Bernoulli Product Research Articles

Dynamic Modeling and Analysis of Bernoulli Production Lines With Small-Lot Orders and Product Quality Problems

Dynamic Modeling and Analysis of Bernoulli Production Lines With Small-Lot Orders and Product Quality Problems

A Predictive Control Model of Bernoulli Production Line with Rework Loop for Real-Time WIP Optimization in Permutation Flowshop

Permutation flowshop design and optimization are crucial in industry as they have a direct impact on production scheduling and efficiency. The ultimate goal is to model the production system (PSM) based on revealing the fundamental principles of the production process, and to schedule or reschedule production release plans in real time without interrupting work-in-progress (WIP). Most existing PSMs are focused on static production processes which fail to describe the dynamic relationships between machines and buffers. Therefore, this paper establishes a PSM to characterize both the static and transient behaviors of automatic and manual machines in the permutation flowshop manufacturing system. Building upon the established PSM, based on Bernoulli’s theory, discrete event model predictive control is proposed in this paper; its aim is to realize real-time optimization of production release plans without interfering with work-in-progress. According to the results of numerical examples, the discrete event model predictive control proposed in this paper is feasible and effective. The model established in this paper provides a theoretical basis for optimizing the effective operation of work-in-progress and replacement process systems.

Transient Response of Homogenous and Nonhomogenous Bernoulli Production Lines

The transient response of production systems is of significant importance especially if present advancements in Digital Twinning technology are taken into account. While the steady-state response enables long-term strategic decision making, the transient response enables more detailed simulation concerning aspects like production losses and preventive maintenance. This is especially relevant if nonhomogenous aspects of production systems are taken into account. An analytical and approximative solution to the problem of the transient response of homogenous and nonhomogenous Bernoulli production systems is developed in this paper based on the eigendecomposition of transition matrices, the eigenvalue problem, and the finite-state method. In particular, sub-resonant and resonant nonhomogeneous production lines are introduced for the first time. Also, the most significant key performance indicators are developed as functions of the time elapsed from the first cycle. Finally, the relationship between the number of eigenvalues and the accuracy of the results is inspected by employing a sensitivity analysis. The presented theoretical framework was employed in the case of a wood processing facility to present the potential application of the theory in the case of long- and short-term management of production systems.

Zero-Temperature Stochastic Ising Model on Planar Quasi-Transitive Graphs

We study the zero-temperature stochastic Ising model on some connected planar quasi-transitive graphs, which are invariant under rotations and translations. The initial spin configuration is distributed according to a Bernoulli product measure with parameter p∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ p\\in (0,1) $$\\end{document}. In particular, we prove that if p=1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ p=1/2 $$\\end{document} and the graph underlying the model satisfies the planar shrink property then all vertices flip infinitely often almost surely.

Network Inference Using the Hub Model and Variants

Statistical network analysis primarily focuses on inferring the parameters of an observed network. In many applications, especially in the social sciences, the observed data is the groups formed by individual subjects. In these applications, the network is itself a parameter of a statistical model. Zhao and Weko propose a model-based approach, called the hub model, to infer implicit networks from grouping behavior. The hub model assumes that each member of the group is brought together by a member of the group called the hub. The set of members which can serve as a hub is called the hub set. The hub model belongs to the family of Bernoulli mixture models. Identifiability of Bernoulli mixture model parameters is a notoriously difficult problem. This article proves identifiability of the hub model parameters and estimation consistency under mild conditions. Furthermore, this article generalizes the hub model by introducing a model component that allows hubless groups in which individual nodes spontaneously appear independent of any other individual. We refer to this additional component as the null component. The new model bridges the gap between the hub model and the degenerate case of the mixture model—the Bernoulli product. Identifiability and consistency are also proved for the new model. In addition, a penalized likelihood approach is proposed to estimate the hub set when it is unknown. Supplementary materials for this article are available online.

