{"id":2259,"date":"2026-04-16T21:16:04","date_gmt":"2026-04-16T14:16:04","guid":{"rendered":"https:\/\/upskills.id\/insights\/?p=2259"},"modified":"2026-04-16T21:16:06","modified_gmt":"2026-04-16T14:16:06","slug":"integrating-weibull-distribution-inter-failure-analysis-and-reliability-growth-for-engineering-decision-making","status":"publish","type":"post","link":"https:\/\/upskills.id\/insights\/integrating-weibull-distribution-inter-failure-analysis-and-reliability-growth-for-engineering-decision-making\/","title":{"rendered":"Integrating Weibull Distribution, Inter-Failure Analysis, and Reliability Growth for Engineering Decision-Making"},"content":{"rendered":"\n<h3 class=\"wp-block-heading\">1. Introduction: From Failure Data to Engineering Decisions<\/h3>\n\n\n\n<p>In complex industrial environments\u2014whether in power plants, petrochemical facilities, utilities, or rotating equipment systems\u2014failures are not isolated events. They are manifestations of underlying system behavior shaped by design, operation, maintenance practices, and environmental conditions. Engineers responsible for maintaining system performance are not only tasked with restoring operation but also with understanding why failures occur, how they evolve over time, and what actions can prevent recurrence.<\/p>\n\n\n\n<p>Three analytical perspectives are central to this effort: <strong>Weibull distribution<\/strong>, <strong>inter-failure time (TBF) analysis<\/strong>, and <strong>reliability growth modeling<\/strong>. Each represents a different lens through which failure data can be interpreted. Individually, they provide valuable insights. When integrated, they form a powerful framework for transforming raw failure data into actionable engineering decisions.<\/p>\n\n\n\n<p>This article develops a structured, systems-level understanding of these methods, their relationships, and their practical application in reliability engineering.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. Weibull Distribution: Understanding Failure Mechanisms<\/h3>\n\n\n\n<p>The <strong>Weibull distribution<\/strong> is widely regarded as the foundational model for analyzing time-to-failure data. Its strength lies in its flexibility to represent different failure behaviors through a single parameter: the shape parameter (\u03b2).<\/p>\n\n\n\n<p>The Weibull probability density function is:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"397\" height=\"237\" src=\"https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-3.png\" alt=\"\" class=\"wp-image-2260\" srcset=\"https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-3.png 397w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-3-300x179.png 300w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-3-250x149.png 250w\" sizes=\"auto, (max-width: 397px) 100vw, 397px\" \/><\/figure>\n\n\n\n<p>2.1 Interpretation of the Shape Parameter (\u03b2)<\/p>\n\n\n\n<p>The shape parameter defines the failure behavior:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u03b2 &lt; 1 (Infant Mortality Region)<\/strong><br>Failures are concentrated early in life. Common causes include manufacturing defects, installation errors, or commissioning issues.<\/li>\n\n\n\n<li><strong>\u03b2 = 1 (Random Failure Region)<\/strong><br>The failure rate is constant. Failures occur independently of age, often due to external stressors or random disturbances.<\/li>\n\n\n\n<li><strong>\u03b2 > 1 (Wear-Out Region)<\/strong><br>Failure rate increases with time. This indicates aging, fatigue, corrosion, or material degradation.<\/li>\n<\/ul>\n\n\n\n<p>2.2 Engineering Significance<\/p>\n\n\n\n<p>Weibull analysis answers a critical question:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u201cWhat is the dominant failure mechanism in the system?\u201d<\/p>\n<\/blockquote>\n\n\n\n<p>This insight directly supports:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Maintenance strategy selection<\/li>\n\n\n\n<li>Replacement interval optimization<\/li>\n\n\n\n<li>Root cause analysis (RCA) validation<\/li>\n<\/ul>\n\n\n\n<p>However, Weibull assumes that each failure represents a complete lifecycle, which is not always valid in real operating systems.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3. Inter-Failure Time (TBF): Observing System Behavior<\/h3>\n\n\n\n<p>In operating systems, especially those undergoing repair, engineers often work with <strong>inter-failure times<\/strong> rather than complete lifecycles.