Exponential Distribution Research Articles

An analytical method for reliability evaluation of power distribution system with time-varying failure rates

Reliability evaluation is crucial in assessing the power supply capacity and guiding the planning and operation of the distribution system. Traditional analytical reliability evaluation usually assumes a constant failure rate for components, implying that the life distribution follows an exponential distribution. However, in reality, the failure rate of components varies over time due to external disturbances and aging. As a result, traditional analytical reliability evaluation methods are no longer applicable. To address this issue, this paper presents an analytical method for evaluating the reliability of the distribution system with time-varying failure rates. The method considers short-term and long-term time-varying failure rates using density evolution and non-homogeneous Poisson process techniques, respectively. Both steady-state and instantaneous indices are calculated to reflect the level of system reliability. Finally, the proposed method is applied to the IEEE RBTS Bus 6 test system, demonstrating its correctness and efficiency.

A Bayesian cure rate regression model using Hamiltonian Monte Carlo methods

In this paper, we propose a flexible cure rate survival model by assuming the number of competing causes, related to the event of interest, to follow generalized Poisson distribution. The generalized Poisson distribution has several important properties, e.g. it can effectively model both underdispersed (mean > variance) and overdispersed (mean < variance) mechanisms present in the data. Further, it includes the promotion time cure model as a special case. In this work, for a non-cure subject, we model the time to event of interest with the piecewise exponential distribution. We emphasize that the piecewise exponential distribution has a greater flexibility to model failure times, and can be considered as a distribution-free alternative to the parametric distributions based assumptions. Inferences for the model parameters are considered via full Bayesian analysis. Further, a detailed simulation study is performed with different parameter settings, sample sizes and censoring percentages to study the behavior of the estimators. Finally, a data set from the medical area is analyzed for further illustration.

Queuing System Simulation to Optimize Waiting Time using the Quality Function Deployment Approach at KFC Pontianak

The aim of this research is to find the best scenario to optimize the queuing system so that it can reduce waiting time during customer service at KFC Pontianak. Queue systems are prevalent in public services. The research employs Quality Function Deployment (QFD) and Discrete Event Simulation (DES) methodologies. Alternatives generated are simulated using ARENA Simulation. 70 respondents participated in the KFC Pontianak customer questionnaire, comprising 48.6% females and 51.4% males aged between 18-45 years. The results of the questionnaire showed that 51.4% of respondents were dissatisfied with the speed of service at KFC Pontianak. Customer needs identified through QFD include service speed, order accuracy, ease of use, customer satisfaction, queue comfort, food quality, and payment integration. Analysis reveals three queue system scenarios, two of which propose improvements. The second scenario involves 2 cashiers and 2 self-service cashiers. Self-service allows customers to order by scanning a QR code integrated with the service provider's application. Arrival type employs an exponential distribution with an average of 61 seconds, unlimited max arrival, and first creation with a value of 0. In self-service, there is a value difference of 1 and a standard deviation of 0.2. Simulation results indicate that 51 customers choose cashier 1, 41 choose cashier 2, 39 opt for self-service cashier 1, and 53 prefer self-service cashier 2. The total successfully served customers are 94, with a success rate of 51%. In the third scenario involved 4 self-service cashiers only, the values of 1 and standard deviation of 0.2. The total number of successfully served customers is 173, with a success rate of 80%. In conclusion, this research offers valuable insights for optimizing KFC Pontianak's queue system, addressing customer dissatisfaction with service speed. The proposed scenarios, especially those involving self-service options, demonstrate potential improvements in customer satisfaction and operational efficiency.

Properties, quantile regression, and application of bounded exponentiated Weibull distribution to COVID-19 data of mortality and survival rates

Well-known continuous distributions such as Beta and Kumaraswamy distribution are useful for modeling the datasets which are based on unit interval [0,1]. But every distribution is not always useful for all types of data sets, rather it depends on the shapes of data as well. In this research, a three-parameter new distribution named bounded exponentiated Weibull (BEW) distribution is defined to model the data set with the support of unit interval [0,1]. Some fundamental distributional properties for the BEW distribution have been investigated. For modeling dependence between measures in a dataset, a bivariate extension of the BEW distribution is developed, and graphical shapes for the bivariate BEW distribution have been shown. Several estimation methods have been discussed to estimate the parameters of the BEW distribution and to check the performance of the estimator, a Monte Carlo simulation study has been done. Afterward, the applications of the BEW distribution are illustrated using COVID-19 data sets. The proposed distribution shows a better fit than many well-known distributions. Lastly, a quantile regression model from bounded exponentiated Weibull distribution is developed, and its graphical shapes for the probability density function (PDF) and hazard function have been shown.

