A comparison study of control charts for Weibull distributed time between events

Hardware‐software co‐design systems abound in diverse modern application areas such as automobile control, telecommunications, big data processing, and cloud computing. Existing works on reliability modeling of the co‐design systems have mostly assumed that hardware and software subsystems behave independently of each other. However, these two subsystems may have significant interactions in practice. In this paper, an analytical approach based on paths and integrals is proposed to analyze reliability of nonrepairable hardware‐software co‐design systems considering interactions between hardware and software during the system performance degradation and failure process. The proposed approach is verified using the Markov‐based method. As demonstrated by case studies on systems without and with warm standby sparing, the proposed approach is applicable to arbitrary types of time‐to‐failure or degradation distributions. Effects of different transition and fault detection/recovery parameters on system performance are also investigated through examples.

“…Based on the general expressions in , if the state transition time follows the Weibull distribution with PDF f i,j ( t )=

((), k / λ_{i, j}) {(t / λ_{i, j})}^{k - 1} e^{- {(t / λ_{i, j})}^{k}}

and CDF F i,j ( t ) = 1‐

e^{- {(t / λ_{i, j})}^{k}},

then the occurrence probabilities of the three paths can be evaluated as

Path 1 : \int_{0}^{t} \int_{0}^{t - τ_{1}} \frac{k}{λ_{d} p_{d} q_{r}} {(\frac{τ_{1}}{λ_{d} p_{d} q_{r}})}^{k - 1} e^{- {(τ_{1} / λ_{d} p_{d} q_{r})}^{k}} e^{- {(τ_{1} / λ_{f})}^{k}} e^{- {(τ_{1} / λ_{d} q_{d})}^{k}} \frac{k}{λ_{t} + λ_{f_{a}}} {(\frac{τ_{2}}{λ_{t} + λ_{f_{a}}})}^{k - 1} e^{- {(τ_{2} / (λ_{t} + λ_{f_{a}}))}^{k}} d τ_{2} d τ_{1},

Path 2 : \int_{0}^{t} \frac{k}{λ_{f}} {(\frac{τ_{1}}{λ_{f}})}^{k - 1} e^{<...>}

…”

Section: Proposed Integral‐based Approachmentioning

confidence: 99%

Section: Proposed Integral-based Approachmentioning

confidence: 99%

An analytical method for reliability analysis of hardware‐software co‐design system

Zeng

Xing

Zhang

et al. 2018

“…Qu et al adopted the sequential analysis, a highly efficient method for sampling inspection, to improve the performance of TBE monitoring schemes. For further reading, interested readers may also see, Ali, Kumar and Chakraborti, and Wang et al For a detailed review of the TBE schemes, we refer Ali et al…”

Section: Introductionmentioning

confidence: 99%

A combination of max‐type and distance based schemes for simultaneous monitoring of time between events and event magnitudes

Sanusi

Mukherjee

2018

Traditionally, two isolated sequential stopping rules are employed for monitoring the time of occurrence of an event (T) and the magnitude of an event (X). Recently, several researchers recommend monitoring T and X together using some unified approach. A unified approach based on combinations of two statistics, one for monitoring T and the other for X, is often more efficient. Likewise, a new approach of simultaneous monitoring of location and scale parameters of a process, combining a max and a distance based statistics, is recently introduced in literature. Motivated by such emerging concepts, we design a new scheme combining a Max‐type and a Distance‐type schemes, referred to as the MT scheme, to monitor T and X simultaneously and efficiently. It retains the advantages of both the Max‐type and the Distance‐type schemes for joint inference. The proposed scheme is very competent in detecting a shift in the process distribution of T or X or both. Moreover, it is computationally simpler. It has nice exact expressions for design parameters. Therefore, it is easier to implement. It has a distinct advantage over its traditional counterparts in detecting moderate to large shifts. Finally, we illustrate the implementation of the proposed scheme with a real dataset of damage caused by outbreak of fire disaster.

