An Alternative to the Exponential and Weibull Reliability Models

The two reliability models most widely used in engineering applications are the exponential and the Weibull. Both models are deficient because their failure rates do not conform to the bathtub curve nor specify the infant mortality period. In addition, Weibull applications require lifetime data which entails testing or field observations and is usually proprietary. The advantage of the exponential model is that it can be supported by both lifetime and MIL handbook data, the latter being publicly available. In this article, an alternative single parameter model—called lifetime—is formulated whose failure rate conforms to the shape of an asymmetric bathtub curve. It is shown that, unlike present multiparameter bathtub curve models, the alternative model can be supported by MIL handbook data. The reliability ramifications of the alternative model will be demonstrated through three applied examples: engine fan, microcomputer, and crystal oscillator. The data supportability claim is substantiated by constructing an estimation procedure demonstrated in the microcomputer and crystal oscillator examples. Calculations will show that both the exponential and Weibull models predict twice as many engine fan failures as the article model. For the microcomputer example, the exponential model gives three and one-half times as many faults.


I. INTRODUCTION
In this article, a reliability model will correspond to a probability distribution. In addition to the mathematical function -exponential, Weibull, etc. -the parameters associated with that function will determine the reliability consequences of the model. Specific mathematical issues concerning the models will be addressed in the following sections.
The most widely used reliability model in engineering practice is arguably the exponential. There are two fundamental reasons why this is so. First, mathematical utility, and second, it readily interfaces with the MIL handbooks such as the NPRD-2016. The latter property is critically important because specific test or field data is often unavailable or proprietary in commercial applications. MIL handbook data is the only practical recourse.
From a reliability perspective, the defining attribute of exponential models is their ''memoryless'' feature. What this means physically is that there is no aging going on. This is at variance with the second law of thermodynamics.
Mathematically, the non-aging property of the exponential model is manifested as a constant failure rate. This is contrary to the empirical fact that physical items over their product lifecycles exhibit failure rates metaphorically described as bathtub-like curves -not constants. A typical illustration of a bathtub curve from the source Wikipedia is shown in the figure below.
To address the non-aging issue, analysts have assumed that a useful life phase ensues after the infant mortality phase. During this period, it is assumed that the failure rate is sufficiently non-varying such that the exponential model is an acceptable approximation. Hence, concerning the reliability analysis, the infant mortality and wear-out phases become irrelevant.
The Weibull model is characterized by its scale and shape parameters. When the shape parameter is less than one, the failure rate decreases. If the shape parameter is greater than one, the rate increases (when the shape parameter equals one, the model becomes identical to the exponential). Clearly, neither of these outcomes will reproduce the bathtub curve.
However, the main issue in applying the Weibull model is computational complexity. First, the model requires an item's time to failure or an accurate approximation thereof. For this reason, the model is incompatible with the MIL handbooks. Second, an accurate statistical estimation of the model parameters usually requires test or field data sets, necessitating considerable resources or observation time to assemble.
In this article, the issues surrounding the exponential and Weibull models are addressed as follows: 1) The alternative model is postulated as an Arcsine function with a single parameter, hereafter referred to as the lifetime parameter.
2) The bathtub curve characteristic is derived analytically from the Arcsine function and exhibited graphically in figures about each of the equipment examples in this article.
3) The infant mortality period is derived from the Arcsine function and found to be approximately 28% of the lifetime parameter. 4) Comparisons between the predicted distributions of item failures over time with respect to the models are given for the engine fan and microcomputer application examples. 5) Estimation procedures for determining the Arcsine lifetime parameter from field and MIL handbook data sets are developed. Article organization is as follows: Section II gives a brief account of how the bathtub curve failure rate is presently addressed in reliability practice. Section III provides the basic reliability model mathematical results such as the mean, standard deviation, and confidence interval formulas. Physical arguments address model infinities. Section IV shows how the model applies to a mechanical fan's field (or life) data using standard statistical estimation methods. The Arcsine and the exponential modeling results are compared in this section by applying them to a hypothetical warranty scenario. Section V involves the application of the model to a microcomputer device using MIL handbook data. Procedures are developed to estimate the lifetime parameter without device start times. Also, in this section, the estimated device infant mortality life span is compared with the expected commercial product lifetimes. Section VI involves the application of the model to a crystal oscillator based on MIL handbook data. In this section, an extension of the model is proposed to deal with all the known failure modes of the oscillator. In the discussion of findings in Section VII, it is concluded that FMECAs are no longer applicable in the Arcsine model as they were in the exponential. However, FMEAs remain unaffected.

II. BACKGROUND
In the introduction, it is stated that the reliability bathtub curve is characterized by three distinct phases or periods. First is the infant mortality phase, where item failure is attributed to construction material imperfections, out-of-tolerance manufacturing processes, and quality control issues. The second is the useful life phase when failures are random such as foreign object damage or improper usage. Third is the wear-out phase, where items begin to physically degrade at increasing failure rates.
As shown by [1], determining the reliability of the bathtub failure rate enables the direct estimation of several important logistic parameters. Among these would be: i) the required burning in or debugging period; ii) the number of spare parts required during any prescribed time period, and iii) the service life of an item.
Technical literature gives a number of theoretical derivations of a bathtub failure rate. Such analytical formulations often use mixtures of Weibull distributions. In [1], curve fitting constructions are performed with respect to five selected life data sets to obtain corresponding empirical failure rate curves -each having three or more parameters. The defining feature of each of the data sets is that the times to failure of each of the subject test items were given, or a sufficiently accurate approximation was given.
As noted in the introduction, MIL handbooks, where item time-to-failure data is unavailable, are often the only data source available to analysts. Hence, the development of a model that does not require time-to-failure type data constitutes the goal of this article.