Bayesian estimation of the mean time between failures of subsystems with different causes using interval‐censored system maintenance data

AbstractEnsuring an acceptable level of reliability stands as a primary imperative for any mission‐focused operation since it serves as a critical determinant of success. Inadequate reliability can lead to severe repercussions, including substantial expenses for repairs and replacements, missed opportunities, service disruptions, and in the worst cases, safety violations and human casualties. Within national defense organizations such as the USAF, the precise assessment and maintenance of system reliability play a pivotal role in ensuring the success of mission‐critical operations. In this research, our primary objective is to model the reliability of repairable subsystems within the framework of competing and complementary risks. Subsequently, we construct the overall reliability of the entire repairable system, utilizing day‐to‐day group‐censored maintenance data from two identical aircraft systems. Assuming that the lifetimes of subsystems follow non‐identical exponential distributions, it is theoretically justified that the system reliability can be modeled by homogeneous Poisson processes even though the number of subsystems of any particular type is unknown and the temporal order of multiple subsystem failures within a given time interval is uncertain due to interval censoring. Using the proposed model, we formulate the likelihood function for the mean time between failures of subsystems with different causes, and subsequently establish an inferential procedure for the model parameters. Given a considerable number of parameters to estimate, we explore the efficacy of a Bayesian approach, treating the contractor‐supplied estimates as the hyperparameters of prior distributions. This approach mitigates potential model uncertainty as well as the practical limitation of a frequentist‐based approach. It also facilitates continuous updates of the estimates as new maintenance data become available. Finally, the entire inferential procedures were implemented in Microsoft Excel so that it is easy for any reliability practitioner to use without the need to learn sophisticated programming languages. Thus, this research supports an ongoing, real‐time assessment of the overall mission reliability and helps early detection of any subsystem whose reliability is below the threshold level.

Efficiency of a New Parametric Cox Proportional Hazard Model Using Monte Carlo Simulation Study

In clinical studies, statistical models have proved useful in inferential analysis. This study examines the parametric Cox PH model using simulation studies. Through simulation studies, the study demonstrated the usefulness of the parametric Cox model by comparing its statistics, such as the concordance, confidence interval, redundancy or otherwise of the covariates, and median survival time, with those of classical Cox and logistic models. By extension, the ROC was plotted to show similarity with the parametric Cox model. The results show that the parametric Cox PH model has a higher concordance ratio of 0.9810 while the classical Cox PH model has 0.7810 concordance ratio. In both model scenarios, the variable Disease-Free survival indicator did not produce any value. The mean square error of the parametric Cox PH is lower than that of the classical Cox PH model. More covariates are significant in the parametric Cox PH model than in the classical Cox PH model. This tells that the proposed parametric Cox PH model improves the classical Cox PH model. The confidence interval for both models is seemingly the same. Because the assumptions of the Cox PH model were not violated in this study, given that the exponential distribution has a constant hazard rate, it is therefore recommended that other choices of non-constant hazard rate functions be made and deployed in the classical Cox PH model to attain some variant parametric Cox PH models.

Gaps Between Consecutive Primes and the Exponential Distribution

Based on the primes less than 4 × 10 18 , Oliveira e Silva et al. (Math. Comp., 83(288):2033–2060, 2014) conjectured an asymptotic formula for the sum of the kth power of the gaps between consecutive primes less than a large number x. We show that the conjecture of Oliveira e Silva holds if and only if the kth moment of the first n gaps is asymptotic to the kth moment of an exponential distribution with mean log n , though the distribution of gaps is not exponential. Asymptotically exponential moments imply that the gaps asymptotically obey Taylor’s law of fluctuation scaling: variance of the first n gaps ∼ (mean of the first n gaps)2. If the distribution of the first n gaps is asymptotically exponential with mean log n , then the expectation of the largest of the first n gaps is asymptotic to ( log n ) 2 . The largest of the first n gaps is asymptotic to ( log n ) 2 if and only if the Cramér-Shanks conjecture holds. Numerical counts of gaps and the maximal gap Gn among the first n gaps test these results. While most values of Gn are better approximated by ( log n ) 2 than by other models, seven exceptional values of n with G n > 2 e − γ ( log n ) 2 suggest that lim sup n → ∞ G n / [ 2 e − γ ( log n ) 2 ] may exceed 1.

