Monitoring time-between-events data: evaluating the effectiveness of a new control chart

1 Introduction

Traditional attribute control charts for discrete data, such as the p-, np-, c-, and u-charts, are designed to monitor processes where a quality characteristic follows a Binomial or Poisson distribution. Xie and Goh [1] and Chan et al. [2, 3], however, indicated that these attribute control charts do not work well in high-yield manufacturing processes like nearly zero-defect processes. As an alternative approach, cumulative control charts have been developed, which plot either the cumulative count of conforming (CCC) items or the cumulative quantity of conformities (CQC) over time. The charts are also interpreted as the time elapsed to the subsequent defect. Additional contributions and literature to the cumulative control charts are in [4–10].

In manufacturing environments where the frequency of events occurs at a constant rate, represented by a homogeneous Poisson process, the time until the occurrence of events follows an Exponential distribution. Chan et al.[2] introduced the CQC-chart for this setting: the cumulative quantity of conformities, Q, is plotted directly, and control limits are obtained via the inverse cumulative distribution function (CDF) of the underlying distribution rather than the traditional Shewhart 3σ approach, since Q is typically right-skewed under near-zero-defect conditions. In such processes, the cumulative quantity of conformities grows as successive inspections occur within the same process; consequently, the total cumulative quantity of conformities can exceed the upper bound of the original chart, which somewhat limits its usefulness in extremely high-yield manufacturing settings. Chan et al.[3] later proposed an alternative formulation in which the cumulative probability of conformities, F(Q), rather than Q itself, is plotted directly, remaining bounded within [0, 1] regardless of cumulative inspection count. Subsequent extensions of the CQC-chart to non-Exponential TBE distributions have, however, predominantly retained the original, Q-based formulation together with inverse-CDF-calibrated control limits: Xie et al.[5, 11] developed the Weibull CQC-chart for wear-and-tear and aging processes, and Sindhu et al.[12] developed the Rayleigh CQC-chart, both plotting Q (or a transform of Q used purely for display) against inverse-CDF-calibrated control limits rather than the [0, 1]-bounded F(Q). The GRCQC-chart proposed in this study follows this same, more widely adopted convention: it uses the cumulative quantity of conformities Q as its plotting statistic, with control limits calibrated via the inverse-CDF of the Generalized Rayleigh (GR) distribution, consistent with [2, 5, 12].

According to Surles and Padgett [13], the Burr Type X distribution, or the GR distribution, is a special case of the generalized Weibull family [14]. The hazard function of a GR distribution can have a non-monotonic bathtub profile, which is rare in commonly employed Gamma and Weibull distributions [15, 16]. The adaptability of the GR distribution makes it an excellent candidate for the failure behavior of components that exhibit early-life failures, followed by a decent period of stability and finally wear-out. The closed form of the probability density function, cumulative distribution function, and survival function also allow one to estimate parameters or analyze data sets with censored observations, which are important in industrial reliability studies or clinical survival analyses. These features of the GR distribution make it especially suitable for TBE data from high-yield manufacturing processes that may not be satisfactorily represented by conventional lifetime distributions with their complex and possibly monotonic failure mechanisms. Consequently, the GR distribution integrated into a cumulative quantity control chart, eases the complicatedness of process monitoring with a more varied reliability characteristic while sustaining a control-chart design that is analytically tractable.

The contribution of this study goes beyond just replacing the GR distribution with a pre-existing template for a CQC-chart. It is therefore important to clarify the specific contributions of the proposed approach. Firstly, since the GR is the distribution among the benchmark distributions considered in this study that can provide a non-monotonic bathtub hazard function, the GR has fundamentally distinct theoretical underpinnings from those of the alternative monotonic hazard functions (Exponential; Weibull with a specified shape parameter; Gamma; Log-Normal) previously used for TBE surveillance. Secondly, the GR distribution provides closed-form expressions for its cumulative distribution function (CDF) and inverse CDF, which are required for obtaining the analytical closed-form expressions for average run length (ARL) performance indicator, thereby avoiding the need to utilize Monte Carlo methods for most of the results provided, thus providing a real computational advantage over distributions whose quantiles cannot be expressed analytically. Third, and most importantly, this study presents additional methodological contributions that go beyond simply substituting a distribution into a CQC-chart. That is, instead of identifying the advantages of the GRCQC-chart as being unconditional, we demonstrate that it lacks robustness against distributional violations, meaning its deployment depends on a goodness-of-fit assessment.

A number of recent reviews and articles help position this contribution within contemporary trends in high-quality and TBE process monitoring. For instance, Ali et al. [17] provide a broad overview of control charts for high-quality processes (including CCC-, CQC-, and TBE-types) and their design trade-offs. Our work extends that literature potentially toward non-monotone bathtub hazard TBE processes, which their review points out are less explored. Furthermore, Zwetsloot, Mahmood, and Woodall [18] tackle the complexities surrounding TBE monitoring in a broader context that includes the choice of performance metric (ARL vs. ATS) and multivariate extensions of TBE data. Mahmood and Xie [19] review zero-inflated and high-yield process monitoring models, which relates closely to the problem setting (namely, near-zero defects rates) we study to motivate our high-yield TBE processes. Saghir and Lin [20] similarly review dispersed count-data control charts, which is a parallel literature that addresses over- and under-dispersion of discrete counts rather than continuous TBE data. Riaz, Abbas, and Mahmood [21] develop a distributional relationship between the geometric and binomial formulations of the CCC-chart.

