Narayanaswamy Balakrishnan - Accelerated Life Testing of One-shot Devices

It is useful to note that in all the preceding examples of one‐shot devices, we will not observe the actual lifetimes of the devices. Instead, we would only observe either a success or a failure at the inspection times, and so only the corresponding binary data would be observed, consequently resulting in less precise inference. In this manner, one‐shot device testing data differ from typical data obtained by measuring lifetimes in standard life‐tests and, therefore, poses a unique challenge in the development of reliability analysis, due to the lack of lifetime information being collected from reliability experiments on such one‐shot devices. If successful tests occur, it implies that the lifetimes are beyond the inspection times, leading to right‐censoring. On the other hand, the lifetimes are before the inspection times, leading to left‐censoring, if tests result in failures. Consequently, all lifetimes are either left‐ or right‐censored. In such a setting of the lifetime data, Hwang and Ke (1993) developed an iterative procedure to improve the precision of the maximum likelihood estimates for the three‐parameter Weibull distribution and to evaluate the storage life and reliability of one‐shot devices. Some more examples of one‐shot devices in the literature include missiles, rockets, and vehicle airbags; see, for example, Bain and Engelhardt (1991), Guo et al. (2010), and Yun et al. (2014).

As one‐shot devices (such as ammunition or automobile airbags) are usually kept for a long time in storage and required to perform its function only once, the reliability required from such devices during their normal operating conditions would naturally be high. So, it would be highly unlikely to observe many failures on tests under normal operating conditions within a short period of time. This renders the estimation of reliability of devices to be a challenging problem from a statistical point of view. In this regard, ALTs could be utilized to mitigate this problem. In ALTs, devices are subject to higher‐than‐normal stress levels to induce early failures. In this process, more failures could likely be obtained within a limited test time. As the primary goal of the analysis is to estimate the reliability of devices under normal operating conditions, ALT models would then typically extrapolate (from the data obtained at elevated stress levels) to estimate the reliability under normal operating conditions. ALTs are known to be efficient in capturing valuable lifetime information, especially when there is a need to shorten the life‐testing experiment. For this reason, ALTs have become popular and are commonly adopted in many reliability experiments in practice. One may refer to the detailed reviews presented by Nelson (1980), Cramer and Kamps (2001), Pham (2006), and Meeker and Escobar (2014), and the excellent booklength account provided by Nelson (2009).

Constant‐stress accelerated life‐tests (CSALTs) and step‐stress accelerated life‐tests (SSALTs) are two popular ALT plans that have received great attention in the literature. Under a CSALT, each device gets tested at only one prespecified stress level. To mention a few recent works, for example, Wang et al. (2014) considered CSALTs with progressively Type‐II right censored samples under Weibull lifetime distribution; for pertinent details on progressive censoring, see Balakrishnan (2007) and Balakrishnan and Cramer (2014). Wang (2017) discussed CSALTs with progressive Type‐II censoring under a lower truncated distribution. Lin et al. (2019) studied CSALTs terminated by a hybrid Type‐I censoring scheme under general log‐location‐scale lifetime distributions. SSALTs are an alternative to apply stress to devices in a way that stress levels will increase at prespecified times step‐by‐step. For SSALTs, there are three fundamental models for the effect of increased stress levels on the lifetime distribution of a device: The tampered random variable model proposed by DeGroot and Goel (1979), the cumulative exposure model of Sedyakin (1966) and Nelson (1980); see also (Nikulin and Tahir, 2013), and the tampered failure rate model proposed by Bhattacharyya and Soejoeti (1989). All these models of SSALTs have been discussed extensively by many authors. Gouno (2001) analyzed data collected from SSALTs and presented an optimal design for SSALTs; see also Gouno (2007). Zhao and Elsayed (2005) analyzed data on the light intensity of light emitting diodes collected from SSALTs with four stress levels under Weibull and log‐normal distributions. For the case of exponential lifetime distribution, by considering a simple SSALT under Type‐II censoring, Balakrishnan et al. (2007) developed exact likelihood inferential methods for the model parameters; see also Balakrishnan (2008) for details, while Xiong et al. (2006) considered the situation when the stress changes from a low‐level stress to a high‐level stress at a random time.

Fan et al. (2009) considered data, presented in Table 1.1, on 90 electro‐explosive devices under various levels of temperature at different inspection times. Ten devices under test at each condition were inspected to see whether there were any failures or not at each inspection time for each temperature setting. These data were then used to estimate the reliability of electro‐explosive devices at different mission times under the normal operating temperature.

Table 1.1 Failure records on electro‐explosive devices under CSALTs with temperature (K).

Source: Fan et al. (2009).

Zelen (1959) presented data from a life‐test of glass capacitors at four higher‐than‐usual levels of temperature and two levels of voltage. At each of the eight combinations of temperature and voltage, eight items were tested. We adopt these data to form one‐shot device testing data by taking the inspection times (hours) as which are summarized in Table 1.2. These data were then used to estimate the mean lifetime of glass capacitors for 250 V and 443 K temperature.

Lau et al. (1988) considered data on 90 solder joints under three types of printed circuit boards (PCBs) at different temperatures. The lifetime was measured as the number of cycles until the solder joint failed, while the failure of a solder joint is defined as a 10% increase in measured resistance. A simplified dataset is derived from the original one and presented in Table 1.3, where two stress factors considered are temperature and a dichotomous variable indicating if the PCB type is “copper‐nickel‐tin” or not.

Table 1.2 Failure records on glass capacitors under CSALTs with two stress factors: temperature (K) and voltage (V).