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Basic Reliability Concepts and Models

1.1 Introduction

Reliability is a statistical approach to describing the dependability and the ability of a system or component to function under stated conditions for a specified period of time in the presence of uncertainty. In this chapter, we provide the statistical definition of reliability, and further introduce the concepts of failure rate, hazard rate, bathtub curve, and their relation with the reliability function. We also present several lifetime metrics that are commonly used in industry, such as mean time between failures, mean time to failure, and mean time to repair. For repairable systems, failure intensity rate, mean time between replacements and system availability are the primary reliability measures. The role of line replaceable unit and consumable items in the repairable system is also elaborated. Finally, we discuss the parametric models commonly used for lifetime prediction and failure analysis, which include Bernoulli, binomial, Poisson, exponential, Weibull, normal, lognormal, and gamma distributions. The chapter is concluded with the reliability inference using Bayesian theory and Markov models.

1.2 Reliability Definition and Hazard Rate

1.2.1 Managing Reliability for Product Lifecycle

Reliability engineering is an interdisciplinary field that studies, evaluates, and manages the lifetime performance of components and systems, such as automobile, wind turbines (s), aircraft, Internet, medical devices, power system, and radars, among many others (Blischke and Murthy 2000; Chowdhury and Koval 2009). These systems and equipment are widely used in commercial and defense sectors, ranging from manufacturing, energy, transportation, healthcare, communication, and military operations.

The lifecycle of a product typically consist of five phases: design/development, new product introduction, volume shipment, market saturation, and phase-out. Figure 1.1 depicts the inter-dependency of five phases. Reliability plays a dual role across the lifecycle of a product: reliability as engineering () and reliability as services (s). RAE encompasses reliability design, reliability growth planning, and warranty and maintenance. RAS concentrates on the planning and management of a repairable inventory system, spare parts supply, and recycling and remanufacturing of end-of-life products. RAE and RAS have been studied intensively, but often separately in reliability engineering and operations management communities. The merge of RAE and RAS is driven primarily by the intense global competition, compressed product design cycle, supply chain volatility, environmental sustainability, and changing customer needs. There is a growing trend that RAE and RAS will be seamlessly integrated under the so-called product-service system, which offers a bundled reliability solution to the customers. This book aims to present an integrated framework that allows the product manufacturer to develop and market reliable products with low cost from a product's lifecycle perspective.

Figure 1.1The role of reliability in the lifecycle of a product.

In many industries, reliability engineers are affiliated with a quality control group, engineering design team, supply chain logistics, and after-sales service group. Due to the complexity of a product, reliability engineers often work in a cross-functional setting in terms of defining the product reliability goal, advising corrective actions, and planning spare parts. When a new product is introduced to the market, the initial reliability could be far below the design target due to infant mortality, variable usage, latent failures, and other uncertainties. Reliability engineers must work with the hardware and software engineers, component purchasing group, manufacturing and operations department, field support and repair technicians, logistics and inventory planners, and marketing team to identify and eliminate the key root causes in a timely, yet cost-effective manner. Hence, a reliability engineer requires a wide array of skill sets ranging from engineering, physics, mathematics, statistics, and operations research to business management. Last but not the least, a reliability engineer must possess strong communication capability in order to lead initiatives for corrective actions, resolve conflicting goals among different organization units, and make valuable contributions to product design, volume production, and after-sales support.

1.2.2 Reliability Is a Probabilistic Measure

Reliability is defined as the ability of a system or component to perform its required functions under stated conditions for a specified period of time (Elsayed 2012; O'Connor 2012). It is often measured as a probability of failure or a possibility of availability. Let T be a non-negative random variable representing the lifetime of a system or component. Then the reliability function, denoted as R(t), is expressed as

It is the probability that T exceeds an expected lifetime t which is typically specified by the manufacturer or customer. For example, in the renewable energy industry, the owner of the solar park would like to know the reliability of the photovoltaic () system at the end of t =?20 years. Then the reliability of the solar photovoltaic system can be expressed as R(20) = P?{T?>?20}. As another example, as more electric vehicles (s) enter the market, the consumers are concerned about the reliability of the battery once the cumulative mileage reaches 100?000?km. In that case, t =?100?000?km and the reliability of the EV battery can be expressed as R(100 000) = P?{T?>?100 000}. Depending on the actual usage profile, the lifetime T can stand for a product's calendar age, mileage, or charge-recharge cycles (e.g. EV battery). The key elements in the definition of Eq. 1.2.1 are highlighted below.

• Reliability is predicted based on "intended function" or "operation" without failure. However, if individual parts are good but the system as a whole does not achieve the intended performance, then it is still classified as a failure. For instance, a solar photovoltaic system has no power output in the night. Therefore, the reliability of energy supply is zero even if solar panels and DC-AC inverters are good.
• Reliability is restricted to operation under explicitly defined conditions. It is virtually impossible to design a system for unlimited conditions. An EV will have different operating conditions than a battery-powered golf car even if they are powered by the same type of battery. The operating condition and surrounding environment must be addressed during design and testing of a new product.
• Reliability applies to a specified period of time. This means that any system eventually will fail. Reliability engineering ensures that the system with a specified chance will operate without failure before time t.

The relationship between the time-to-failure distribution F(t) and the reliability function R(t) is governed by

In statistics, F(t) is also referred to as the cumulative distribution function (). Let f(t) be the probability density function (); the relation between R(t) and f(t) is given as follows:

Example 1.1

High transportation reliability is critical to our society because of increasing mobility of human beings. Between 2008 and 2016 China has built the world's longest high-speed rail with a total length of 25?000?km. The annual ridership is three billion on average. Since the inception, the cumulative death toll is 40 as of 2016 (Wikipedia 2017). Hence the annual death rate is 40/(2016?-?2008) = 5. The reliability of the ridership is 1?-?5/(3 × 109) = 0.999 999 998. As another example, according to the Aviation Safety Network (ASN 2017), 2016 is the second safest year on record with 325 deaths. Given 3.5 billion passengers flying in the air in that year, the reliability of airplane ridership is 1?-?325/(3.5 × 109) = 0.999 999 91. This example shows that both transportation systems achieve super reliable ridership with eight "9"s for high-speed rail and seven "9"s in civil aviation.

1.2.3 Failure Rate and Hazard Rate Function

Let t be the start of an interval and ?t be the length of the interval. Given that the system is functioning at time t, the probability that the system will fail in the interval of [t, t +??t] is

The result is derived based on the Bayes theorem by realizing P{A, B} = P{A}, where A is the event that the system fails in the interval [t, t +??t] and B is the event that the...