Burn-In

Statistical Science 1997, Vol. 12, No. 1, 1–19 Burn-In Henry W. Block and Thomas H. Savits Abstract. A survey of recent research in burn-in is undertaken. The emphasis is on mixture models, criteria for optimal burn-in and burn-in at the component or system level. Key words and phrases: Bathtub failure rate, mixture models, TTT plot, TTT transform, cost functions. 0. INTRODUCTION Burn-in is a widely used engineering method to eliminate weak items from a standard population. The standard population usually consists of various engineering systems composed of items or parts, or components which are assembled together into sys- tems. The components operate for a certain amount of time until they fail, as do the systems composed of these components. The systems might be electronic systems such as circuit boards and the components would be various types of chips and printed circuits. A typical mechanical system is an air conditioner and the components are condensor, fan, circuits and so forth. Usually within any population of com- ponents there are strong components with long lifetimes and weak components with much shorter lifetimes. To insure that only the strong components reach the customer, a manufacturer will subject all of the components to tests simulating typical or even severe use conditions. In theory, the weak components will fail, leaving only the strong com- ponents. This type of testing can also be carried out on systems after they are assembled in order to de- termine the weak or strong systems or to uncover defects incurred during assembly. These tests are known as burn-in. One important issue is to deter- mine the optimal time for burn-in. Burn-in is more typically applied to electronic than to mechanical systems. We give a survey of recent burn-in research with emphasis on mixture models (which are used to describe populations with weak and strong compo- nents), criteria for optimal burn-in and whether it is better to burn in at the system or component level. After some background, we give a brief description Henry W. Block and Thomas H. Savits are Pro- fessors, Department of Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 (e-mail: hwb@stat.pitt.edu, ths@stat.pitt.edu). of the types of statistical distributions which model the lifetimes of components for which burn-in is rel- evant. The remainder of the paper is devoted to ex- plicating recent promising developments in burn-in. Because of the authors’s interests, most emphasis will be placed on probability modeling for burn-in, but some statistical topics will also be covered. We will not review the fairly extensive engineering liter- ature on burn-in since this has been done in several review articles which we cite at the end of Section 1. Section 1 contains several illustrative examples and an introduction to some references for addi- tional background on burn-in. The distributions which are used to describe the lifetimes of com- ponents which can benefit from burn-in are given in Section 2. An important family of distributions is one in which the failure rate functions have a bathtub shape. In particular, distributions which arise as mixtures are singled out for emphasis since many bathtub-shaped failure rates arise in this way. In Section 3, various criteria are described which have been used to determine optimal burn-in times. Section 3.1 considers general criteria and Section 3.2 covers various cost structures. Sec- tion 4 discusses two recent mixture models. The first of these (Section 4.1) examines a typical het- erogeneous population to which burn-in is often applied and how this translates into renewal in- tensity behavior. The second of these proposes a general mixture model. A related result involves the asymptotic failure rate of a mixture model in terms of the asymptotic failure rates of the compo- nents of the mixture. The question of whether it is better to burn in at the component or the system level is discussed in Section 5. In Section 6, we con- sider an important tool, the TTT transform, which is used for approximating burn-in times. Section 7 gives a brief introduction to some recent sequential burn-in procedures involving optimal control. Sec- tion 8 gives a discussion with an indication of some future research directions. 