III. MATHEMATICAL MODEL DERIVATION
The starting point is the Arcsine distribution, presented in [2] in the standard form as The density function associated with the Arcsine distribution is given as The support for which the distribution and density functions are defined on is (0, 1). The next step is to transform to a general support (a, b). This transformation is accomplished utilizing the standard probability transform theorem [3]. Proceeding with a linear transform gives the result Solving for x = g −1 (t) gives Hence, according to the transform theorem, the probability density transforms as From the results (1) and (4), the distribution is The Arcsine probability distribution or Arcsine model is a function of the time-to-failure random variable T with support (0, L). In this case, there is only one prescribed parameter to determine -the lifetime parameter. Hence, for 0 < t < L or t ∈ (0, L) the Arcsine reliability model is The corresponding probability density distribution is given as Next follows the failure rate Inspection of the failure rate leads to the first model infinity. From (9), the following result is obtained From (9), the failure rate becomes Therefore, the interpretation is that as the item population ages approaching the lifetime limit L, the proportion of the population failures, F (t), approaches 100%, causing the failure rate to increase without bound. Hence, according to the Arcsine model, an item population has a well-defined finite lifetimethe distribution parameter Lestimated from the empirical data.
The second model infinity arises from consideration of the initiation stage of the lifecycle as follows Unlike the case where t → L, the situation where t → 0 does not readily admit to a meaningful physical interpretation. How to explain the mathematical result that as devices VOLUME 10, 2022 appear to become ''newer,'' their corresponding failure rates increase without bound. To this end, it is the case for a population of size N , the number of failed units in the time interval t approaching zero is Therefore, according to the Arcsine model, the number of failures vanishes even though the failure rate itself becomes unbounded.

A. RELIABILITY DISTRIBUTION PARAMETERS
First, the expectation, i.e., the counterpart of the exponential distribution mean-time-between-failure (MTBF) parameter, is computed. Making the change of variable x = t/L in (8), writing in terms of the Gamma function (x) From (10) and computing the variance, the result is Therefore, the standard deviation is

B. SHAPE OF THE FAILURE RATE CURVE
The applications that follow plots of the Arcsine failure rate provide evidence of the bathtub curve shape. Analytically, the characteristic of that curve is the existence of a unique minimum. This is calculated below. In computing the Arcsine failure rate extremum, (9) is rewritten as Using the relation F (t) = 1 − R (t) , it follows from the first equation in (9) after performing the necessary algebraic Making the change of variable x = t L, the preceding result becomes Solving this equation for x, which is a minimum, is shown as Next, employing (9) and (12) to calculate numerical results is useful when applying the Arcsine reliability model in the application example. One such being Utilizing (12) and the fact that L > 0, it is established that the extremum is a minimum by the standard calculus criterion as follows Substituting the result from (12) into (14) gives the result 8.119437 L 3 > 0. Per the above analysis, a unique minimum for the Arcsine failure rate is established. Interpreting the results leads to the following conclusions: i) an infant mortality phase is defined as 0 < t ≤ 0.278945L, during which the failure rate decreases; ii) at the failure rate minimum, the maximum reliability robustness is achieved which is approximately 28% of the total lifetime; and iii) during the life phase t min < t < L, the failure rate is monotonically increasing.
Using data from the engine fan example in the next section, a plot of the failure rate curve h (t) exhibiting the bathtub curve shape is shown in Fig. 1 below.

C. CONFIDENCE INTERVALS
The objective is to derive upper and lower bound estimates for the lifetime distribution parameter of the Arcsine model based on the results established for the exponential distribution. The previous section determined that the Arcsine failure rate curve had one extremum: a minimum. The interpretation is that the tangent to the curve at this point, a constant, corresponds to an exponential failure rate λ * . See Fig. 2 below.
In the exponential case, it is shown [4] that the random variable 2rθ θ * is Chi-square distributed with 2r degrees of freedom, and the observation conditions -either test or field -are such that termination occurs with the rth failure. Here, θ * is the usual exponential MTBF = 1 λ * parameter for which two-sided 100γ % confidence limits is determined. It is the case that the statistical estimate for the exponential MTBF isθ = T a r, where T a is the total operating hours (OH) as accumulated by all units during the observation or test interval. Given these results, it is the case that the Upper Bound (UB) and Lower Bound (LB) parameters can be expressed in the following probability relations The Chi-square terms are the percentage points or quantiles of the Chi-square distribution in the probability equations. Revising the previous equation in terms of the failure rate θ * = 1 λ * yields the following result Using the engine fan example data again, the procedure can be represented graphically, as shown in Fig. 2 below.
Given the interpretation associated with the defined exponential failure rate λ * and based on (12) and (13), the expression for the number of failures in the time span T a is as follows From (15), the confidence limits concerning λ * are given as From (16), (17), and (18), the confidence limits of the lifetime parameter are These results as a probability expression are as follows

IV. ENGINE FAN APPLICATION
The first application is the only one for which the time-tofailure of the units can be estimated from the available field data.