The rest of the article is organized as follows. Section 2 discusses the GR distribution and some of its important characteristics. Section 3 presents the design of the proposed GRCQC-chart and the seven benchmark CQC-type charts. Section 4 defines the performance indicators that will be used in the comparative evaluation. Section 5 reports the comparative findings. Section 6 examines the robustness of the proposed chart under distributional assumption violations. Section 7 investigates the uncertainty of Phase-I estimation for the proposed and benchmark control charts. Section 8 includes two case studies as illustrations. Finally, Section 9 concludes the article with a summary of the main findings, recommendations, and directions for future research.

2 Generalized Rayleigh (GR) distribution

The Generalized Rayleigh (GR) or the Burr Type X distribution was introduced by Surles and Padgett [13] and has since become a well-established model in life testing and reliability engineering [22]. It is particularly useful for modeling lifetime and strength data due to its flexibility and robustness. The distribution is parameterized by a scale parameter λ and a shape parameter θ. Let the random variable Q follow a GR distribution with parameters λ and θ. Then its cumulative distribution function (CDF) is

F(q;λ,θ)=(1-e-(λq)2)θ, q>0,(1)

its probability density function (PDF) is

f(q;λ,θ)=2θλ2qe-(λq)2(1-e-(λq)2)θ-1, q>0,(2)

its survival function is

S(q;λ,θ)=1-(1-e-(λq)2)θ,(3)

and finally its hazard function is

h(q;λ,θ)=f(q;λ,θ)S(q;λ,θ)=2θλ2qe-(λq)2(1-e-(λq)2)θ-11-(1-e-(λq)2)θ.(4)

Raqab and Kundu [15] showed that the PDF is a decreasing function of q when θ ≤ 0.5 and exhibits a right-skewed unimodal shape when θ>0.5. Furthermore, the non-monotonic hazard function displays a bathtub shape depending on the shape parameter θ. In this section, we introduce the notation used throughout the article to simplify the interpretation of the mathematical expressions. For the convenience of the readers, the list of notation is in Table 1.

Table 1. Summary of the notation.

3 Design of the proposed and benchmark control charts

3.1 Design of the GRCQC-chart

Let Q denote the cumulative quantity of conformities inspected between two successive nonconforming events in a process governed by a GR distribution with scale parameter λ and shape parameter θ. Following the original CQC-chart framework of Chan et al.[2], the plotting statistic is the cumulative quantity of conformities Q itself, and the control limits are calibrated via the inverse-CDF of the GR distribution. For a two-sided chart with nominal false-alarm rate φ, the control limits are obtained by equating the CDF to the false-alarm probability between the two tails. Specifically, it is divided equally between the two tails for the two-sided chart, whereas for one-sided charts, the entire false-alarm probability is assigned to one tail. The upper control limit (GRQU), central line (GRQC), and lower control limit (GRQL) are

GRQU=F-1(1-φ2;λ0,θ0),GRQC=F-1(0.5;λ0,θ0),GRQL=F-1(φ2;λ0,θ0),(5)

where F⁻¹(·) is the quantile function of the GR distribution. Note that φ is used consistently for both the one-sided and two-sided structures; each one-sided chart uses φ in the corresponding tail. The in-control parameter values (λ₀, θ₀) are assumed to be fixed. The process are classified as follows: (i) stable if GRQL ≤ Q ≤ GRQU; (ii) deteriorating if Q<GRQL; (iii) improve if Q>GRQU; and (iv) two-sided out-of-control where the process exceeds either control limit. The average run length (ARL) expressions for the GRCQC-chart are derived using the CDF of GR distribution, which yields analytical closed-form expressions. For a scale shift of magnitude δ, such that λ = λ₀/δ, the ARL for signals above GRQU and below GRQL, and for the two-sided structure are, respectively,

ARL+=11-F(GRQU;λ,θ0),(6)

ARL-=1F(GRQL;λ,θ0),(7)

ARL=1F(GRQL;λ,θ0)+[1-F(GRQU;λ,θ0)].(8)

These formulas are analogous to those introduced by Chan et al.[2] for the Exponential CQC-chart and are straightforward applications of the general result that ARL = 1/p, where p is the probability of a signal at any given observation [23].

3.2 Benchmark CQC-type control charts

3.2.1 Exponential CQC-chart

The traditional CQC-chart was originally proposed by Chan et al.[2] and modified by Chan et al.[3]. It was designed for TBE data arising from a homogeneous Poisson process, under which interarrival times follow an Exponential distribution with constant rate μ. It has been commonly used in reliability analysis of components and manufacturing systems [6, 24, 25]. Its CDF is

F(q;μ)=1-e-μq.(9)

Control limits are obtained by the inverse-CDF approach with in-control rate μ₀ = 1.

3.2.2 Weibull CQC-chart

When the event-occurrence rate is non-constant, for instance due to aging or wear-and-tear conditions, the Weibull distribution with shape parameter β and scale parameter η provides a flexible alternative. Xie et al.[5] introduced the CQC-chart based on the Weibull CDF,

F(q;β,η)=1-exp[-(qη)β].(10)

Control structure and ARL expressions are derived analogously to the original CQC-chart. Two fixed-shape variants are examined throughout this study, β = 0.5 and β = 1.5, each with in-control scale η₀ = 1.

3.2.3 Rayleigh CQC-chart

The Rayleigh distribution is a special case of the Weibull distribution with shape parameter β = 2 and scale parameter η=2λ. It is particularly useful when failure rates exhibit a unimodal profile. Sindhu et al.[12] introduced the RCQC-chart, whose CDF is

F(q;λ)=1-e-λ2q2.(11)

Its control limits and ARL expressions are obtained by following the original CQC-chart, with in-control scale parameter λ₀ = 1 (equivalently Weibull shape 2, scale η0=2).