Optimal designs of accelerated degradation tests with random shock failures and measurement errors

AbstractAccelerated degradation tests (ADTs) are widely used for assessing the reliability of long‐life products. During an ADT, accelerated stresses not only expedite the degradation of test products but also increase the likelihood of encountering traumatic shocks. Moreover, it is important to acknowledge that measurement errors can be inevitable during the observation process of an ADT. Unfortunately, these errors are often overlooked in the optimal design of the ADT, especially when multiple competing failure modes are present. In this article, we propose a new approach to design ADTs when measurement errors exist and test products suffer from degradation failures and random shock failures. We utilize the Wiener process to model the degradation path, incorporating normally distributed measurement errors, and an exponential distribution to fit the time between random shock failures. Given the number of test products and the termination time, we optimize the ADT plans under three common design criteria. The equivalence theorem is used to verify the optimality of the optimal ADT plans. A real‐life example and sensitivity analysis are provided to illustrate our proposed method. The results demonstrate that when competing failure modes are present, the optimal ADT plans, which account for measurement errors, differ significantly from those that do not consider such errors.

Measuring the propagation effect of production networks: is China more susceptible to shocks?

ABSTRACT The production network, formed by inter-industrial input-output linkages, propagates and amplifies shocks. Accurately measuring the propagation effect is crucial for risk management. However, most empirical research measuring the effect rely on specific assumptions on parameters and production function forms, such as Cobb-Douglas, constant elasticity of substitution (CES) functions, leading to varying results depending on the setting of assumptions. This paper derives a flexible modified measuring method that does not make assumptions on specific production function forms, reducing subjectivity in the measuring results. The new method indicates that the propagation effect mainly depends on the heterogeneity of Domar weights across sectors in the production network, consistent with our empirical findings from the United States, China, Japan, Germany, and the United Kingdom. Due to stronger heterogeneity in its production networks, China has a significantly higher propagation effect compared to the other four countries. Additionally, we found that the weighted degree of production networks follows exponential distributions instead of the commonly assumed power-law distributions. This study contributes to a quantitative understanding of systemic risk within production networks.

Statistical inference of the exponentiated exponential distribution based on progressive type-II censoring with optimal scheme

Statistical inference of the exponentiated exponential distribution based on progressive type-II censoring with optimal scheme

Generalized Moment Estimators Based on Stein Identities

For parameter estimation of continuous and discrete distributions, we propose a generalization of the method of moments (MM), where Stein identities are utilized for improved estimation performance. The construction of these Stein-type MM-estimators makes use of a weight function as implied by an appropriate form of the Stein identity. Our general approach as well as potential benefits thereof are first illustrated by the simple example of the exponential distribution. Afterward, we investigate the more sophisticated two-parameter inverse Gaussian distribution and the two-parameter negative-binomial distribution in great detail, together with illustrative real-world data examples. Given an appropriate choice of the respective weight functions, their Stein-MM estimators, which are defined by simple closed-form formulas and allow for closed-form asymptotic computations, exhibit a better performance regarding bias and mean squared error than competing estimators.

THE COMPARISON SOME MODELS DAILY WIND SPEED DATA IN PEKANBARU CITY USING TWO, THREE AND FOUR PARAMETERS GAMMA DISTRIBUTIONS

Accurate wind speed modeling is critical in estimating wind energy potential for harnessing wind power effectively. One of the bases for assessment of wind energy potential for a specified region is the probability distribution of wind speed, therefore wind speed data is needed to produce statistical modeling, especially in determining the best probability distribution. For this purpose, several modified distributions will be used and tested to determine the best model to describe wind speed in Pekanbaru. Six continuous probability density function such as Two Parameters Gamma (GM), Three Parameters Slashed Quasi-Gamma (SQG), Three Parameters Quasi Gamma (QG), Three Parameters Generalized Gamma (GG), Four Parameters Poly Weighted Exponentiated Gamma (PWEG) and Four Parameters Exponentiated Gamma Exponential (EGE) distributions were fitted for these tasks using the maximum likelihood technique. To determine the model’s fit to the daily wind speed data, several goodness-of-fit tests were applied, including the graphical methos test (pdf plot) and numerical criteria method test (AIC). The QG and GG distributions are found to be the best-fitted probability distribution based on goodness-of-fit tests for the daily wind speed data at the Pekanbaru station.