A. LLFETIME PARAMETER ESTIMATION
The field data given in Table 1 below is from [5]. Unbracketed table entries refer to recorded fan run-times. Items with the + sign denote the run time or OH of fans that did not fail. Bracketed entries in Table 1 refer to an item's VOLUME 10, 2022 enumeration in the population. Fans that did not fail (DNF) but whose run times are the same are assumed to have been fielded simultaneously.
From Table 1, the period of observation is taken as 11500 OHs completed by the 70th table entry. The fans are numbered chronologically by OH or time to failure and sorted into two sets defined as follows In this scheme, the 70th item ran for 11500 hours, and the attached + sign indicates that it did not fail; hence, it is a member of the DNF set. The field deployment of the 70th item marks the start time of the period of observation which is assumed to be when the 70th fan was shut down operationally. In Table 1, the first item in the engine fan data set, k = 1, ran for 450 OH, at which time it failed. Unlike item k = 70, whose start time is designated as zero, the start time for k = 1 is unknown. All that is known about the first item operating start time is the following Given that the deployment of the first item was a managerial/logistic decision, it is assumed that any start time up to 11050 OH was equally likely. Hence, an assumed probability distribution of the operating hour start time (OHST) is taken to be a uniform distribution specified in this case as In the above expression, I [a,b] is the usual indicator function.
Since the mode is the most probable distribution value, this value is taken as the statistical estimate of the starting point for the first item. Hence, for the first fan referring to Table 1 Mode In this case, the support is â 1 , L +â 1 , and according to (5), the corresponding density is In the case of fans that did not fail in their recorded run times, the probability that the fan did not fail in this time span will be required. For example, when fan k = 5, For the mode, the result refers to Table 1 Mode Therefore, with support â 5 , L +â 5 , it follows from (6) that the corresponding reliability distribution is Following these procedures, both the start times and times to failure for all other engine fans can be constructed. The parameter estimation procedure is the standard MLE and construction of the likelihood function is as follows Next, taking the logarithm of the likelihood function given in (24), and with (22) and (23), the result is Maximizing the above expression with respect to the Arcsine lifetime parameter L gives the result From the preceding equation, the expression to be solved for L Utilizing the engine fan field data given in Table 1, the preceding equation can be solved numerically, resulting in a lifetime parameter estimate asL = 70623 OH . From [5] and the result (10), the comparison of the MTTF values corresponding to this example's Arcsine, exponential, and Weibull models are presented in Table 2 below.

B. FAILURE RATE CUTOFF, TRUNCATION
From Table 1, it is seen that the time increments with respect to run-time or times-to-failure are 50 OH except for the k = 2, 5, and 7 fans. In the case of the first fan, its time-to-failure was 450 OH or 9, 50-OH increments. However, the run-time was 460 OH or 9, 50-OH increments plus 10 OH for fan number two. From this fact, interpretation of the fan field data shows that the resolution limit of the time recording instrumentation was 10 OH. Therefore, for those fans whose timesto-failure are identical in Table 1, they failed within 10 OH of each other. In this application, the resolution limit will serve as a cutoff of the failure rate or truncation point c. The truncated distribution is found to be [3] Calculations utilizing the above result show that a nonzero cutoff amounts to a 1.01% percent increase in the value of the Arcsine model MTTF. For this reason, taking into account, a truncation correction was considered negligible.
The question of when and how the cutoff or truncation factor must be taken into account will depend on the physical system being analyzed. The truncated failure rate issue will be encountered when dealing with start time concerns in the crystal oscillator example. Table 1 shows that N 0 = 58 engine fans did not fail. According to [5], after the fan data was collected, the question was whether to redeploy the unfailed fans in the field with an 8000 OH warranty. Alternatively, replace them with a new design. The predicted proportion of remaining fans failing over the warranty period would be one of the inputs management would use to decide the issue.

C. WARRANTY ISSUES
Because of the memoryless feature of the exponential model, the expected number of warranty failures would be based on the data from Table 2 according to the relation Hence, the exponential model predicts that 14 out of 58 or 24% of the remaining fan population would fail over the fan warranty period.
Over the warranty period W = 8000 OH , the Arcsine model predicts the number of failures as Next, taking L =L = 70263 OH and the data in Table 1 for the OH k values, use is made of (25) to calculate the number of failures N F predicted by the Arcsine model as In the case of the Arcsine model, 6 or 10.3% of the fans would fail over the warranty term.
As a final prediction in [5], the Weibull model with scale parameter α = 26297 and shape parameter β = 1.0584 Using the data in Table 1, and (26) shows that the Weibull model predicts 13 fan failures over the warranty period.
Following the same calculation procedures the number of failures associated with other lifecycle milestones can be computed. These results are presented in Table 3 below. A 133% differential in the warranty failure predictions between the Arcsine model versus the exponential and 117% for the Weibull might be acceptable for a population of size 70. However, if the engine fan population is, for example, 70000, these results could be much more consequential to management.
From the Arcsine failure rate curve shown in Fig. 1, it would be expected that most of the engine fan failures occur early in their lifecycles. Inspection of the data given in Table 1 yields the following results: i) of the 30 fans that logged run times of at least 5000 OH, there were two failures; and ii) of the 12 recorded fan failures, 10 occurred in less than 5000 OH. Of these, 9 were less than 4000 OH. These results are viewed as consistent with the Arcsine model.

D. COMPARISON WITH THE WEIBULL AND EXPONENTIAL
The first comparison is quantitative and concerns the number of expected failures each model predicts over the warranty time span. Table 3 results make clear the difference between the three models.
A second means of comparing the three models is qualitative. This is accomplished by plotting the failure rate curves of the three models in Fig. 3.
The interpretation of the graphs in Fig. 3 is that infant mortality issues are of significance only in the case of the Arcsine model.

V. MICROCOMPUTER APPLICATION
The second application is taken from [6], Vol. I, on the IC, Digital Microcomputer, or Microcomputer. The application environment that pertains to this device is classified as Ground Benign Commercial (GBC). The data source for [6] was warranty repair data from a commercial electronic computer manufacturer whose identity is listed as Proprietary in [6], Vol. IV. The microcomputer populations, with respect to each period of record, are construed as being from different manufacturing production runs.

A. ENTRY EXPLANATIONS
From the part detail data provided in [6], Vol. II, Table 4 was constructed.
Explanations and calculations pertaining to each of the table categories are as follows: 1) Period of Record (POR): This column gives the calendar dates over which the data was collected as well as the period length in terms of Calendar Days (CD) and Calendar Hours (CH) units.