3.2.4 Log-Normal CQC-chart

The Log-Normal distribution is appropriate when the logarithm of the failure time follows a normal distribution, as commonly occurs in fatigue and degradation processes. Its CDF is

F(q;μ,σ)=Φ(ln q-μσ),(12)

where Φ(·) denotes the standard normal CDF, μ is the location parameter and σ is the shape (dispersion) parameter on the log scale. The parameter values of μ₀ = 1.2 and σ = 0.8 were selected to represent a moderately skewed lifetime distribution, consistent with the parameter settings commonly considered in reliability modeling studies (Nelson, 1990; Meeker and Escobar, 1998). Control limits and ARL expressions are derived analogously to the original CQC-chart.

3.2.5 Gamma CQC-chart

The Gamma distribution, parameterized by shape k and scale υ, accommodates increasing (k>1), decreasing (k < 1), and constant (k = 1) hazard profiles, making it broadly applicable in waiting-time and failure modeling. Its CDF involves the lower regularized incomplete gamma function,

F(q;k,υ)=γ(k,q/υ)Γ(k).(13)

For the comparative analysis, the shape and scale parameters were set to k = 2 and υ₀ = 1, respectively, yielding a flexible increasing-hazard lifetime model commonly considered in reliability studies. Control limits and ARL expressions are obtained by following the original CQC-chart.

3.2.6 Log-Logistic CQC-chart

The Log-Logistic distribution is frequently employed in survival analysis and reliability studies when the hazard function is non-monotone, initially increasing and then decreasing. Its CDF is

F(q;a,b)=11+(q/a)-b.(14)

For the present study, parameter values a₀ = 2.5 and b = 2.0 were selected to generate a non-monotone hazard profile and provide a representative reliability scenario (Bennett, 1983; Lawless, 2003). Control limits and ARL are derived analogously to the original CQC-chart.

3.2.7 Inverse Gaussian CQC-chart

The Inverse Gaussian (Wald) distribution was included because of its established applications in first-passage-time modeling, degradation analysis, and reliability studies. Its CDF is

F(q;μ,τ)=Φ[τq(qμ-1)]+exp(2τμ)Φ[-τq(qμ+1)],(15)

where μ is the mean parameter and τ is the shape parameter. The parameters μ₀ = 2.5 and τ = 2.5 were selected to produce a representative Inverse Gaussian scenario consistent with its established use in reliability and first-passage-time modeling (Chhikara and Folks, 1989; Whitmore, 1979). This is the one benchmark chart among the seven for which the control limits themselves require numerical computation (due to no closed-form solution).

Control limits and closed-form ARL expressions for each of the benchmark charts are given in Appendix. It is important to note that the closed-form ARL expressions depends on both the shift in scale parameter and distribution-specific parameters controlling shape or dispersion. We chose a single parameter setting of each benchmark distribution (used through Sections 5 to 7) as one operating scenario, rather than exhaustive search across the full parameter space. This implies, readers should not interpret the observed differences as being independent of the selected parameter values.

4 Performance indicators

It is difficult to accurately specify the shift magnitude of the process in practical applications. Therefore, control charts should perform satisfactorily for a wide range of possible shifts, rather than specifically being optimally designed for a certain shift magnitude, as demonstrated by [26, 27]. Three performance indicators were used to measure the shift detection capability of a control chart. These include the average run length (ARL), the average extra quadratic loss (AEQL), and the average ratio of ARL (ARARL) [9, 10, 27–30].

4.1 Average run length (ARL)

The average run length (ARL) counts the expected number of samples plotted until a control chart signals. Under in-control conditions, it represents the expected number of samples before a false alarm occurs (denoted by ARL₀). This implies ARL₀ = 1/φ, where φ is the false-alarm probability. Under out-of-control conditions, ARL measures the expected detection delay after a process shift. For TBE charts, separate ARL values are computed for shifts in deterioration, and shifts in improvement across three states of the process.

4.2 Average time to signal (ATS)

ARL counts the expected number of samples plotted until a signal. However, it does not directly indicate the quantity or the time that accumulates until the signal. Thus, Zwetsloot et al.[18] specifically recommend reporting average time to signal (ATS) for TBE monitoring if there are different scales for the inter-arrival times and/or magnitude shifts in the distribution among different charts. We discuss ATS here in direct response to this recommendation. Because Phase II observations are i.i.d., the run length is a geometric random variable, and by Wald's Identity the expected cumulative quantity accumulated until a signal is exactly

ATS(δ)=ARL(δ)×μ(δ),(16)

where μ(δ) is the mean of Q under the shifted distribution. A caveat governs how this raw ATS should be interpreted across charts. For instance, a chart having a larger assumed mean shows a larger raw ATS irrespective of the ARL behavior. We, therefore, also present relative ATS, RATS(δ) = ATS(δ)/ATS(1), that is the fair basis for cross-chart comparison; raw ATS remains meaningful within a single chart. Furthermore, the ARL curves in Section 5 are visually indistinguishable from (RATS) curves on every chart and every shift that we examined, apart from a minor rescaling. We show below that this is a mathematical necessity under the assumptions that are already intact throughout this study. Establishing this formally lets us state precisely what RATS does and does not add beyond the ARL results.

4.3 RATS–ARL rank equivalence

Let there be K control charts indexed by k = 1, …, K, each monitoring the same quantity Q under a possible shift δ in scale parameter. Let ARL_*k*(δ) denote the average run length of chart k at shift δ, and μ_*k*(δ) = E_δ[Q] the mean of Q under chart k's assumed distribution at that shift. Furthermore, we have the following two assumptions:

(A1) Pure scale family. For each chart k,

μk(δ)=δ·μk(1) for all δ>0.(17)

(A2) Common calibration. All K charts are designed to the same nominal false-alarm rate φ, so

ARLk(1)=1φ=ARL0 for every k.(18)

Under (A1)–(A2), define RATS_*k*(δ) = ATS_*k*(δ)/ATS_*k*(1). Then for every δ>0 and every pair of charts i, j,

RATSi(δ)≤RATSj(δ) ⇔ ARLi(δ)≤ARLj(δ).(19)

That is, ranking charts by RATS at any fixed δ produces exactly the same ordering as ranking them by ARL.