<\/p>\n\n\n\n<p>Inter-failure time is defined as:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"514\" height=\"131\" src=\"https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-4.png\" alt=\"\" class=\"wp-image-2261\" srcset=\"https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-4.png 514w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-4-300x76.png 300w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-4-250x64.png 250w\" sizes=\"auto, (max-width: 514px) 100vw, 514px\" \/><\/figure>\n\n\n\n<p>3.1 Nature of TBF Data<\/p>\n\n\n\n<p>Unlike Weibull life data:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>TBF represents <strong>intervals between failures<\/strong><\/li>\n\n\n\n<li>It captures <strong>system-level behavior<\/strong><\/li>\n\n\n\n<li>It reflects the <strong>combined effect of multiple failure modes<\/strong><\/li>\n<\/ul>\n\n\n\n<p>3.2 Limitations of Simple TBF Analysis<\/p>\n\n\n\n<p>A common mistake is to treat TBF as independent and identically distributed data. In reality:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Repairs may not restore the system to \u201cas good as new\u201d<\/li>\n\n\n\n<li>Degradation accumulates over time<\/li>\n\n\n\n<li>Operational conditions may change<\/li>\n<\/ul>\n\n\n\n<p>Therefore:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Mean TBF alone is insufficient<\/li>\n\n\n\n<li>Trend analysis becomes essential<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">4. Reliability Growth: Understanding System Evolution<\/h3>\n\n\n\n<p>Reliability growth analysis addresses a fundamentally different question:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u201cIs the system becoming more reliable or less reliable over time?\u201d<\/p>\n<\/blockquote>\n\n\n\n<p>This is particularly relevant for:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Systems under continuous operation<\/li>\n\n\n\n<li>Assets undergoing iterative improvement<\/li>\n\n\n\n<li>Facilities experiencing recurring failures<\/li>\n<\/ul>\n\n\n\n<p>4.1 Crow-AMSAA Model (NHPP Power Law)<\/p>\n\n\n\n<p>The most widely used model for reliability growth is the <strong>Crow-AMSAA model<\/strong>, expressed as:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"391\" height=\"210\" src=\"https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-5.png\" alt=\"\" class=\"wp-image-2262\" srcset=\"https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-5.png 391w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-5-300x161.png 300w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-5-250x134.png 250w\" sizes=\"auto, (max-width: 391px) 100vw, 391px\" \/><\/figure>\n\n\n\n<p>4.2 Interpretation of \u03b2 in Reliability Growth<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u03b2 &lt; 1 (Improvement \/ Growth)<\/strong><br>Failures occur less frequently over time. Corrective actions are effective.<\/li>\n\n\n\n<li><strong>\u03b2 = 1 (Random Behavior)<\/strong><br>No improvement or degradation. System operates at steady state.<\/li>\n\n\n\n<li><strong>\u03b2 > 1 (Degradation)<\/strong><br>Failures accelerate. Indicates aging, design weakness, or systemic issues.<\/li>\n<\/ul>\n\n\n\n<p>4.3 Engineering Value<\/p>\n\n\n\n<p>Reliability growth analysis provides:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Evidence of improvement effectiveness<\/li>\n\n\n\n<li>Early warning of degradation<\/li>\n\n\n\n<li>Quantitative support for CAPEX decisions<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">5. Connecting Weibull, TBF, and Reliability Growth<\/h3>\n\n\n\n<p>5.1 Conceptual Integration<\/p>\n\n\n\n<p>Each method addresses a different dimension of reliability:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Weibull Distribution \u2192 Failure Mechanism (Why failures occur)<\/strong><\/li>\n\n\n\n<li><strong>TBF Analysis \u2192 Failure Pattern (What is happening)<\/strong><\/li>\n\n\n\n<li><strong>Reliability Growth \u2192 System Trend (Where the system is heading)<\/strong><\/li>\n<\/ul>\n\n\n\n<p>Together, they form a complete diagnostic framework.<\/p>\n\n\n\n<p>5.2 Micro vs Macro Perspective<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Weibull operates at the <strong>component level<\/strong><\/li>\n\n\n\n<li>Crow-AMSAA operates at the <strong>system level<\/strong><\/li>\n\n\n\n<li>TBF bridges both perspectives<\/li>\n<\/ul>\n\n\n\n<p>This distinction is critical in complex systems where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Multiple failure modes coexist<\/li>\n\n\n\n<li>Repairs are imperfect<\/li>\n\n\n\n<li>System behavior evolves over time<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">6. Practical Engineering Interpretation<\/h3>\n\n\n\n<p>6.1 Scenario 1: Early Failures (\u03b2 &lt; 1)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Weibull indicates infant mortality<\/li>\n\n\n\n<li>TBF shows short intervals initially<\/li>\n\n\n\n<li>Crow-AMSAA \u03b2 &lt; 1 confirms improvement<\/li>\n<\/ul>\n\n\n\n<p>Engineering action:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Improve commissioning practices<\/li>\n\n\n\n<li>Eliminate design or installation defects<\/li>\n<\/ul>\n\n\n\n<p>6.2 Scenario 2: Random Failures (\u03b2 \u2248 1)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Failures are unpredictable<\/li>\n\n\n\n<li>TBF shows no clear