2) Duty Cycle (DC):
In general, logistic hours -e.g., the total time an equipment unit is in the ''field'' -refer to the calendar or clock hours. The time equipment in the field is spent performing its intended function is measured in Operating Hours (OH). The duty cycle is the conversion between these two time units. These factors, as listed in Table 4, are calculated as follows The duty cycle in a sometimes more convenient procedure can be computed as follows The general formula for DC is expressed as follows 3) Population (Pop.): Population refers to microcomputers manufactured with respect to each POR. The Electronic Parts Reliability Data (EPRD) source gives several manufacturers' population and field failure numbers. However, only the National Semiconductor Corp. population and failure data were included in the analysis. Therefore, only 9890 microcomputers are listed in the Population category of Table 4 Table 4 shows that the EPRD data records do not give device run times or OH. For each POR, the total OH in terms of the OH for each microcomputer is given as The first step in the estimation procedure requires finding a method for estimating the operating or run times of the microcomputers.
Begin by defining the following random variables Therefore, it follows: As noted above in Table 4, the data in the EPRD Failed column are the observed equipment failures with respect to each POR. Theoretically, the number of failures per POR, N Fk , is given as the sum of the random variables X k as defined previously and given as In the above expression, k = 1, . . . , 8 refers to each POR in Table 4. Identification of the empirical failure data in terms of the expected values of the N Fk random variables is as follows Next, the device run times (27) are specified. This is done by assuming two application profiles. The first is a mission scenario whereby each device is specified by a user program to operate at the same DC associated with the POR. An example would be equipment items aboard a military aircraft having a prescribed number of flying hours per month. It follows from (27), then From (28) and (29), the following is obtained The second application profile will be referred to as a commercial scenario. In this case, the equipment run times are assumed to be independent and identically normally distributed. An example would be cell phone users in a general population.
Starting on Day one of the first POR of Table 4, where j = 1, . . . , CD 1 Since for a 24-hour day the start time is always zero and 99.7% of a normal population is within the 3σ limit, the standard deviation is approximated as follows For the distribution over the entire first POR or 184 CD Having specified the run time distribution for a POR, the run times associated with (27) are next determined In the above equation, α m is the proportion of the normal population within the mth and (m-1) standard deviationse.g., α 1 = 0.682. To perform calculations in the above equations, considerable latitude in specifying trial values for the lifetime parameter is assumed -e.g., 10 7 < L k < 10 9 . Given the above results, the specification of run times for a commercial scenario would be Applying the results obtained above to the first POR of Table 4 yields a percent difference of less than 2.5% between (28) for normally distributed run times given by (31) versus (30), where all run times are the mean value of the distribution. Performing the analogous calculations for the third POR of Table 8 pertaining to the crystal oscillator example below yields a percent difference of 0.58%. Based on these outcomes, a heuristic argument is made for using (29) and (30) to estimate the lifetime parameter.
The results can be interpreted as analogous to the information-theoretic result of the amount of information associated with a signal consisting of m symbols each having a designated probability. It is shown [7] that the information of such a signal is a maximum if the probability of each symbol is the same. Here, device operating time is viewed as proportional to probability.
Estimates are now constructed for the lifetime parameter for each POR by solving (30) for L k , k = 1, . . . , 7 From Table 4, it is noted that for the 8th POR, there are no recorded field failures (i.e., E (N F8 ) = 0), and therefore the use of (32) for estimating the lifetime parameter is invalid. This situation is often encountered in the Nonelectronic Parts Reliability Data (NPRD) and EPRD handbooks, and often there are no recorded failures over all the PORs for many hardware items. Under these circumstances, the MIL handbooks assume just one failure corresponding to each POR. Doing this, the estimate would now be assuming E (N F8 ) = 1 However, this is not be the estimation method used. In estimating the lifetime parameter for the Arcsine distribution, the empirical probabilities are defined as follows for k = 1, . . . , 7 Even in the null case when E (N Fk ) = 0, an existing nonzero probability is still interpreted, although it cannot be estimated from the EPRD data record as was done before. In this case, the null probability is calculated from the geometric average from the non-null empirical probabilities shown in Table 4 Therefore, the estimate of the lifetime parameter for the 8th POR.L 8 = t avg.8 These computations are the numbers in the L Parameter Estimates column of Table 4. Taking the geometric mean of these values, the estimate of the lifetime parameter distribution of the microcomputer is determined to bê The tradeoff in employing the Arcsine model versus the exponential is computational simplicity. That claim is shown by considering the estimation of the exponential failure rate from Table 4

C. UPPER/LOWER BOUND ESTIMATES AND RESULTS
Referring to (16) and the Total and Avgs. row of Table 4, the total accumulated OH is T a = 132.2854 × 10 6 from which follows To proceed with the calculations, the Excel χ 2 function default value of one is assumed for the degree-of-freedom parameter 2h (t min ) T a . Referring to (19) and (20) In the case of the normal distribution symmetric about the mean, 68.2% of the population falls within one standard deviation of the mean. The Arcsine distribution is symmetric about the mean time to failure (MTTF) parameter or L 2. From (11) Making the change of variable x = t L it follows from (8) For future reference, an estimate of the UB, LB, and onesided LB (OLB) for L at the 50% Confidence Level (CL) Also for future reference, the MTTF at the 50% CL is computed MTTF OLB / 50% = L OLB / 50% 2 = 3.195879 × 10 8 OH .
In Fig. 4, a comparison of the failure rate plots at the 50% OLB CL with the point estimate is shown.
From (38) and (39), calculations show the infant mortality period corresponding to the LB/50% estimate is 0.2561 of the point estimate infant mortality period while that of the UB/50% is 3.3378 times the same period. The implication of this result is that the analyst must exercise due caution as to the impact overestimation or underestimation of the infant mortality period will have on the project objectives. The next section provides further clarification of this matter.