By Wald's Identity, since the run length N is a stopping time on the i.i.d. sequence Q₁, Q₂, … under the shifted distribution,

ATSk(δ)=ARLk(δ)·μk(δ).(20)

By (A1),

μk(δ)=δ·μk(1),(21)

so

RATSk(δ)=ARLk(δ)·δ·μk(1)ARLk(1)·μk(1)=δ·ARLk(δ)ARLk(1).(22)

By (A2), ARL_*k*(1) = ARL₀ for every k, giving

RATSk(δ)=δARL0·ARLk(δ).(23)

For fixed δ, δ/ARL₀>0 is a positive constant not depending on k. Multiplying every element of {ARL_*k*(δ)} by the same positive constant preserves their relative order. Therefore, the chart minimizing (or maximizing) ARL(δ) among the K charts also minimizes (or maximizes) RATS(δ), at every δ. Consequently, under (A1)–(A2), RATS cannot serve as evidence independent of ARL. It is the same comparison re-expressed in cumulative-quantity units/time. Furthermore, in case of violation of either assumption stated above, ARL- and RATS-based rankings diverge.

4.4 Average extra quadratic loss (AEQL)

The average extra quadratic loss (AEQL) is the second indicator to know how well a control chart performs in detecting a wide range of possible shift magnitudes [δ_min, δ_max]. This is done by averaging each ARL value using the squared shift magnitude as weights:

AEQL=1δmax-δmin∫δminδmaxδ2ARL(δ)dδ, δmin<δ<δmax.(24)

An important clarification on interpretation of AEQL is required for the one-sided chart. In this study, we compute AEQL⁻ and AEQL⁺ over the complete shift range rather than just along their targeted directions. Consequently, “smaller is better” applies only when comparing charts within the state they are designed to monitor. Therefore, comparisons using AEQL⁻ should be restricted to deterioration shifts (δ < 1), whereas comparisons using AEQL⁺ should focus on improvement shifts (δ>1). Large AEQL values reflect the intended one-sided chart behavior which is designed to detect shifts in one specific direction while being relatively insensitive to shifts in the opposite direction.

4.5 Average ratio of ARL (ARARL)

The average ratio of ARL (ARARL) is obtained by averaging the ratio of the benchmark chart's ARL to that of the proposed chart over the specified shift range:

ARARL=1δmax-δmin∫δminδmaxARLbenchmark(δ)ARLproposed(δ)dδ.(25)

If this indicator has a value of 1.0 then both charts require approximately the same number of samples to detect a given shift. If the value is larger than 1.0, it means that the benchmark chart requires a larger number of samples on average to detect a given shift than does the proposed chart.

5 Performance evaluation

An important methodological clarification about Section 5 is worth stating explicitly at the outset. No Monte Carlo data are generated at any stage of the numerical comparisons that follow in this or the next section. Instead, each of the nine control charts considered in this study is evaluated using a fixed set of in-control parameter values without performing a Phase I parameter estimation. Furthermore, Phase II observations are assumed i.i.d. so the expected number of samples until a signal occurs is a geometric random variable. Therefore, by substituting the fixed in-control control limit(s) directly into the closed-form CDF of the respective distribution, one obtains the closed-form ARL expressions (see Appendix). Performance is evaluated via three indicators listed in the previous section under four nominal false-alarm rates, φ∈{0.0005, 0.00135, 0.0027, 0.05} and a wide range of process shift values δ∈{0.10, 0.30, 0.50, 0.70, 0.90, 1.00, 1.50, 2.00, 2.50, 3.00}, for three monitoring states. In each monitoring state, δ < 1 indicates that the process is deteriorating (i.e., the scale parameter has decreased, causing the interval between events to shorten), while δ>1 indicates that the process is improving (i.e., the scale parameter has increased, and event intervals have increased). The associated in-control average run length (ARL₀ = 1/φ) is therefore approximately {2000, 740, 370, 20} for the four false-alarm rates, respectively.

5.1 Performance in lower state

The lower-sided average run length (denoted ARL⁻) for the proposed and the benchmark CQC-type charts are given in Figure 1. For each deterioration shift (δ < 1), the GRCQC-chart has the lowest ARL⁻ value, showing early detection of deterioration in the process. In the in-control case (δ = 1), all charts have the same ARL⁻ value, calibrated to the given false-alarm rate. As anticipated, ARL⁻ increases when a lower-sided chart signals in the improvement region (δ>1), since it was not designed to do so. The relative efficiency of the proposed and benchmark charts is summarized by ARARL⁻ in Figure 2, with the GRCQC-chart used as the baseline (ARARL⁻ = 1). Each of the benchmark CQC-charts has ARARL⁻ values larger than one for all combinations of false-alarm rate; the proposed chart therefore requires fewer observations on average to indicate a signal than each of the benchmark charts. The AEQL⁻ for each chart is reported in Figure 3, calculated over the full shift interval [0.1, 3.0]. Although the GRCQC-chart has a higher AEQL⁻ value than each of the other benchmark charts, this does not necessarily mean that the GRCQC-chart performs worse: since AEQL⁻ is large due to the high ARL⁻ values for improvement shifts (δ>1), these large values are expected for a properly performing lower-sided chart, one that infrequently signals during process improvement.