“…Chakraborty et al 25 proposed a one-sided generally weighted moving average control chart based on gamma distribution to monitor the TBE (GWMA-TBE). For details about control charts based on Weibull distribution, the reader is referred to the works of Zhang and Chen, 30 Khoo and Xie, 31 Pascual, 32 Akhundjanov and Pascual, 33 Shafae et al, 34 and Wang et al 35 Many other modifications of EWMA and CUSUM charts, as well as another memory-type control charts have been studied by researchers in order to detect more quickly process shifts. Alevizakos and Koukouvinos 27 and Alevizakos et al 28 proposed a double EWMA (DEWMA) and a double generally weighted moving average (DGWMA) control chart, respectively, with a lower time-varying control limit based on the gamma distribution to monitor TBE.…”

mentioning

confidence: 99%

“…When the failure rate is not constant, then, TBE observations may be modeled by a Weibull distribution. For details about control charts based on Weibull distribution, the reader is referred to the works of Zhang and Chen, 30 Khoo and Xie, 31 Pascual, 32 Akhundjanov and Pascual, 33 Shafae et al, 34 and Wang et al 35 Many other modifications of EWMA and CUSUM charts, as well as another memory-type control charts have been studied by researchers in order to detect more quickly process shifts. Abbas et al 36 proposed a progressive mean (PM) chart for monitoring the mean of a process, and it was found very effective in detecting small and moderate shifts.…”

mentioning

confidence: 99%

A progressive mean control chart for monitoring time between events

Alevizakos

Koukouvinos

2019

Control charts based on time between events (TBE) are widely used in highquality manufacturing processes where the events occur rarely. For monitoring such data, a new two-sided TBE control chart called progressive mean control chart (regarded as PM-TBE chart) is proposed in the present study. Through a simulation study and using the average run-length (ARL) measure, it is shown that the proposed chart outperforms the t r , ARL-unbiased gamma, GWMA-TBE, and DEWMA-TBE charts in detecting very small to moderate downward shifts, as well as in some cases small upward shifts. Moreover, the PM-TBE chart is very robust for moderate to large shifts when the true distribution of the TBE observations is a Weibull or a lognormal. Finally, examples are given to display the application of the proposed chart. KEYWORDSaverage run length, Erlang distribution, progressive mean, robustness, time between events | INTRODUCTIONControl charts based on time between events (TBE) are frequently used in high-quality processes where the rate of occurrences is low. The traditional Shewhart control charts for attributes, such as c chart, u chart, p chart, and np chart, are used to monitor the number or the proportion of nonconformities or defective products, 1 whereas TBE control charts are used to monitor the inter-arrival times between the occurrences of events, which are assumed to be exponentially distributed. 2 Many researchers have focused on the study of TBE control charts. Lucas 3 studied the cumulative sum (CUSUM) chart for exponentially distributed observations, while Vardeman and Ray 4 used integral equations to calculate the average run-length (ARL) of the exponential CUSUM chart. Gan 5 investigated the optimal design of the exponential CUSUM chart, and Gan and Choi 6 presented a program for calculating the ARL of the above control chart. Gan 2 proposed the exponentially weighted moving average (EWMA) control chart for exponential data computing its ARL using differential equations, and he found it slightly less sensitive than the exponential CUSUM chart. Chan et al 7 proposed a Shewhart-type chart, called cumulative quantity control chart (CQC chart) for monitoring exponential TBE data. Jones and Champ 8 studied the phase I of Shewhart-type TBE control charts. Xie et al 9 proposed a Shewhart-type control chart, namely, t r chart, based on the Erlang distribution, to monitor the time between r (r ≥ 1) failures. Borror et al 10 studied the robustness of the exponential CUSUM chart assuming that the TBE observations follow a Weibull or a lognormal distribution, and they showed that it is very robust for both small and large shifts of parameters. Liu et al 11 compared various one-and two-sided exponential TBE control charts. Liu et al 12,13 proposed CUSUM and EWMA schemes, respectively, for monitoring exponential data after transforming it to an approximate normal distribution using the double square root transformation. Zhang et al 14 proposed a TBE control chart based on gamma distribution using the Let Y t ∼ iid ExpðθÞ, t=1,2,… r...