D. INFANT MORTALITY AND WARRANTY ISSUES
Using Table 4, two estimates of the infant mortality period can be computed at the 29.64% DC. The first calculation uses  From the estimated infant mortality periods above, it follows that all failures during a realistic product lifecycle of the microcomputer will be in the infant mortality regime. In Fig. 5 below, a plot of the failure rate infant mortality phase of the lifecycle is given for a point estimate of L: To assess the impact of the lifecycle of the microcomputer being entirely in the infant mortality phase, the following hypothetical warranty scenario is considered. Assume a manufacturer intends to produce 20000 devices per year having a five-year warranty. The expected lifecycle is ten years, after which the microcomputer will be considered obsolescent. This means that devices will have to be tracked for nine years because the 20000 devices produced in the 5th year will still be under warranty for another five years.
In performing the calculations on the distribution of failures, it is assumed items that fail in the field are not replaced. To further simplify the calculations, the mission usage scenario can be assumed -i.e., the OH per year per device is the same. Using the data from Table 4, warranty failures can be calculated according to the Arcsine model over a product lifecycle of ten years. The results in Table 5 can also be tabulated.  Comparing the Arcsine model's warranty predictions versus the exponential Table 6 is compiled in the same fashion as Table 5 employing the Table 4 failure rate λ = 0.666468 × 10 −6 OH −1 .
A comparison of the scenarios shows that the exponential model predicts more than three and one-half times as many failures as the Arcsine model over the same product lifecycles.

E. COMPARISON WITH 217PLUS AND EXPONENTIAL
For many electronic devices, the EPRD HDBK will not have any associated field failures entries (i.e., only zero data entries). The Arcsine model estimation procedure is inapplicable for these types of failure outcomes and a different method will be required.
A widely used reliability model for estimating electronic device failure rates is the 217Plus TM model [8]. Referring to Table 2.3-2 of the handbook for a Ground Stationary Indoors environment, (converting from CH to OH as proscribed in [8]) Since the data used in the Arcsine model was from 05/1984 to 10/1990, the growth factor in the 217Plus TM model was zeroed-out by scaling back to 1993, the base year in the model.

VI. CRYSTAL OSCILLATOR/DIGITAL CLOCK APPLICATION
The third application is for a crystal oscillator. The Wiley Electrical and Electronics Engineering Dictionary defines it as, ''An oscillator circuit that utilizes a piezoelectric crystal, usually quartz, to control the oscillation frequency.''

A. DEFINITIONS OF STATE AND FAILURE MODES
Crystal oscillator applications require the specification of two fundamental figures-of-merit (FOM), Startup (SU) time and frequency tolerance. The latter FOM is characterized by two components: i) frequency accuracy, which is the permitted deviation of the frequency from a standard value at 25 • C; and ii) stability which corresponds to the ability of the oscillator frequency to resist variation as a function of environmental operating parameters. The following is the definition of the operating states of the device from the perspective of these two FOMs.
First is the SU state, where oscillation is initiated by the active part of the circuit, e.g., the microcomputer, and the signal is produced. The SU time required to achieve a stable signal for commercial oscillators is generally in the microsecond range.
Next is the transmission state, where the generated signal continues until fault disrupted or the device shuts down and then restarts. The device is considered functional only during the transmission state and the Arcsine model is assumed to only apply to this state. In applications, this state entails the specification of the frequency accuracy in terms of engineering technical standards. Examples of such are the IEEE 802.3 or the radio frequencies regulations of the country in which the device is intended to operate. Failure mode data on a crystal oscillator is presented in Table 7 below. This data was provided by [9] and came from two proprietary sources. The first source was an agency that performed failure analysis and destructive physical analysis reports. The second source conducted component testing to simulate an airborne uninhabited environment (AUF).
In [9], mechanical and bond failure modes were interpreted as not applicable to the transmission operating environment and therefore eliminated by applying the renormalizing procedure given in [9]. The only failure modes construed as applying to the Startup state are No Operation and Unknown, while the Transmission state is assumed to exhibit all possible modes. Issues concerning SU time will be deferred until the final section of this article.

B. TABLE 8 ENTRY EXPLANATIONS
The field data source for parameter estimation for this device is [10], Vol. 1, operating in a naval environment (interpreted as naval sheltered). In the handbook, the device is titled a ''crystal unit, quartz,'' and the original data source was the US Navy maintenance data on Destroyer class ships.
As in the microcomputer analysis, the following table constructed from the NPRD data source is first explained. 1) N Pop.k : The data in this column refers to the acquisition of devices for deployment in the field by the Navy during each POR. Since there was no information on actual deployment dates, it is assumed all devices were commissioned at the start of the period. For calculation purposes, it is assumed that there is a spare on hand for immediate replacement of each device that fails in the field. 4) TOH×10 6 : This is the accumulated OH in units of millions of the total crystal oscillator device population per POR. Hence, for the first period OH 1 = 1.923432 × 10 6 OH .

5) NPRD Failed:
The number of failures per POR. 6) Average OH (per device): Same as in the microcomputer case where each entry is designated OH k for the kth POR. 7) Arcsine Parameter (L) Estimate:Analogous to the same argument employed in the microcomputer application, all device operating hours during any POR are assumed to be the average OH per POR. Unlike the microcomputer example, however, the operating population for each POR consist of subpopulations with different deployment ages. The data in Table 8 is directly related to additional part data given in Table 9 below. This data is obtained from the NPRD source, Section 4, Part Details, from which Table 8 was constructed. From the data in this table, it can be concluded that all the oscillators were manufactured according to MIL-STD-683. Since oscillator reliability was not directly dependent on the nominal frequency, it was assumed that the several populations' units could be considered homogeneous units in the reliability modeling and not mixtures.
The first POR covers 1675 Days or 4.59 CY starting 06/01/1991, during which there were no recorded failures. The POR empirical probability method used in the microcomputer application to compute the observed probability q 1 will be employed.
For the first POR, there were 138 devices deployed at project start ''0'', whose distribution support was (0, L), and the corresponding governing distribution In Table 8, there are no recorded failures. Therefore, the empirical probabilities approach used in the microcomputer example must be employed. The empirical probability is computed from the equation preceding (34) but now using the data in Table 8   resulting in a total population of Pop.2 = 529. There were seven recorded failures during this period, but how many were contributed by each subpopulation is unknown. Referring to the derivation of (6) and Table 8, the support corresponding to the distribution associated with N Pop.2 is (t 1 , L + t 1 ). The distribution applicable to N Pop2 is obtained from (6) as In the calculations that follow, an expression for the time span after a POR will be needed and is given as The following expression calculates the number of failures for the second POR From (41) and (43), the equation for the number of failures with respect to the second POR is found to be Referencing the data in Table 8, solving (45) for L 2 yields the resultL 2 = 3.218251 × 10 7 . In general, for the kth POR, the support for the governing distribution would be (t k−1 , L + t k−1 ) , in which case With the data in Table 8 and (46), the remaining lifetime parameters, L k can be calculated from the following equation From the calculated lifetime parameter estimates, the estimation of the lifetime parameter of the crystal oscillator from Table 8 is as followŝ Again, considering the trade-off with the exponential model, its failure rate parameter from Table 8 data can be estimated as followŝ