trend<\/li>\n\n\n\n<li>Growth model \u03b2 \u2248 1<\/li>\n<\/ul>\n\n\n\n<p>Engineering action:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Enhance monitoring systems<\/li>\n\n\n\n<li>Introduce redundancy<\/li>\n<\/ul>\n\n\n\n<p>6.3 Scenario 3: Wear-Out Failures (\u03b2 &gt; 1)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Weibull indicates aging<\/li>\n\n\n\n<li>TBF decreases over time<\/li>\n\n\n\n<li>Growth model \u03b2 > 1 confirms degradation<\/li>\n<\/ul>\n\n\n\n<p>Engineering action:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Implement preventive replacement<\/li>\n\n\n\n<li>Justify equipment overhaul or CAPEX<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">7. Beyond Classical Methods: Advanced Models<\/h3>\n\n\n\n<p>7.1 Non-Homogeneous Poisson Process (NHPP)<\/p>\n\n\n\n<p>Generalizes reliability growth models:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Allows time-varying failure rates<\/li>\n\n\n\n<li>Suitable for evolving systems<\/li>\n<\/ul>\n\n\n\n<p>7.2 Renewal Process<\/p>\n\n\n\n<p>Applicable when:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>System is restored to \u201cas good as new\u201d after repair<\/li>\n<\/ul>\n\n\n\n<p>Bridges Weibull and TBF analysis.<\/p>\n\n\n\n<p>7.3 Proportional Hazard Model (PHM)<\/p>\n\n\n\n<p>Incorporates operating conditions:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"512\" height=\"129\" src=\"https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-6.png\" alt=\"\" class=\"wp-image-2263\" srcset=\"https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-6.png 512w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-6-300x76.png 300w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/image-6-250x63.png 250w\" sizes=\"auto, (max-width: 512px) 100vw, 512px\" \/><\/figure>\n\n\n\n<p>Applications:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Predictive maintenance<\/li>\n\n\n\n<li>Condition-based monitoring<\/li>\n<\/ul>\n\n\n\n<p>7.4 Bayesian Reliability Models<\/p>\n\n\n\n<p>Used when:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Data is limited<\/li>\n\n\n\n<li>Uncertainty is high<\/li>\n<\/ul>\n\n\n\n<p>Enables continuous updating as new data becomes available.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">8. Engineering Workflow for Integrated Analysis<\/h3>\n\n\n\n<p>A structured approach ensures that analysis leads to actionable decisions.<\/p>\n\n\n\n<p>8.1 Step 1: Data Collection<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Failure timestamps<\/li>\n\n\n\n<li>Operating conditions<\/li>\n\n\n\n<li>Maintenance actions<\/li>\n<\/ul>\n\n\n\n<p>8.2 Step 2: TBF Calculation<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Compute inter-failure intervals<\/li>\n\n\n\n<li>Identify trends<\/li>\n<\/ul>\n\n\n\n<p>8.3 Step 3: Reliability Growth Analysis<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Fit Crow-AMSAA model<\/li>\n\n\n\n<li>Determine growth or degradation<\/li>\n<\/ul>\n\n\n\n<p>8.4 Step 4: Weibull Analysis<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Fit life distribution<\/li>\n\n\n\n<li>Identify failure mechanism<\/li>\n<\/ul>\n\n\n\n<p>8.5 Step 5: Decision-Making<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Maintenance strategy selection<\/li>\n\n\n\n<li>Root cause validation<\/li>\n\n\n\n<li>CAPEX justification<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">9. Application in Industrial Systems<\/h3>\n\n\n\n<p>Consider a nitrogen generation system experiencing declining performance.<\/p>\n\n\n\n<p>Observed data:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Increasing failure frequency<\/li>\n\n\n\n<li>Reduced output capacity<\/li>\n\n\n\n<li>Instability in operating pressure<\/li>\n<\/ul>\n\n\n\n<p>Analysis:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>TBF shows decreasing intervals<\/li>\n\n\n\n<li>Crow-AMSAA \u03b2 > 1 indicates degradation<\/li>\n\n\n\n<li>Weibull \u03b2 > 1 suggests wear-out mechanism<\/li>\n<\/ul>\n\n\n\n<p>Interpretation:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>System is not experiencing random failures<\/li>\n\n\n\n<li>Degradation is systematic, possibly due to material fatigue or control instability<\/li>\n<\/ul>\n\n\n\n<p>Engineering decision:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Evaluate component replacement (e.g., CMS material)<\/li>\n\n\n\n<li>Investigate control system mismatch<\/li>\n\n\n\n<li>Justify capital investment based on reliability decline<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">10. Strategic Importance in Reliability Engineering<\/h3>\n\n\n\n<p>The integration of these methods aligns with international best practices, including:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Reliability-centered maintenance (RCM)<\/li>\n\n\n\n<li>MIL-HDBK-189 (Reliability Growth Management)<\/li>\n\n\n\n<li>IEC reliability standards<\/li>\n<\/ul>\n\n\n\n<p>These