Unavailability Analysis for k‐out‐of‐n:G Systems with Multiple Failure Modes Based on Micro‐Markov Models

Abstract

Markov models are commonly used for unavailability analysis of redundant systems. However, due to the exploding states of Markov models for redundant systems, the states need to be merged to simplify the computation, which is called micro-Markov models. However, how to derive the failure rates and repair rates of the newly developed micro-Markov models has not been studied thoroughly. Therefore, this paper proposes detailed explanations and rules to derive the static unavailability by the micro-Markov models for thek-out-of-n:G systems with multiple failure modes. Firstly, two properties about applying the Markov models to the repairable system with independent multiple failure modes are presented. Based on these two properties, two rules are proposed for implementing the micro-Markov models. The micro-Markov models provide the exact same results for the repairablek-out-of-n:G system with multiple independent failure modes and repair mechanisms and approximate results for systems with multiple hybrid failure modes. A case study of safety integrity verification for safety instrumented systems is provided to illustrate the application of the proposed method. The conceptual comparison and numerical examples demonstrate the reasonability and usefulness of the proposed micro-Markov models.

Extending the Possibilities of Quantitative Determination of SIL - a Procedure Based on IEC 61508 and the Markov Model with Common Cause Failures

“…For example, PFD, probability of failure on demand. in reference 15 the maximum difference amounts to 27% and in one case even 37%. The reason why different results are obtained can be explained by different approximations and assumptions in developed models.…”

“…Significantly larger differences can be noticed in the results obtained by various methods presented in other references. For example, in reference the maximum difference amounts to 27% and in one case even 37%. The reason why different results are obtained can be explained by different approximations and assumptions in developed models.…”

“…In accordance with IEC 61508, in the case of the single element, this value is assumed as equal to T 1 /2, while for the architectures 1oo2 and 2oo3 as equal to T 1 /3. In , the authors presented the mean time ( t a ) when 1oo n architecture failure, due to the undetected failures, occurs over the interval [0, T 1 ]. Because the condition λ DU · Τ 1 < < 1 is fulfilled, t a can be approximately obtained and was written as follows: t a ≈ ( n /( n + 1))· T 1 .…”

“…In this way, the transition diagram's size and calculation complexity was reduced. Moreover, in the mean time ( t a ) when k oo n system failure, due to the undetected failures, occurs over the interval [0, T 1 ], it was approximately obtained that: t a = ( n /( n + 1))· T 1 , because the condition λ DU · Τ 1 < < 1 is fulfilled. This approximation of t a will be used in this paper to estimate the mean downtime ( t GE ) of k oo n systems.…”

“…Various calculating techniques were analyzed in the paper showing that different techniques lead to different results and, in effect, to different SILs. The generalized equations for k ‐out‐of‐ n ( k oo n ) architectures were also studied in . In , the generalized model for calculating PFD and probability failure per hour was presented.…”