Generalized variables approach to generalized inverted exponential distribution reliability analyses with progressively type II censored data

Reliability analysis plays a crucial role in various fields, including engineering, manufacturing, and qualitycontrol. It provides valuable insights into the failure behavior of systems and products. One commonly used distribution in reliability modeling is the generalized inverted exponential distribution (GIED). The GIED distribution is known for its flexibility and adaptability to a wide range of failure data. This paper presents a new method for estimating confidence intervals and testing hypotheses for GIED distribution reliability functions based on a generalized value approach. By transforming the reliability function into the generalized value domain and calculating the generalized lower confidence limit, the proposed method offers enhanced accuracy and precision. Furthermore, the generalized p-value approach for hypothesis testing provides a robust and computationally efficient method for analyzing reliability data. The results from a real data set and Monte Carlo simulations confirm the superiority of the proposed approach over classical methods. The proposed method offers improved accuracy and computational efficiency, making it a valuable tool for reliability analysis using GIED distributions.

Study of data replication process using Raft replication algorithm to maintain consistency in server cluster

The article considers the issues of ensuring availability and fault tolerance of a server cluster. The analysis of methods and technologies for increasing system reliability and improving performance, such as data replication, automatic switching between servers and fast recovery after failures, is carried out. It is indicated that the key element in modern distributed systems is replication-based clustering. The types of replication are analyzed. It is shown that during the operation of the system, there is a choice between the stability of the received data and the speed or scaling limitation. It is shown that the Raft algorithm is one of the reliable mechanisms with a high level of availability for achieving consensus and data management. An analysis of the literature on the use of the Raft consensus algorithm is carried out. The replication process using the Raft algorithm is analyzed. It is substantiated that it is possible to ensure strong consistency and high availability of the cluster using the management and control mechanism on the Kubernetes platform while maintaining cluster configuration synchronization. The Raft consensus process is studied using the M/M/s QS model. The exponential distribution of time between client requests and servicing with a certain number of servers is considered. The program code for the mathematical model in Python has been developed. Modeling has been performed in the Visual code environment in Python. The following parameters have been studied: the number of servers that operate at a certain average speed, the expected time of a client request in the system, and the probability of message delay. It has been proven that as the arrival speed increases, the expected time of a client request increases. The analysis provides an understanding of how the Raft algorithm works in different contexts and is applicable to optimizing system design processes.

A Bayesian adaptive phase III design for multi-arm trials with time-to-event endpoint for nonproportional hazards utilizing the generalized gamma distribution

In case of trials with time-to-event endpoints, sample size calculations are well-studied under the assumption of proportional hazards or when the endpoint of interest follows an exponential distribution. When prior evidence suggests otherwise, using traditional approaches may lead to inefficiently designed and underpowered studies. In such situations, a recently introduced frequentist approach proposes a sample size calculation for a fixed two-arm trial based on the accelerated failure time model thereby allowing nonproportional hazards and can be utilized for any distribution from the generalized gamma family. Advances in the field of clinical trials research have focused on the need for adaptive phase III designs with complex features, however, existing methods in literature have ignored the nonproportional hazards scenario. In this article, we propose a Bayesian adaptive design for a multi-arm phase III trial with nonproportional hazards allowing many complex features such as incorporation of prior knowledge of early phase studies, arms dropping, and response adaptive randomization. We extend the frequentist approach utilizing a generalized gamma distribution to a Bayesian setting while simultaneously addressing one of the key limitations of the frequentist approach. Extensive simulations are performed to study the operation characteristics of the proposed design using two examples representing real-life applications.

Bias-Correction Methods for the Unit Exponential Distribution and Applications

The bias of the maximum likelihood estimator can cause a considerable estimation error if the sample size is small. To reduce the bias of the maximum likelihood estimator under the small sample situation, the maximum likelihood and parametric bootstrap bias-correction methods are proposed in this study to obtain more reliable maximum likelihood estimators of the unit exponential distribution parameters. The procedure to implement the bias-corrected maximum likelihood estimation method is derived analytically, and the steps to obtain the bias-corrected bootstrap estimators are presented. The simulation results show that the proposed maximum likelihood bootstrap bias-correction method can significantly reduce the bias and mean squared error of the maximum likelihood estimators for most of the parameter combinations in the simulation study. A soil moisture data set and a numerical example are used for illustration.