C. ENVIRONMENTAL CONVERSION AND CONFIDENCE LEVEL RESULTS
As stated above, the field environment pertaining to the oscillator was listed as ''naval,'' which was interpreted as Naval Sheltered (NS). To compare results with those in other technical publications, it is necessary to convert to a GBC environment, which will be the assumed operating environment. The conversion is accomplished in [11] as depicted in Table 6 Also, computing the 90% and 50% one-sided confidence levels of lifetime parameters gives results needed later in the analysis. Calculating the number of degrees of freedom of the Chi-square distribution begins with the development from In this case, the degree of freedom parameter is rounded up to two.
Next, referring to (19), to convert from LB to OLB, (1 + γ ) /2 is replaced by CL. In the conversion from UB to OUB, it follows from (20) that (1 − γ ) /2 is replaced by 1 − CL, resulting in the following Oscillator failure rate curves corresponding to the previously computed lifetime parameters L GBC and L OLB / GBC / 90% are compared and exhibited in Fig. 6 below.

D. RELIABILITY COMPARISONS AND INFANT MORTALITY
According to [12], in comparing microelectromechanical systems (MEMS) devices with crystals oscillators, the standard MTBF reliability FOM for the oscillators is cited as 30000 years compared with 130000 for the MEMS. According to [8], Table 2.3-1, the DC for a GBC environment is 80%. From (10) and the previous result forL GBC , the outcome is For the infant mortality lifespan, it is the case t min = 0.278945 × 2MTTF GBC = 2360 CY As noted in [12], a design life for a typical electronic product ranges from five-year to ten-year lifecycles, with long designs being fifteen years. Hence, based on preceding calculations, two conclusions can be drawn: i) the Arcsine model crystal oscillator MTTF is considerably less than the MTBF of the exponential modeled oscillator cited in [12]; and ii) the product lives of most electronic devices, components, etc., will not exceed the infant mortality phases of their lifecycles.

E. FREQUENCY VARIATION
Crystal oscillator frequency variation is comprised of several sources, listed as follows: 1) Frequency Tolerance is the initial plus/minus deviation of the crystal piezoelectric material itself from the nominal value at a specific standard temperature, usually 25 • C.
2) Frequency Stability -when referenced to the nominal frequency at 25 • C -is the maximum plus/minus frequency deviation from the accepted standard value over the temperature range relevant to the intended application. 3) Long Term Stability (Aging) refers to the gradual variation of oscillator frequency over time due to various physical effects of the oscillator circuit and housing, such as outgassing, mechanical mounting stresses, and leakage into the enclosure. These effects will be modeled in terms of the undesired parasitic or stray capacitances arising between adjacent device components. 4) Circuit Components, i.e., the lumped capacitances appearing in Fig. 7 and on which the oscillator resonance frequencies are directly dependent, exhibit parameter change-of-value failure modes, causing oscillator frequency variations. The total accuracy of a crystal oscillator as constituted by the effects of 1-4 above will be referred to as the frequency variation in what follows.
Based on the previous discussions, the total frequency variation can be expressed as In the crystal oscillator technical literature, the quantity ν s is referred to as the series-resonant-frequency of the oscillator. In the analysis that follows, the approximation ν s ∼ = ν 0 , where ν 0 is the nominal frequency of the device, will often be needed.

F. COMPONENT VARIATION CONTRIBUTION
The starting point is presenting a basic crystal oscillator circuit diagram taken from [13] as shown in the figure below. Referring to Fig. 7, the circuit parameters are defined as follows: i) C 1 is called the motional arm capacitance and represents the elasticity of the crystal and other physical characteristics transformed into electrical outputs via the piezoelectric effect; ii) L 1 , the motional arm inductance, represents VOLUME 10, 2022 the vibrating mechanical mass of the quartz; and iii) R 1 is the motional arm resistance which represents the frictional resistive losses within the crystal. The motional arm components model the mechanical behavior of the crystal when excited by the active circuit of the oscillator and, as such, are not actual physical circuit components. These motional arm parameters are usually not included in manufacturers' data sheets and hence must be obtained by user measurement or from the manufacturer.
Additional parameters shown in the circuit diagram are as follows: i) C 0 is the shunt capacitor and is associated with the packaging of the device, such as the crystal electrodes and the mechanical holder; ii) C X 1 and C X 2 are discrete or lumped capacitors necessary to integrate the quartz crystal into the active part of the circuit; iii) C Pin (not shown in Fig. 7) corresponds to the capacitance between the two pins of the integrated circuit part of the overall active circuit; and iv) C Stray is the parasitic capacitance due to the circuit board traces connecting the crystal terminals to the active circuit. Based on Fig. 7, the load capacitance C L is given as For a crystal oscillator unit, there are two resonance operating frequencies, ν s and ν a -e.g., [14]. The equation for ν s is As stated previously, this frequency is referred to as the series resonant frequency. The second resonance frequency is expressed as In the crystal oscillator technical literature, ν a is designated as the anti-resonance frequency.
In (51), the following standard approximation is used Using the above result with respect to (51), the expression for the anti-resonant frequency ν a is obtained In what follows, the fractional frequency variation due to circuit capacitances is expressed as It follows from (53) that the dependency of the fractional frequency variation on the variation of the load capacitance C L is expressed as follows ν (comp. tol.) To carry out example calculations, it is given from [14] that C 1 = 0.018 pF, from [15] C Stray = 2 pF and C Pin = 1 pF, and from [13], 3 pF ≤ C 0 ≤ 7 pF. Hence, given these values and data in Table 9 corresponding to ν 0 = 8.575 MHz, it follows from (5)