5.2 Performance in upper state

As depicted in Figure 4, the upper-sided average run length (ARL⁺) for the proposed GRCQC-chart and benchmark CQC-type charts is calculated for four nominal false-alarm rates and a range of process shifts. As anticipated, the proposed chart exhibits the largest ARL⁺ for deterioration shifts (δ < 1), indicating that the upper-sided chart will not signal process deterioration. In contrast, for improvement shifts (δ>1), the proposed chart provides the smallest ARL⁺ values, thus detecting process improvement quickly. At the in-control state (δ = 1), all charts have the same ARL⁺, meaning that they are calibrated to that false-alarm rate. As shown in Figure 5, the relative performance is displayed via ARARL⁺, where the proposed GRCQC-chart serves as the reference (ARARL⁺ = 1). Across all false-alarm rates, the ARARL⁺ values given by the benchmark charts exceed one, which indicates that more observations are required, on average, to detect process improvements than the proposed chart. Although the size of the performance advantage varies with the false-alarm rate, the rank ordering of the benchmark charts has largely remained unchanged. Furthermore, as shown in Figure 6, AEQL⁺ is computed for the upper state of the process across the full shift interval [0.1, 3.0]; its interpretation is analogous to AEQL⁻. Collectively, Figures 4–6 show that the proposed chart detects process improvements earlier than the benchmark charts.

5.3 Performance in both states

Figure 7 displays the two-sided log(ARL) profiles for the proposed GRCQC-chart and benchmark CQC-type control charts across varying false-alarm rates and process shifts. A common characteristic among all charts in the four panels is the inverted-V shape of the two-sided ARL profiles. The results show that the proposed chart has the steepest decline in ARL on both sides of the in-control point, indicating early detection of both deterioration and improvement shifts. The Rayleigh and Weibull (β = 1.5) charts are found to have similar overall performance, while the Weibull (β = 0.5) and Log-Normal charts always show the late detection, especially for larger improvement shifts. Figure 8 shows the relative performance by means of ARARL. The results show that benchmark charts need more observations to detect a shift than the proposed chart. The performance advantage may decline as the nominal false-alarm rate increases, but the relative ranking of the competing charts does not change. Finally, Figure 9 shows the AEQL findings when the process operates across both states. The results differ from when the process was in only the lower or upper state, because AEQL balances the average losses across both the upper and lower process states.

6 Robustness analysis

A natural question raised by the comparisons in Sections 5.1 to 5.3 is whether the performance advantages of the GRCQC-chart are attributable to the assumed Generalized Rayleigh (GR) distribution rather than to intrinsic methodological advantages. This section, therefore, investigates the robustness of the proposed chart to the underlying distribution violation by assuming the true in-control process follows one of six alternative TBE distributions already used in Section 3.2 (Weibull β = 0.5, Weibull β = 1.5, Gamma, Log-Normal, Log-Logistic, Inverse Gaussian), while the monitoring chart assumes a GR distribution. To isolate the effect of distributional form alone, the GR shape parameter is held fixed at θ = 3; it is not estimated, because doing so would necessitate an additional Phase I parameter-estimation step, contradicting the estimation-free framework used throughout Section 5. The only parameter changed relative to the true process is the GR scale parameter λ, which is adjusted so that the mean TBE of the GR distribution equals that of the true process:

λ=mean(GR(1,3))mean(true family).(26)

Thus, differences in performance are due entirely to differences in the distributional form and not to differences in the mean of the process. Since the data points continue to be independently and identically distributed, the run length is again geometrically distributed, and its expected value can therefore be determined analytically as in Section 5. The resulting in-control ARLs for φ = 0.0027, with a nominal ARL of 370.4, are reported in Table 2.

Table 2. The in-control ARL at φ = 0.0027 (nominal ARL = 370.4) under each true family.

The results indicate that matching solely the average TBE is not adequate to maintain the desired false-alarm rate under distributional violation. The actual in-control ARLs across the six distributions vary from being as low as 1.56 to 5.84, merely 0.42% to 1.58% of their nominal target. As a result, the GRCQC-chart would produce false alarms far more frequently, even when the process is in control. Differences in tail behavior are the potential cause of this deterioration, rather than differences in central tendency. As an instance, the Weibull distribution with β = 0.5 has a heavy right tail with survival probability falling off approximately as exp(-x). By contrast, the mean-matched GR distribution (θ = 3; λ = 0.6452) has a much thinner/lighter tail that decreases approximately as exp(−x²). As a result, the GR chart's upper control limit (about 4.3) is notably lower than the Weibull one (about 43.7), causing many in-control Weibull observations to cross the GR limit, which significantly inflates the false-alarm rate. Similar mechanisms mark the decline in ARL performance for the other distributional forms as well. The recommendation for implementing a GRCQC-chart is therefore: control limits should not be constructed without a preceding goodness-of-fit assessment.

6.1 Distributional robustness beyond the GRCQC-chart

The sensitivity of the GRCQC-chart to the underlying distribution demonstrated above raises a natural follow-up question: is this sensitivity specific to the proposed chart, or does it reflect a broader limitation of all benchmark CQC-type charts discussed in this study? A complete inquiry would involve taking every chart under every alternative distribution into account; instead, we perform a bounded cross-chart analysis. Each chart was designed using its respective design distribution outlined in Section 3.2, with the shape parameter held constant and the scale parameter calibrated using the mean-calibration method of Equation 26, mirroring exactly the procedure used for the GRCQC-chart above. The in-control ARL was then calculated under several alternative true distributions for each chart. The results are presented in Table 3.

Table 3. Bounded cross-chart analysis.