1 2 H. W. BLOCK AND T. H. SAVITS 1. BACKGROUND AND SIMPLE EXAMPLES Many manufacturers and users of electronic com- ponents and systems, as a matter of course, subject these systems and/or components to initial testing for a fixed period of time under conditions which range from typical to those which approximate a worst-case scenario. A typical regimen is to intro- duce for a period of time some vibration and tem- perature elevation for a device. In a particular con- text this is sometimes known as “shake and bake.” At the end of this period, those components and/or systems which do not survive this period of testing may be discarded (scrapped), analyzed for defects and/or repaired. Those which survive this period may be sold, placed into service or subjected to fur- ther testing. Although these procedures have a va- riety of names depending on the area of application, we use the term burn-in to describe them all. The time period will be called the burn-in period. We il- lustrate some of these ideas with the following three examples. Example 1. Rawicz (1986) considers 30-watt long- life lamps manufactured by the Pacific Lamp Cor- poration (Vancouver, Canada) which were designed “for 5,000 hours of constant work in severe envi- ronmental conditions at 120 V.” These are installed on billboards where it is difficult and expensive to replace them. It turns out that a certain small percentage of these lamps tend not to last the requisite 5,000 hours but fail relatively early. Ob- viously it would be beneficial if this subpopulation of lamps could be identified and eliminated before being placed on a billboard. The procedure rec- ommended involves stressing all of the lamps at a high voltage (240 V) for a short period of time, which causes the weak lamps to fail rather quickly while the stronger lamps do not fail during this period. The lamps which do not fail are the lamps potentially capable of surviving the 5,000 hours of constant work. Often the burn-in weakens the surviving devices. In this particular application, however, the surprising result is that the surviving lamps are actually improved. This was thought to occur since the high thermal treatment seemed to relax structural stresses caused by the fabrication process. Example 2. In the AT&T Reliability Manual (Klinger, Nakada and Menendez, 1990) an elec- tronics switching system (the 5ESS Switch) is discussed. Immediately after manufacture this sys- tem is operated at room temperature (25◦C) for 12 hours, during which “volume-call” testing is performed; that is, 1,000 calls are simulated and passed through each of the five to eight modules of the switch. The system is then subjected for up to 48 hours to the high temperature (50◦C) which can occur within the switch if the air conditioning should fail. The first part of this procedure is to find and eliminate early system failures, and the second part simulates use in an extreme case which might occur. The objective of the second part is to accelerate aging, so that weak systems fail. It also provides data which can be used to see how this equipment compares to certain standards set for it. Example 3. Jensen and Petersen (1982) consider a piece of measuring equipment made up of approx- imately 4,000 components. They focus on several critical types of these components. One of these, called an IC-memory circuit, accounts for 35 of the 4,000 components. The bimodal Weibull distribution (i.e., a mixture of two Weibulls) is used to model this type of component and has the following survival function: F¯t‘ = p exp’−t/n1‘β1“ + 1− p‘ exp’−t/n2‘β2“: From the data, the values p = 0:015, β1 = 0:25, n1 = 30, β2 = 1 and n2 = 10 have been determined, but an explicit method is not given. We illustrate the results of Block, Mi and Savits (1993) (which is discussed in Section 4.2) to obtain the optimal burn-in time for a reasonable cost func- tion (we use CF1 of Section 3.2 in this example). Assume that we would like to plan a burn-in for components of this type so that those surviving burn-in should function for a mission time of τ = 60 units. If a circuit fails before the end of burn-in a cost c0 = q0C, where 0 < q0 < 1, is incurred. If it fails after burn-in but before the mission time is over, a cost of C is incurred. If an item survives burn-in and the mission time, a gain of K = kC is obtained. For illustrative purposes, we choose q0 = 0:5 and k = 0:05. We apply Theorem 2.1 of Block, Mi and Savits (1993). Let f be the density of the bimodal Weibull given above. It is not hard to show that gt‘ = ft + τ‘/ft‘ is increasing in t (either directly or by standard results) and goes from 0 (as t→ 0) to 1 (as t→∞). By the cited results an optimal burn-in time 0 < b∗ <∞ exists and satisfies gb∗‘ = C− c0 C+K: For the values above we obtain the equation gb∗‘ = 0:476, and solving graphically yields b∗ = 102:9. BURN-IN 3 Even though we present Example 2 as an ex- ample of burn-in, in the AT&T Reliability Manual (Klinger, Nakada and Menendez, 1990), Example 2 is called a system reliability audit. Other terms which are often used are “screen” and “environ- mental stress screening” (ESS). The AT&T Manual (Klinger, Nakada and Menendez, 1990, page 52) de- fines a screen to be an application of some stress to 100% of the product to remove (or reduce the number of) defective or potentially defective units. Fuqua (1987, pages 11 and 44) concurs with the 100% but states that this may be an inspection and stress is not required. Fuqua (1987, page 11) describes ESS as a series of tests conducted un- der environmental stresses to disclose latent part and workmanship defects. Nelson (1990, page 39) is more specific and describes ESS as involving ac- celerated testing under a combination of random vibration and thermal cycling and shock. Burn-in is described by the AT&T Manual (Klinger, Nakada and Menendez, 1990, page 52) as one effective method of screening (implying 100%) using two types of stress (temperature and elec- tric field). Nelson (1990, page 43) describes burn-in as running units under design or accelerated con- ditions for a suitable length of time. Tobias and Trindade (1995, page 297) restrict burn-in to high stress only and require that it be done prior to shipment. Bergman (1985, page 15) defines burn-in in a more general way as a pre-usage operation of components performed in order to screen out the substandard components, often in a severe envi- ronment. Jensen and Petersen (1982) have more or less the same definition as Bergman. For the purposes of this paper we use the term burn-in in a general way, similar to the usage of Jensen and Petersen (1982) and of Bergman (1985). We think of it as some pre-usage operation which involves usage under normal or stressed conditions. It can involve either 100% of the product or some smaller subgroup (especially in the case of complex systems as in Example 2) and it is not limited to eliminating weak components. Many of the traditional engineering ideas con- cerning burn-in are discussed in the handbook of Jensen and Petersen (1982). This book is intended as a handbook for small or moderate-size electronics firms in order to develop a burn-in program. Conse- quently the book should be viewed in this spirit. Em- phasis is on easy-to-apply methods and on graphi- cal techniques. One important contribution of the book is to popularize the idea that components and systems to which burn-in is applied have lifetimes which can be modeled as mixtures of statistical dis- tributions. Specifically components either come from “freak” or “main” populations and their lifetimes can be modeled as mixtures of Weibull distributions. Systems are assumed to inherit this dichotomous behavior, but the weaker population is called an “in- fant mortality” population. This population arises partly because of defects introduced by the manu- facturing process. Most reliabilty books familiar to the statistics community do not discuss burn-in. We mention three applied reliability books which discuss this topic. The first of these is the book by Tobias and Trindade (1995), which has a section on burn-in cov- ering some basics. An engineering reliability book by Fuqua (1987) delineates the uses of burn-in (see Section 2.4 and Chapter 14) for electronic systems at the component, module (intermediate between component and system) and system level. Most useful is the AT&T Reliability Manual (Klinger, Nakada and Menendez, 1990), which discusses a particular burn-in distribution used at AT&T along with a variety of burn-in procedures and several examples of burn-in. Two papers which review the engineering literature on burn-in are Kuo and Kuo (1983) and Leemis and Beneke (1990). 2. BURN-IN DISTRIBUTIONS For which components or systems is burn-in ef- fective? Another way of posing this question is by asking, “For which distributions (which model the lifetimes of components or systems) is burn-in effec- tive?” First, it seems reasonable to rule out classes of distributions which model wearout. The reason for this is that objects which become more prone to fail- ure throughout their life will not benefit from burn- in since burn-in stochastically weakens the residual lifetime. Consequently, distributions which have in- creasing failure rate or other similar aging proper- ties are generally not candidates for burn-in. For burn-in to be effective, lifetimes should have high failure rates initially and then improve. Since those items which survive burn-in have the same failure rate as the original, but shifted to the left, burn-in, in effect, eliminates that part of the lifetime where there is a high initial chance of fail- ure. The class of lifetimes having bathtub-shaped failure rates has this property. For this type of dis- tribution the failure rate starts high (the infancy period), then decreases to approximately a constant (the middle life) and then increases as it wears out (old age). As suggested by the parenthetical remarks, this distribution is thought to describe human life and other biological lifetimes. Certain other mechanical and electronic lifetimes also can be approximated by these distributions. This type 4 H. W. BLOCK AND T. H. SAVITS Fig. 1. Burn-in improvement example K = 1;000 hours; PPM/K = parts per million per 1;000 hours‘. of distribution would seem to be appropriate for burn-in, since burn-in eliminates the high-failure infancy period, leaving a lifetime which begins near its former middle life (see Figure 1). It turns out that there are reasons why many sys- tems and components have bathtub-shaped failure rates. As described