Weibull Approximation Research Articles

Modelling and analysis of repulsive cellular networks using Matérn hard‐core point processes

AbstractThe repulsive nature of Matérn hard‐core point process makes it well‐suited for modelling base station placements, where proximity constraints are critical. This letter reexamines the spatial properties of the model and proposes an empirical Weibull approximation for the contact distance distribution, where the shape and scale parameters can be reliably estimated through tractable regression models. Additionally, the probability generating functional is approximated using an exponential upper bound, simplifying the computation of the Laplace transform of interference. From these results, analytically tractable expressions are derived for the probability of coverage and average data rate. Simulations validate the accuracy of these analytical estimates.

Cost-effectiveness analysis of camrelizumab plus paclitaxel and carboplatin versus sintilimab plus gemcitabine and cisplatin or carboplatin for the first-line treatment of local advanced or metastatic squamous NSCLC in Chinese mainland.

Both camrelizumab plus paclitaxel and carboplatin (CTC) and sintilimab plus gemcitabine and cisplatin or carboplatin (SGP) have been approved by the National Medical Products Administration of China (NMPA) for the first-line treatment of local advanced or metastatic sqNSCLC. However, the comparison of the two treatments as first-line treatments in efficacy or pharmacoeconomics has barely been studied. To deeply understand the costs and outcomes of the two treatments, this work directly compared the cost-effectiveness for the first-line treatment of local advanced or metastatic squamous NSCLC in the Chinese mainland. A network meta-analysis was first performed based on the three clinical trials, namely, CameL-Sq, ORIENT-12, and C-TONG1002, to compare the clinical benefits of the two treatments. The Weibull approximation was applied to further calculate the life expectancy of the two treatments. The partitioned survival model (PSM) was next established, and one-way sensitivity analysis and probabilistic sensitivity analysis were also performed to evaluate the stability of the underlying parameter values and assumptions within the model. CTC treatment gained 0.68 QALYs and cost $14,764. SGP treatment gained 0.54 QALYs and cost $14,584. The CTC arm gained 0.14 additional QALYs and cost $179 more than the SGP arm, and the ICERs was $1,269/QALY, which was lower than one-fold GDP per capita in the Chinese mainland ($12,734 GDP per capita in 2022). In probabilistic sensitivity analysis, when the WTP ranged from $12,734-38,202 (1-3 folds, 2022 GDP per capita in China), the CTC group had higher probabilities than the SGP group for being cost effective, which ranged from 85.65% to 88.38%. From the perspective of the payers, camrelizumab plus chemotherapy was cost-effective compared with sintilimab plus chemotherapy for the first-line treatment of local advanced or metastatic squamous NSCLC in the Chinese mainland.

Sustainability of manufacturing technology: friction stir welding in focus

The sustainability of quality products is the primary concern of today's manufacturing entrepreneurs. The desire to produce cost effective and competitive products by metal forming or welding process has increased spontaneously. The authors examined the sustainability of friction stir welding (FSW) and evaluated weld joints reliability for health, safety, environment and economic purposes. An interplay between welding speed and tool rotational speed led to the formation of viable structures. The welding energy, materials deformation, emissions and life cycle assessment were analysed in line with the mechanical properties. A bivariate Weibull approximation model was used to predict weld joints reliability. A reliable and sustainable joints were obtained when the Weibull modulus increases as heat input reduces. The authors concluded by linking the tensile property of welded joints, heat input and Weibull modulus to the sustainability of FSW. A closed energy system was recommended for this green technology to be worthwhile.

Time-dependent breakdown of fiber networks: Uncertainty of lifetime.

Materials often fail when subjected to stresses over a prolonged period. The time to failure, also called the lifetime, is known to exhibit large variability of many materials, particularly brittle and quasibrittle materials. For example, a coefficient of variation reaches 100% or even more. Its distribution shape is highly skewed toward zero lifetime, implying a large number of premature failures. This behavior contrasts with that of normal strength, which shows a variation of only 4%-10% and a nearly bell-shaped distribution. The fundamental cause of this large and unique variability of lifetime is not well understood because of the complex interplay between stochastic processes taking place on the molecular level and the hierarchical and disordered structure of the material. We have constructed fiber network models, both regular and random, as a paradigm for general material structures. With such networks, we have performed Monte Carlo simulations of creep failure to establish explicit relationships among fiber characteristics, network structures, system size, and lifetime distribution. We found that fiber characteristics have large, sometimes dominating, influences on the lifetime variability of a network. Among the factors investigated, geometrical disorders of the network were found to be essential to explain the large variability and highly skewed shape of the lifetime distribution. With increasing network size, the distribution asymptotically approaches a double-exponential form. The implication of this result is that, so-called "infant mortality," which is often predicted by the Weibull approximation of the lifetime distribution, may not exist for a large system.

Authors' response: Understanding variation in disease risk.

Authors' response: Understanding variation in disease risk.

Stochastic Modeling of Cooperative Multi-Hop Strip Networks With Fixed Hop Boundaries

In this paper, a strip-shaped cooperative multi-hop wireless network is modeled stochastically with quasi-stationary Markov chain. The network is considered to be a fixed boundary decode-and-forward Opportunistic Large Array (OLA), where each level is of the same size and contains the same number of nodes placed randomly. The state of the system is represented by the number of nodes that decode the message in the current level. The distribution of the received power at a node is derived to formulate the transition probability matrix. For the distribution of power, a closed-form expression of the distribution of distance between a pair of nodes in disjoint levels is derived. It is seen that the distribution of distance can be well-approximated by the Weibull distribution. The Weibull approximation is then carried forward to find the distribution of the received power at a node assuming all nodes transmit with the same power and the channel has Rayleigh fading and path loss with an arbitrary exponent. The coverage and outage characteristics for various network sizes and path loss exponents are quantified. The signal-to-noise ratio (SNR) margin required for a given network coverage is determined by using the Perron-Frobenius theorem of non-negative matrices. Numerical simulations are performed to validate the theoretical results.