Two key findings emerge from this bounded cross-chart analysis. First, all CQC-type charts considered in this study are sensitive to their underlying distributional assumption; the sensitivity observed for the GRCQC-chart above is not unique, and no benchmark CQC-type chart evaluated here can provide reliable performance without prior validation of the assumed distribution. Second, the magnitude and direction of performance degradation are not determined solely by the overall difference between the assumed and true distributions; instead, they are primarily driven by differences in tail behavior. For distributions with more similar tail behavior (the non-GR pairs in Table 3), degradation continues but is substantially less marked, with in-control ARLs taking values between 15% and 68% of the nominal. These results therefore reinforce, rather than contradict, the findings above. Collectively, the take-home message of the robustness study is that not only the proposed GRCQC-chart, but all benchmark CQC-type charts, suffer from the underlying distributional assumption violation, with the extent of degradation depending largely on the discrepancy between the tail behavior of the assumed and true distributions. This demonstrates the importance of a goodness-of-fit assessment before the construction of control limits, irrespective of the monitoring chart in hand.

7 Phase I estimation uncertainty

Since all the preceding comparisons assume that the in-control parameters are known exactly. This raises a natural question: is the GRCQC-chart more, less, or similarly vulnerable to Phase I estimation uncertainty than the seven benchmark CQC-type charts examined in this study? To this end, this section estimated, via Monte Carlo simulation, the false-alarm-rate inflation that finite-sample Phase I estimation introduces for all charts investigated in this study. We adopted the identical protocol for every chart like in the previous comparisons. For each of the nine charts, and for Phase I sample sizes n∈{20, 30, 50, 100, 200, 500, 1, 000}, we selected 1,000 independent Phase I samples of size n from the chart's true in-control distribution (parameter values as given in Section 3.2), fitted distributional parameters by maximum likelihood estimation (MLE) since we do not know them in advance. Closed-form maximum-likelihood estimators were used for Exponential, Rayleigh, Log-Normal, Inverse Gaussian; numerical optimization of the log-likelihood was used for Weibull, Gamma, Log-Logistic, GR. Hence, the control limits of all charts are built in via the fitted (not true) values at φ = 0.0027, and the actual in-control ARL that these estimated limits produce (under the chart's true in-control distribution) are computed. Every fitting distribution was independently verified to accurately recover known parameters from large simulated samples (n=20,000) prior to use them. All 63 (chart × sample size) combinations had a roughly 100% convergence during all 1,000 replications. Table 4 summarizes the results by reporting, for each chart and Phase I sample size, the probability that the estimated control limits produce an actual in-control ARL less than half the nominal target (i.e. at least a twofold inflation of the intended false-alarm rate) and the mean actual ARL as a percentage of nominal in Figure 10.

Table 4. Phase I estimation-uncertainty by chart and sample size (probability of ≥ 2× false-alarm inflation; N = 1, 000 Monte Carlo replications per cell; φ = 0.0027, nominal ARL₀ = 370.4).

Two findings emerge. First, the GRCQC-chart is not distinguishable from the benchmark charts in its sensitivity to Phase I estimation uncertainty. At every sample size, its risk of ≥2 × false-alarm inflation falls within the range spanned by the benchmark charts (for example, at n = 20: 0.486 for GRCQC, vs. 0.444–0.554 across the six two-parameter benchmarks), and by n = 500 every two-parameter chart, GRCQC included, has converged to a risk below 2.5%. The vulnerability documented for the GRCQC-chart alone is therefore not a weakness specific to the proposed chart; it is shared by every benchmark CQC-type chart. Second, and more informatively, risk at small sample sizes is determined primarily by the number of parameters requiring estimation, not by which distributional family is assumed. The two single-parameter charts, Exponential and Rayleigh, show substantially lower inflation risk than every two-parameter chart at every sample size examined (e.g. at n = 100: 0.030–0.032 for the single-parameter charts vs. 0.118–0.235 for the two-parameter charts) a direct consequence of estimating one quantity from the data rather than two, which reduces sampling variance in the fitted control limits. This distinction is visible as a clear, persistent separation between the two single-parameter curves and the remaining seven curves in Figure 10, rather than any chart-specific pattern. These results refine, rather than contradict, the practical recommendation of Section 7: a Phase I reference sample of at least 200–500 observations is advisable before deploying any two-parameter CQC-type control chart, GRCQC included, with somewhat smaller samples tolerable for the single-parameter Exponential and Rayleigh charts specifically.

8 Case studies

8.1 Simulated integrated circuit manufacturing data

To illustrate the practical operation of the GRCQC-chart, we present a hypothetical example motivated by integrated circuit (IC) chip manufacturing, a prototypical high-yield process in which maintaining near-zero defect rates is critical. Quality inspections are performed on each chip, and the cumulative quantity inspected (Q) is reset to zero following each detected defect. This application uses in-control parameters λ₀ = 0.5, θ₀ = 2, and φ = 0.10. The resulting control limits for the proposed and some of the benchmark CQC-type charts, after applying the logarithmic transformation log₁₀(Q) to stabilize the display, are shown in Table 5.

Table 5. Control structure for the simulated integrated circuit manufacturing data.

Table 6 presents selected monitoring results including the cumulative quantity, process indication, and reset events. The notation follows Chan et al.[3]: “n.d.” = no decision (below the center line following a reset), “i.c.” = in-control, and “im.” = improved. Figure 11 compares the proposed and three benchmark CQC-type charts on this dataset.

Table 6. Selected monitoring results for the simulated IC manufacturing dataset.