frameworks emphasize:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Data-driven decision-making<\/li>\n\n\n\n<li>Continuous improvement<\/li>\n\n\n\n<li>System-level thinking<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">11. Common Pitfalls and Misinterpretations<\/h3>\n\n\n\n<p>11.1 Over-Reliance on MTBF<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>MTBF hides variability and trends<\/li>\n\n\n\n<li>Does not indicate improvement or degradation<\/li>\n<\/ul>\n\n\n\n<p>11.2 Misapplication of Weibull<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Using Weibull for repairable systems without validation<\/li>\n\n\n\n<li>Ignoring dependency between failures<\/li>\n<\/ul>\n\n\n\n<p>11.3 Ignoring System Context<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Failure data must be interpreted within operational conditions<\/li>\n\n\n\n<li>Statistical results without engineering context can mislead<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">12. Toward Predictive and Prescriptive Reliability<\/h3>\n\n\n\n<p>Modern reliability engineering is evolving toward:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Predictive analytics using machine learning<\/li>\n\n\n\n<li>Integration of sensor data<\/li>\n\n\n\n<li>Real-time condition monitoring<\/li>\n<\/ul>\n\n\n\n<p>In this context:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Weibull provides baseline understanding<\/li>\n\n\n\n<li>TBF analysis provides operational insight<\/li>\n\n\n\n<li>Reliability growth models provide strategic direction<\/li>\n<\/ul>\n\n\n\n<p>Together, they form the foundation for advanced reliability systems.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">13. Conclusion<\/h3>\n\n\n\n<p>Understanding failures requires more than statistical analysis. It requires a structured approach that connects data, models, and engineering judgment.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Weibull distribution explains the mechanism of failure<\/strong><\/li>\n\n\n\n<li><strong>Inter-failure analysis reveals system behavior<\/strong><\/li>\n\n\n\n<li><strong>Reliability growth modeling shows the direction of system performance<\/strong><\/li>\n<\/ul>\n\n\n\n<p>Individually, each method answers a specific question. Collectively, they enable engineers to move from reactive maintenance to proactive reliability management.<\/p>\n\n\n\n<p>The ultimate objective is not merely to predict failure, but to design systems where failure becomes less likely, less frequent, and less impactful.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><a href=\"https:\/\/play.google.com\/store\/books\/details?id=vLPPEQAAQBAJ\"><img loading=\"lazy\" decoding=\"async\" width=\"640\" height=\"1024\" src=\"https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/2-640x1024.png\" alt=\"\" class=\"wp-image-2264\" srcset=\"https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/2-640x1024.png 640w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/2-188x300.png 188w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/2-768x1229.png 768w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/2-960x1536.png 960w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/2-1280x2048.png 1280w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/2-600x960.png 600w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/2-156x250.png 156w, https:\/\/upskills.id\/insights\/wp-content\/uploads\/2026\/04\/2.png 1600w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/a><\/figure>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>1. Introduction: From Failure Data to Engineering Decisions In complex industrial environments\u2014whether in power plants, petrochemical facilities, utilities, or rotating equipment systems\u2014failures are not isolated events. They are manifestations of underlying system behavior shaped by design, operation, maintenance practices, and environmental conditions. Engineers responsible for maintaining system performance are not [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":2265,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_daextrevo_audio_file_creation_date":"","_daextrevo_text_to_speech":"","_daextrevo_document_type":"","footnotes":""},"categories":[9,351,350],"tags":[],"class_list":["post-2259","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-and-technical-skills","category-maintenance","category-reliability"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.7 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Integrating Weibull Distribution, Inter-Failure Analysis, and Reliability Growth for Engineering Decision-Making - Insights<\/title>\n<meta name=\"description\" content=\"In complex industrial environments\u2014whether in power plants, petrochemical facilities, utilities, or rotating equipment systems\u2014failures are not isolated events. 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