Fuzzy Modeling of Survival Function from Interval or Censored Observations

In medicine, biology, or actuarial theory the so called survival function is often used, i.e., a probability P(X ＞ x) that the time X from the beginning of an initial event to a final one will not exceed x. The observed data are mostly right-censored what causes that the observed variable is T = min(X, Z) where Z is the so called censoring variable. The celebrated and widely used nonparametric estimator of survival function is the Kaplan-Meier estimator which suffers from a disadvantage that it is stepwise and therefore it may happen that in considerably distinct points the values of the estimator may be equal.Rossa and Zielinski proposed a local smoothing of the Kaplan-Meier estimator based on an approximation by means of the piecewise Weibull survival function. They have shown that Mean Square Error and Mean Absolute Deviation of the smoothed estimator have been significantly smaller. Moreover, the Weibull approximation method has appeared to be a quite simple algorithm based on logarithmic transformations of the data and by applying the standard estimating procedure of the simple regression model y=ax+b. However, the censoring variables introduce uncertainty, which can be treated as the source of fuzziness. Thus, the estimation problem can be transferred into the fuzzy analysis. Another estimator leading to fuzzy model is the semi-parametric model proposed by Rossa. Parametric part of the estimator contains formulae based on estimates of the Weibull parameters, whereas the censored observations yield uncertainty.The proposed approaches have a general character. As it has been pointed out in Rossa and Zielinski, the Kaplan-Meier survival function can be approximated with a prescribed level of accuracy by a piecewise Weibull survival function. Using double logarithms of that Weibull pieces we get the piecewise linear intervals. All the intervals can have the same slop parameters or can have changing points separating different slops. Thus, the basic tool in the approach is the fuzzy linear regression. For simplicity, it will be assumed the fuzzy regression coefficients have symmetric triangular membership functions.

Case study on statistically estimating minimum makespan for flow line scheduling problems

Lower bounds are typically used to evaluate the performance of heuristics for solving combinatorial minimization problems. In the absence of tight analytical lower bounds, optimal objective-function values may be estimated statistically. In this paper, extreme value theory is used to construct confidence-interval estimates of the minimum makespan achievable when scheduling nonsimilar groups of jobs on a two-stage flow line. Experimental results based on randomly sampled solutions to each of 180 randomly generated test problems revealed that (i) least-squares parameter estimators outperformed standard analytical estimators for the Weibull approximation to the distribution of the sample minimum makespan; (ii) to evaluate each Weibull fit reliably, both the Anderson–Darling and Kolmogorov–Smirnov goodness-of-fit tests should be used; and (iii) applying a local improvement procedure to a large sample of randomly generated initial solutions improved the probability that the resulting Weibull fit yielded a confidence interval covering the minimum makespan.

A new approach for determining fully empirical altimeter wind speed model functions

A statistical technique is developed for determining fully empirical model functions relating altimeter radar backscatter (σ0) measurements to near‐surface neutral stability wind speed. By assuming that σ0 varies monotonically and uniquely with wind speed, the method requires knowledge only of the separate, rather than joint, distribution functions of σ0 and wind speed. Analytic simplifications result from using a Weibull distribution to approximate the global ocean wind speed distribution; several different wind data sets are used to demonstrate the validity of the Weibull approximation. The technique has been applied to 1 year of Geosat data. Validation of the new and historical model functions using an independent buoy data set demonstrates that the present model function not only has small overall bias and RMS errors, but yields smaller systematic error trends with wind speed and pseudowave age than previously published models. The present analysis suggests that generally accurate altimeter model functions can be derived without the use of collocated measurements, nor is additional significant wave height information measured by the altimeter necessary.

Lower tail analysis of the distribution of the strength of load-sharing systems

The Weibull distribution for the strength of a long fiber is reviewed, with emphasis on its derivation via extreme-value theory. It is shown that a corresponding Weibull distribution, but with different scale and shape parameters, is appropriate for the lower tail distribution of a bundle of parallel fibers under general load-sharing assumptions. This result is applied to series-parallel systems. The paper contains discussion of the application of these results to the strength of fibrous materials. It is concluded that the Weibull approximation, developed for series-parallel systems, is appropriate for long thin materials consisting of a small number of parallel fibers.

Lower tail analysis of the distribution of the strength of load-sharing systems

The Weibull distribution for the strength of a long fiber is reviewed, with emphasis on its derivation via extreme-value theory. It is shown that a corresponding Weibull distribution, but with different scale and shape parameters, is appropriate for the lower tail distribution of a bundle of parallel fibers under general load-sharing assumptions. This result is applied to series-parallel systems. The paper contains discussion of the application of these results to the strength of fibrous materials. It is concluded that the Weibull approximation, developed for series-parallel systems, is appropriate for long thin materials consisting of a small number of parallel fibers.

Applications of the Weibull Distribution in Inventory Control

This paper is intended to show that the Weibull distribution can effectively be used to approximate the lead time demand in several cases of distributed demands coupled with distributed lead times. With this approximation, the computations involved are shown to be very simple, requiring no numerical or iterative procedures for determining the re-order levels or protection levels. Some pertinent properties of the Weibull distribution are summarized, and necessary tables for estimating the parameters of the Weibull distribution are given. Numerical examples are given in three cases of distributed demands and lead times and, in each case, the Weibull approximation results are compared with exact and/or other approximate results.