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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 740936, 12 pages
Research Article

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

1High-Tech Institute of Xi’an, Xi’an, Shaanxi 710025, China
2Suzhou INVO Automotive Electronics Co., Ltd., Suzhou, Jiangsu 215200, China

Received 2 January 2014; Revised 16 March 2014; Accepted 19 March 2014; Published 24 April 2014

Academic Editor: Carsten Proppe

Copyright © 2014 Shengjin Tang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 the k-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 repairable k-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.