by Jensen and Petersen (1982), many industrial populations are heterogeneous and there are only a small number of different subpopu- lations. Although members of these subpopulations do not strictly speaking have bathtub-shaped fail- ure rates, sampling from them produces a mixture of these subpopulations and these mixtures often have bathtub-shaped failure rates. For a simple ex- ample, assume that there are two subpopulations of components each of which is exponential, one with a small mean and one with a large mean. Sam- pling produces a distribution with decreasing failure rate which is a special case of the bathtub failure rate. An intuitive explanation of why this occurs is easy to give. Initially the higher failure rate of the weaker subpopulation dominates until this subpop- ulation dies out. After that, the lower failure rate of the stronger subpopulation takes over so that the failure rate decreases from the higher to the lower level. This type of idea, about the eventual domina- tion of the strongest subpopulation, carries through for very general mixtures. See Block, Mi and Savits (1993, Section 4). A subjectivist explanation of the fact that mixing exponentials produces a decreasing failure rate distribution was given by Barlow (1985), who argued that even though a model may be expo- nential, information may change our opinion about the failure rate. The mixture of two exponentials mentioned above produces a special case of the bathtub failure rate where no wearout is evident. Models of this type with no wearout are thought to be sufficient for mod- eling the lifetimes of certain electronic components, since these components tend to become obsolete be- fore they wear out. Mixing two distributions which are more complex than exponentials yields distri- butions with more typical bathtub-shaped failure rates, as can be seen in the following example. A typical bathtub curve is given in Figure 8.2 of To- bias and Trindade (1995, page 238) which we repro- duce in Figure 1. This distribution is realized as a mixture of a log- normal and a Weibull distribution (both of which are used to model defectives) and another distribution (which models the population of normal devices), Ft‘ = 0:0028 ( lnt/2;700‘ 0:8 ) + 0:001 ( 1− exp [ − ( t 400 )0:5]) + 0:997 ( 1− exp −10−7t‘ · [ 1−8 ( lnt/975;000‘ 0:8 )]) ; where 8 is the standard normal cdf. Notice that the left tail of the distribution is very steep. This tail represents the period where many failures oc- BURN-IN 5 cur. Burn-in is utilized in order to remove this part of the tail. The dotted line represents the result- ing distribution after a burn-in of several hours at an accelerated temperature. The point at which the curve flattens out and stops decreasing is at about 20K. This is called the first change point. Many papers have appeared in the statisti- cal literature providing models and formulas for bathtub-shaped failure rates. See Rajarshi and Ra- jarshi (1988) for a review of this topic and many references. One easy way of obtaining some of these is by mixing standard life distributions such as the exponential, gamma and Weibull. See Vaupel and Yashin (1985) for some illustrations of various dis- tributions or Mi (1991) for an example of a simple mixture of gammas which has a bathtub-shaped failure rate. The AT&T Reliability Manual (Klinger, Nakada and Menendez, 1990) gives another model (called the AT&T model) for the failure rate of an electronics component. The early part of the fail- ure rate is modeled by a Weibull with decreasing failure rate, and the latter part is modeled by an exponential (i.e., constant). It does not have a part describing wearout since the manual claims that the AT&T electronic equipment tends not to wear out before it is replaced. The AT&T model has been used extensively by Kuo and various co-authors (e.g., see Chien and Kuo, 1992) to study optimal burn-in for integrated circuit systems. This model is also called the Weibull–exponential model in the statistical literature (e.g., see Boukai, 1987). Since mixtures are emphasized in this review we point out one apparent anomoly mentioned by Gur- land and Sethuraman (1994). In that paper it is ob- served that when even strongly increasing failure rate distributions are mixed with certain other dis- tributions, their failure rate tends to decrease after a certain point. This is not surprising in the light of the previously mentioned result of Block, Mi and Savits (1993), which gives that asymptotically the failure rate of a mixture tends to the asymptotic fail- ure rate of the strongest component of the mixture. Since the failure rate of the strongest component is the smallest, the failure rate of the mixture is often eventually decreasing to this smallest value. Most definitions of bathtub-shaped failure rates assume the failure rate decreases to some change point t1‘, then remains constant to a second change point t2‘, then increases. The case t1 =