G. AGING CONTRIBUTION
In the analysis, a lumped capacitance is physically concentrated and governed by manufacturing processes and tolerances -these are the discrete C X 1 and C X 2 and for modeling considerations, the motional arm capacitance C 1 . The shunt capacitance, as well as C Pin and C Stray , are modeled as parasitic capacitances and, therefore, energy wasteful. Moreover, because of the unregulated energy processes these capacitances exhibit, it is assumed that the aging conditions are primarily concentrated in these non-lumped capacitances. Hence, the aging of the fractional frequency variation is expressed as Next, it is assumed that capacitor aging curves vary slowly enough per calendar year to justify the following approximations: Therefore, it follows from (49), (53), (55), and the preceding result that d dt Next, the rate of frequency variation due to aging is written as d dt With this, the previous result can be written as The values of C Pin and C Stray given in [15] are assumed. The load capacitance C L and the shunt capacitance C 0 values correspond to those listed in Table 9 for an 8.575 MHz device. If C X 1 = C X 2 = C X 1 is assumed, it follows from (49) Assuming a 1% tolerance for C X , its variance computes as From the two preceding results and the value and variation of the load capacitance C L , for an 8.575 MHz oscillator given in Table 9, it follows Next, using the value of the motion arm capacitance C 1 found in [14] and the previous results, it is calculated (53) .
In the technical literature on crystal oscillators and manufacturers' data sheets, the values specified for the aging rate -between 2 to 5 ppm/yr -are consistent with the above result. However, this result is contingent on the assumption that C 0 ∼ = C s .

H. CALCULATION OF ALLOWED FREQUENCY VARIATION
As stated above, applying a crystal oscillator requires a standard such as the IEEE 802.3. This standard pertains to Ethernet communication in a local area network (LAN) environment. It is adopted for the purpose of the following calculation example.
Incorporating the IEEE standard and a derating margin (DM) into (48) yields the following result In the case of the 8.575 MHz oscillators from Table 9 ν (freq. stability) ν s = 100 ppm.
Manufacturers offer various values for the variation in the nominal value of the material crystal frequency. According to [15], typical values would be ν (freq. tol.) ν s = 40 ppm, 100 ppm, 200ppm. It is generally recommended to set DM values between 20 to 80% in reliability engineering practice. The actual designation is based on operating conditions such as safety requirements. In the example, calculations show that To attain higher DM values requires adjusting either the manufacturer's Frequency Stability or the Frequency Tolerance parameters.
Oscillator operating malfunctions stemming from various capacitance sources of the circuitry and overall housing are well known -e.g., [13]. How these problems explicitly manifest themselves has been shown in the analysis through (55) and (56). Based on the analysis, it can be concluded that the primary frequency variation issue and the drift/aging failure mode in Table 7 are due predominantly to the parasitic capacitance associated with the packaging.

I. STARTUP ISSUES
That the SU time of quartz oscillators is in the millisecond range is a consequence of the fact that the initial conditions are dictated by noise (typically in the nanovolt range) and the negative resistance originating in the active component of the circuit (which mitigates the effect of crystal losses). Both quantities are initially minute. The final amplitude of the output of the oscillation signal is limited by the ability of the crystal to dissipate the power injected into it from the active branch of the circuit -see the Drive Level (µW ) column of Table 9.
Investigators [13] and [16] have identified two main physical mechanisms concerning SU time. First, the proper value of the load capacitors C X 1 and C X 2 , and second, the power supply rise time dV DD dt, respectively. What SU times are achieved is determined by circuit design in conjunction with the operating requirements of the economic and engineering environment, such as the IEEE Standard 803.2 referred to previously. Examples of this spread in the SU parameter t SU can be seen from two sources: i) commercial [17], which gives a value of 1.5 msec for a 25 MHz crystal; and ii) technical literature [18], which lists in their Table 4 SU times of 200, 400, 64, and 15 µ-sec for 24 MHz crystals each corresponding to different circuit design and energy activation techniques.
Once a frequency tolerance standard has been specified, the t SU parameter is estimated by assuming the conjugate variable relationship between time and frequency ( t) SU f ≥ 1-e.g., [7]. As another example, for a ν 0 = 24 MHz oscillator satisfying the IEEE standard, the estimate for the SU time uses the approximation ν 0 ∼ = ν s From the preceding expression, The above result is consistent with 200 and 400 µ-sec SU time values of [18]. In the case of the 64 and 15 µ-sec times, the IEEE standard is not met as the following calculation shows for ( t) SU = 64µ sec In the case of the commercial example with ( t) SU = 1.5 msec, The above calculation shows a considerable margin between the standards requirement and the estimated frequency deviation. Other manufacturers list SU times as 5 to 10 msec, which would mean an even more significant margin. Wider frequency variations generally mean production cost savings -a key design consideration -and the above results bore this out.
Although the conjugate variable computation method is relatively imprecise, it allows for a quantitative evaluation of the tradeoff between SU time and frequency tolerance. Moreover, the calculations show that the expected size of the SU time span t SU compared with the lifetime parameter L would be negligible. This being the case, it can be assumed from Table 7 that over the period t SU , only one failure would occur and only one failure mode would be relevant: No Operation. Therefore, the model for the oscillator SU time is a Bernoulli trial with parameter p = P [failure]. For the case of N oscillators, the probability of x SU failures would follow a binomial distribution with parameter p. As an example of extending this argument to an application environment of N oscillators covering both oscillator states, the following conditions are assumed: i) each device is turned off after the end of each application day and restarted the following day; ii) the statistical weights of the SU state and transmission state are w 1 and w 2 respectively; and iii) failed units observed on the kth day of operation N k are not replaced. Given these circumstances, a mixture probability model with the accumulated OH by the kth day given as t k = k (24DC) would be adopted The binomial distribution parameter p in (58) would have to be estimated in a separate statistical test in order to apply the above model.
It should be noted that the No Operation mode distribution value given in Table 7 cannot be adopted for the value of the statistical weight w 1 . The reason is that, as implied at the end of section V. (A), the No Operation mode could also apply to the Transmission state -e.g., a disruption in the power supply.
In the previous estimation of the lifetime parameter L, the possible contribution of the No Operation failure mode due to a SU failure was neglected due to differences in the operating environment. The data used in the previous estimation of L was from a naval sheltered operating environment where it was assumed the units tracked in logistical accounting were units that did not fail at SU at the start of any new day of operations. Those that may not have restarted were assumed to have been rejected and made no contribution to the operating hour totals. The situation, with respect to Table 7, was different as the data was acquired under test conditions. All unit failures, including those during the SU time, were accounted for in the construction of Table 7.