The GRCQC-chart, as expected, provides the early detection of improvement shift; this was first detected at observation 15, at a time when log₁₀(Q) = 2.18 (Q≈150 cumulative conforming units), or ten sampling periods after the process restart. Detection by the RCQC-chart occurs six samples after that, at observation 21 with Q≈240. The Weibull (β = 1.5)-chart does not indicate until observation 90 with Q≈1, 275, or 75 sampling periods after the GRCQC-chart indicates improvement. In high-yield manufacturing processes, delayed detection has direct cost consequences: each additional inspection period under an unconfirmed process state means more units produced before the improvement is verified. Notably, the CQC-chart (Exponential) never signaled over the entire data set. Its upper control limit of log₁₀(Q) = 4.4722 (or Q≈29, 668) exceeded the greatest number of cumulative units ever recorded during data collection (at log₁₀(1, 608) = 3.21). There is a clear theoretical basis for this: the Exponential distribution is characterized by a constant hazard function, so its probability density function assigns equal probabilities to each side of the median, creating a much higher upper control limit than can be reasonably expected from the upper-tail behavior of a count distribution under the GR model.

8.2 Carbon fiber tensile strength data

The second application uses 66 tensile strength measurements of single carbon fibers over a gauge length of 10 mm, a dataset that has been extensively studied in the reliability literature [13, 15]. This dataset is well suited to evaluating cumulative control charts because prior analyses have established both the relevance of monitoring fiber strength variability and the statistical properties of the measurements. The data are:

0.39, 0.85, 1.08, 1.25, 1.47, 1.57, 1.61, 1.61, 1.69, 1.80, 1.84, 1.87, 1.89, 2.03, 2.03, 2.05, 2.12, 2.35, 2.41, 2.43, 2.48, 2.50, 2.53, 2.55, 2.55, 2.56, 2.59, 2.67, 2.73, 2.74, 2.79, 2.81, 2.82, 2.85, 2.87, 2.88, 2.93, 2.95, 2.96, 2.97, 3.09, 3.11, 3.11, 3.15, 3.15, 3.19, 3.22, 3.22, 3.27, 3.28, 3.31, 3.31, 3.33, 3.39, 3.39, 3.56, 3.60, 3.65, 3.68, 3.70, 3.75, 4.20, 4.38, 4.42, 4.70, 4.90.

Maximum-likelihood fitting of the GR distribution to this dataset (n = 66) gives λ^=0.4381 and θ^=2.3483. The fitted hazard function for this data set has an estimated value for θ at θ^=2.3483 > 1, making it monotonically increasing (consistent with progressive fiber failure due to increased tensile loads), rather than having a non-monotonic “bathtub” shape. This is precisely similar to the type of distributional violation considered in the robustness study of Section 6. Thus, its advantage here should not be attributed to a non-monotonic bathtub hazard shape. Therefore, the deployment of the GRCQC-chart for this particular dataset does not simply depend on the existence of a bathtub-shaped hazard function (i.e., θ < 1) but rather on the ability to demonstrate that the GR distribution fits the data adequately. This goodness-of-fit assessment step is a prerequisite to the construction of the control limits for the use of any CQC-type charts being studied here. Figure 12 presents a full picture of goodness-of-fit diagnostics.

To begin with results in Figure 12, there was a lower AIC (176.14) for the Weibull distribution than for the GR model (AIC = 181.27). This slight increase in AIC for the GR model does not result in inferior monitoring performance. As indicated by its tighter upper control limit (UCL = 1.3785 vs. UCL = 1.4400 using the Rayleigh/Weibull family), it yields better early improvement signals, as shown in this case study. This is precisely the type of distributional mismatch considered in the robustness study of Section 6. Although the fitted GR distribution for this dataset (θ^=2.3483) exhibits a monotone increasing hazard rather than a non-monotonic bathtub-shaped hazard, it provides a an opportunity to evaluate how robust the proposed GRCQC-chart remains when its original motivating hazard-shape assumption is violated. Thus, its advantage here should not be attributed to a matching bathtub hazard shape. The GR model still provides a good statistical representation of the tensile strength data (the AIC value for the GR model = 181.27; AIC for Weibull model = 176.14; the results of KS test and AD test are not rejected; see Figure 12). The parameters for three benchmark distributions were estimated from the same data using their respective maximum likelihood procedures, so that each chart is calibrated to the data under its own distributional assumption. The resulting control limits for the proposed and three of the benchmark CQC-type charts, after applying the logarithmic transformation are shown in Table 7.

Table 7. Control structure for the Carbon fiber tensile strength data.

Figure 13 illustrates how the proposed and benchmark CQC-type charts performed on the tensile strengths of carbon fiber data at φ = 0.0005. There is a significant differences in the detection performance among the four charts. Both the proposed GRCQC-chart (upper control limit 1.3785) and the RCQC-chart (upper control limit 1.4400) indicate an increase in the tensile strength distribution at measurement #5, evidenced by the fact that all 61 subsequent measurements (#6 through #66) exceed their respective upper control limits. Both charts therefore identify an upward trend in tensile strength that persists through nearly all subsequent measurements. The CQC-chart, by comparison, indicates no change until measurement #62, which had a tensile strength of 4.20 against an upper control limit of 4.1473, a delay of 57 measurements compared to the GRCQC-chart, with only five of the sixty-six measurements exceeding its upper control limit. The Exponential chart is not able to identify the large improvement in tensile strength detected by the GRCQC-chart beginning at the fifth sample. Since the UCL for the Weibull (β = 1.5)-chart (4.9242) is slightly higher than the highest tensile strength measured during testing (4.90), it fails to trigger at any time during the test and is therefore insensitive to the change in shift. These results support the earlier findings. The GRCQC-chart's tighter upper control limit, resulting from calibration to the fitted GR distribution, allows it to detect changes at the earliest possible time. Although the fitted GR model for this dataset exhibits a monotone increasing hazard rather than the non-monotonic bathtub shape that originally motivated the proposed chart, it nevertheless provides an adequate fit to the data. This illustrates the robustness of the GRCQC-chart to departures from the motivating hazard-shape assumption. By contrast, the Exponential and Weibull-based CQC-charts produce substantially higher upper control limits for this dataset, resulting in significantly delayed detection or, in the case of the Weibull chart, no signal at all.