The Fav-Jerry Distribution: Another Member in the Lindley Class with Applications

In this paper, we designed another one-parameter distribution using a mixture of exponential and gamma distributions. This new distribution is unique among other members of the Lindley class because the qunatile function has a closed-functional form hence lending itself to analytical study. This distribution is named Fav-Jerry after the names of the authors. The statistical properties and point estimation using some non-Bayesian methods were studied. We deploy tow real datasets to demonstrate the usefulness of the new model. The real data applications using data sets on mortality rate and failure rate in a particular airplane showed that the proposed model fits well compared to its competitors, therefore, the Fav-Jerry distribution is superior to Two parameter Chris-Jerry(TPCJ), Chris-Jerry, Exponentiated Inverted Exponential distribution, and Weibull distributions and then parametric plots showing the histogram, CDF, survival and TTT plots gotten from both data sets are displayed.

Search for baryon junctions in photonuclear processes and isobar collisions at RHIC

During the early development of quantum chromodynamics, it was proposed that baryon number could be carried by a non-perturbative Y-shaped topology of gluon fields, called the gluon junction, rather than by the valence quarks as in the QCD standard model. A puzzling feature of ultra-relativistic nucleus-nucleus collisions is the apparent substantial baryon excess in the mid-rapidity region that could not be adequately accounted for in most conventional models of quark and diquark transport. The transport of baryonic gluon junctions is predicted to lead to a characteristic exponential distribution of net-baryon density with rapidity and could resolve the puzzle. In this context we point out that the rapidity density of net-baryons near mid-rapidity indeed follows an exponential distribution with a slope of -0.61±0.03\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-0.61\\pm 0.03$$\\end{document} as a function of beam rapidity in the existing global data from A+A collisions at AGS, SPS and RHIC energies. To further test if quarks or gluon junctions carry the baryon quantum number, we propose to study the absolute magnitude of the baryon vs. charge stopping in isobar collisions at RHIC. We also argue that semi-inclusive photon-induced processes (γ+p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma +p$$\\end{document}/A) at RHIC kinematics provide an opportunity to search for the signatures of the baryon junction and to shed light onto the mechanisms of observed baryon excess in the mid-rapidity region in ultra-relativistic nucleus-nucleus collisions. Such measurements can be further validated in A+A collisions at the LHC and e+p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e+p$$\\end{document}/A collisions at the EIC.

A Relationship between Classical and Bayesian Estimation Procedures through Fisher Information

The aim of this article is to investigate the association between the classical and Bayesian approaches through Fisher information. For any particular distribution, the computation of Fisher information is quite significant, as it provides the amount of information about the unknown parameter inferred from the observed data and is related to classical methods of estimation. Also, in the light of some prior knowledge, we may estimate the unknown parameter through Bayesian approach. Specifically, we want to see a relationship between information and Bayes estimation. In this article, the scale parameter of the one-parameter exponential distribution is estimated under the weighted squared error and Kullback-Leibler distance loss functions. The information acquired from both the classical and Bayesian methodologies have been connected through the risk intensity and error intensity which have been introduced in this article. The results of extensive simulation studies using these intensity measures show that the Bayes estimator performs more intensely as the amount of Fisher information increases. It is seen that the Fisher information, which is pivotal to many classical estimation methods, has a relationship with the Bayesian method depending on prior distribution, at least in this case, as the intensity measures of the Bayes estimator decrease with the increase in information. Further, to comprehend the theoretical notion of association, two real-life datasets have been included to show usefulness in practical field.

Shift Detection Comparison For Sized Biased And Area Biased Exponential Distributions Based Control Charts

Probability distribution models are very helpful to solving many real-life problems. The development of new or extensions to an existing model is a very attractive area of research. Among many types of probability distribution models, weighted distributions are becoming more important for modeling data. Weighted distributions can help discover defects in manufacturing and outliers by weighting significant observations. Control charts are the most important tools to monitor process efficiency. In recent decades, researchers have focused on weighted distributions, especially those with size and area biased exponential distribution. This research introduces a novel control chart for a characteristic following a size-biased and area-biased exponential distribution to emphasize the relevance of weighted distributions. Charts are effectively determines size-biased and area-biased exponential distribution control chart limits also Monte Carlo simulations are used to assess different Average Run Length (ARL) values.