VII. DISCUSSION OF FINDINGS
Comparisons of the analyses results with corresponding values given in the technical literature are complicated because the operating environments or duty cycles with respect to the citations are often not provided. This was found to be the case with the crystal oscillator. To compare the 4230-year MTTF value with the quoted 30000-year MTBF, four assumptions are made. First, the construction differences between the original equipment manufacturer (OEM) of the MIL handbook data source and the OEM of the 30000-year MTBF oscillator did not affect the comparison of item reliabilities significantly. Secondly, the operating environment of the latter oscillator was GBC, and thirdly, the duty cycles were the same. Finally, because the operating environment that pertained to the analysis was NS, to equate the MTTF result with the MTBF one, it was necessary to convert to a GBC environment. Hence, the fourth assumption is that the conversion method prescribed by [11] was reasonably accurate. For the reasons stated above, agreement within a 95-percent difference between reliability analysis results, with test and field results found in the general reliability technical literature, and device data sheets is often considered satisfactory.
The analysis has shown how to calculate infant mortality time spans. Two consequences follow from this result. First, from Tables 5 and 6, the number of predicted failures and how they are distributed over time differ significantly from a corresponding exponential model analysis. Such results could inform managers on the construction of product warranties.
Second, the calculations have shown that at the 90% lower bound confidence level, the infant mortality period of the microcomputer would have been 714 years. The corresponding result for the crystal oscillator would have been 345 years. Hence, the normal lifecycles of these devices are within the infant mortality period. These results are at variance with the claim -initially stated in the introduction -that an item's useful life occurs after a relatively short infant mortality phase during which its failure rate is constant to an acceptable approximation. The previous assertion has often been given as a justification for using the exponential model.
In performing FMECAs, the proportionality of the failure modes is assumed to be constant throughout the lifespan of the equipment item. This assumption is a direct consequence of the exponential model failure rate being a constant. As indicated in the Arcsine model failure rate figures, the failure rate exhibits a wide degree of variation over the lifetime.
Calculations show that for the engine fan over the infant mortality period, the failure rate declines by 284%. Over the following same time span starting at the end of the infant mortality period, the rate increases by 20%. In the case of the crystal oscillator, the corresponding variations would be a 7368% decrease and 21% increase. Given the oscillator failure modes shown in Table 7 based on the previous analysis and calculations, the following interpretations can be made: i) during the infant mortality phase, the predominant failure mode would be No Operation with secondary contributions from Drift/Aging, Noise, and Shorted (interpreted as induced failure); ii) for the slowly varying failure rate period, the modes would be Drift/Aging, Noise, and Shorted; and iii) in the wear-out phase the Degraded and Drift/Aging modes would be dominant with secondary contributions from Noise and Shorted.
Hence, from the discussion above, except for the availability of unique data sets, the application of the Arcsine model would preclude performing an FMECA. However, carrying out an FMEA would still be a valid exercise.

VIII. CONCLUSION
This article has formulated an alternative to the exponential and Weibull reliability models as a single-parameter Arcsine function. Unlike symmetric bathtub curves shown throughout -e.g., the Wikipedia example at the beginning of the article and [11] -the mathematical derivation resulted in an asymmetric bathtub curve because of the unique minimum position determined to be 28% of the lifetime parameter. Consequently, the alternative model predicts an item's lifetime and infant mortality time spans as natural outcomes of modeling results. In the case of the exponential and Weibull model applications, calculations of these quantities are add-ons to the analysis and are ad hoc in nature.
In applying the alternative model, the analyst will encounter issues that do not typically arise when using a model like the exponential. As an example, consider the crystal oscillator. The failure rate is acquired as a direct lookup in the exponential case in [10]. The alternative model requires the solution of (58) after needed inputs -e.g., the statistical weights -are determined. Furthermore, whether the first term of the equation is included in the analysis depends on whether dead-on-arrival (DOA) incidents are considered valid failures. At the initial outset of oscillator operation, the device's active circuit excites the quartz crystal until stable and sustained oscillations are obtained within a specified SU time window. This parameter is often given in the manufacturer's data sheets. If the active circuit fails to perform this task, a No Operation fault mode is exhibited and a DOA (because SU time is in milliseconds) failure has occurred. However, as was the case in [10], DOAs are disregarded by the user in the evaluation of the operational performance of the device. Outcomes such as this are to be expected in applying the alternative model.