9 Conclusions, recommendations, and limitations

The Generalized Rayleigh (GR) distribution is a flexible model for describing TBE processes. The GR distribution's hazard function can exhibit a non-monotonic “bathtub” shape, a behavior that none of the other benchmark distributions of this study have. This study, therefore, proposes a CQC-chart based on the GR distribution. An analytical evaluation was performed to compare the proposed GRCQC-chart with the existing benchmark CQC-type charts under process deterioration and improvement scenarios, across multiple false-alarm rates and a range of shift multipliers (δ∈[0.10, 3.00]). Separate analyses were performed for the lower state, upper state, and both states of the monitoring process. The numerical findings in terms of ARL showed that the GRCQC-chart shows better detection capability when the process is in a lower state (δ < 1), an upper state (δ>1), or in both states with either direction of δ. The ARARL indicator endorses these findings. At first glance, AEQL appears to give contradictory results for the one- and two-sided states; this, however, is not the case. AEQL for a one-sided chart is large due to the high ARL values for the direction in which the chart is not designed to detect; the two-sided AEQL, by contrast, balances the average loss across both states and provides an additional measure of overall usefulness, confirming the earlier conclusion of better GRCQC-chart performance.

The robustness study results showed that the underlying distributional assumption violation is a common limitation found across all benchmark CQC-type charts and is not unique to the GRCQC-chart. All charts evaluated in this study showed a considerable decline in performance when the presumed distribution did not match the actual in-control distribution, with the degree of degradation determined mainly by differences in tail behavior between the assumed and true distributions. Consequently, before constructing control limits for any benchmark CQC-type chart, one must first reliably validate the distribution via the goodness-of-fit assessment. Since all these preceding comparisons assume that the in-control parameters are known exactly, so we also investigated how sampling uncertainty from a Phase I estimation step propagate into the false-alarm behavior of every chart examined. This is basically extending the evaluation from an idealized comparison to one that reflects realistic conditions. The results indicate that increasing the sample size vastly enhances reliability. At every sample size, its risk of ≥2 × false-alarm inflation falls within the range spanned by the benchmark charts (for example, at n = 20: 0.486 for GRCQC, vs. 0.444–0.554 across the six two-parameter benchmarks), and by n = 500 every two-parameter chart, GRCQC included, has converged to a risk below 2.5%. The vulnerability documented for the GRCQC-chart alone is therefore not a weakness specific to the proposed chart; it is shared by every benchmark CQC-type chart.

Therefore, for practitioners deciding whether to adopt the GRCQC-chart for a specific process, we recommend the following explicit decision sequence, consistent with the guidelines established in Sections 6 to 8: (a). Collect a Phase I reference sample of at least 200-500 observations where feasible (Section 7). (b). Fit candidate distributions (at minimum: GR, Weibull, Exponential, and any family with prior domain support, e.g. Log-Normal for fatigue/degradation processes) to the Phase I data by maximum likelihood. (c). Compare candidate models by AIC/BIC and confirm adequacy with a goodness-of-fit test (KS and/or Anderson-Darling, as in Section 8.2); do not proceed to control-limit construction for any family that is rejected at the chosen significance level. (d). Among families that are not rejected, prefer GR when it is not rejected and either (i) the fitted hazard shows non-monotonic (bathtub) behavior, or (ii) GR's AIC/BIC is competitive with the best-fitting alternative even if not lowest, since Section 8.2 shows GRCQC's monitoring advantage does not require the fitted shape to itself be non-monotonic bathtub. (e). If no candidate family is adequate by goodness-of-fit testing, do not deploy any parametric CQC-type chart (GRCQC included) without further investigation; Section 6.1 shows the false-alarm consequences of a misspecified family are severe regardless of which chart is chosen.

There are few limitations of this study that could potentially lead to promising extensions for future research. The robustness analysis of Section 6 and the Phase I uncertainty analysis of Section 7 are each conducted one factor at a time: Section 6 holds Phase I estimation error at zero, and Section 7 holds the assumed family fixed at the true GR family. A joint analysis, in which the in-control family is simultaneously misspecified and estimated from a finite Phase I sample, would more closely represent practice and may compound the two sources of error identified separately here. We view this joint analysis as a first extension of the present study. Secondly, many real processes have evolving failure characteristics; extending the GRCQC framework through mixture models, state-space approaches, or Bayesian sequential updating would allow monitoring of dynamic TBE processes. Thirdly, the proposed approach can be extended to multivariate TBE monitoring via joint GR models or copula-based approaches for modern manufacturing systems. Finally, of the two case studies in Section 8, only the carbon fiber dataset (Section 8.2) is genuinely real-world data with GR selected among competing families by AIC and validated by goodness-of-fit testing rather than assumed; the IC manufacturing dataset (Section 8.1) is hypothetical under the GR model and so does not, by itself, provide independent validation. A third, fully independent real-world validation dataset, ideally from a different application domain than fiber strength, would strengthen the empirical case further.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fams.2026.1894611/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Reza Pakyari, Qatar University, Qatar

Reviewed by: Tahir Mahmood, University of the West of Scotland, United Kingdom

Shin-Li Lu, Chung Yuan Christian University, Taiwan