISs) which has been adapted by many national regulations as the recommended way to achieve high-reliability systems. Many decisions about the design of SIS rely on the results from reliability assessments. It is therefore important that the reliability assessments are able to capture key properties of the system, such as the consideration of regular partial and full proof tests. IEC 61508 has proposed analytical formulas for commonly used architectures. Unfortunately, these formulas do not explicitly include the contribution of partial tests and consequently their use is mainly restricted to full proof tests. In addition, the already existing formulas dealing with partial tests disregard the different repair times. The aim of this paper is to (i) extend the PFDavg formulas given in IEC 61508 by including partial tests impact and, (ii) investigate their consistency based on multi-phase Markov models related to 1oo1 and 1oo2 architectures and (iii) to establish new generalized formulations in light of the results related to the investigation process, which account for the different repair times. Different comparisons are performed throughout the paper in order to validate the set of the derived formulations.