Dynamics of the Box-Ball System with Random Initial Conditions via Pitman’s Transformation

The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al. recently showed the invariance in distribution of Bernoulli product measures with density strictly less than 1 2 \frac {1}{2} , and gave a soliton decomposition for invariant measures more generally. We study the BBS dynamics using the transformation of a nearest neighbour path encoding of the particle configuration given by ‘reflection in the past maximum’, which was famously shown by Pitman to connect Brownian motion and a three-dimensional Bessel process. We use this to characterise the set of configurations for which the dynamics are well-defined and reversible (i.e. can be inverted) for all times. The techniques developed to understand the deterministic dynamics are subsequently applied to study the evolution of the BBS from a random initial configuration. Specifically, we give simple sufficient conditions for random initial conditions to be invariant in distribution under the BBS dynamics, which we check in several natural examples, and also investigate the ergodicity of the relevant transformation. Furthermore, we analyse various probabilistic properties of the BBS that are commonly studied for interacting particle systems, such as the asymptotic behavior of the integrated current of particles and of a tagged particle. Finally, for Bernoulli product measures with parameter p ↑ 1 2 p\uparrow \frac 12 (which may be considered the ‘high density’ regime), the path encoding we consider has a natural scaling limit, which motivates the introduction of a new continuous version of the BBS that we believe will be of independent interest as a dynamical system.

Accuracy of Semi-Analytical and Numerical Approaches in the Evaluation of Serial Bernoulli Production Lines

The manufacturing industry has a great impact on the economic growth of countries. It is, therefore, crucial to master the skills of the production system by mathematical tools that enable the evaluation of the production systems’ performance measures. Four mathematical approaches toward the modeling of steady-state behavior of serial Bernoulli production lines were considered in this study, namely, the analytical approach, the finite state method, the aggregation procedure, and numerical modeling. The accuracy of the performance measures determined using the semi-analytical methods and the numerical approach was validated using numerous theoretical examples and the results obtained using the analytical model. All of the considered methods demonstrated relevant reliability, regardless of the different theoretical backgrounds.

A finite state method in improvement and design of lean Bernoulli serial production lines

Production systems have a significant impact on national and global economies. It is, therefore, of particular interest to develop, master, and apply new methodologies improving the performance of the production systems. A systematic approach to the improvability and design for lean production in the case of the serial Bernoulli production lines is presented in this research using the finite state method and application of the Markovian modeling approach. A new concept of the bottleneck identification procedure is presented based on the analytical relationship between line properties and performance measures. Also, a new design methodology combining features of the finite state method and genetic evolution algorithm is developed to enable the efficient lean design of the serial Bernoulli production lines. The developed procedures are applied in the case of an automotive paint shop system including bottleneck identification concerning the production rate, the work-in-process, and the probabilities of blockage and starvation. Finally, the lean automotive paint shop system is obtained in an efficient and designer-friendly way.

Quantitative ergodicity for the symmetric exclusion process with stationary initial data

We consider the symmetric exclusion process on the $d$-dimensional lattice with translational invariant and ergodic initial data. It is then known that as $t$ diverges the distribution of the process at time $t$ converges to a Bernoulli product measure. Assuming a summable decay of correlations of the initial data, we prove a quantitative version of this convergence by obtaining an explicit bound on the Ornstein $\bar d$-distance. The proof is based on the analysis of a two species exclusion process with annihilation.

Dynamical phase transition of ASEP in the KPZ regime

We consider the asymmetric simple exclusion process (ASEP) on Z. For continuous densities, ASEP is in local equilibrium for large times, at discontinuities however, one expects to see a dynamical phase transition, i.e. a mixture of different equilibriums. We consider ASEP with deterministic initial data such that at large times, two rarefactions come together at the origin, and the density jumps from 0 to 1. Shifting the measure on the KPZ 1∕3 scale, we show that the law of ASEP converges to a mixture of the Dirac measures with only holes resp. only particles. The parameter of that mixture is the probability that the second class particle, which is distributed as the difference of two independent GUEs, stays to the left of the shift. This should be compared with the results of Ferrari and Fontes from 1994 [6], who obtained a mixture of Bernoulli product measures at discontinuities created by random initial data, with the GUEs replaced by Gaussians.

Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP

We consider the weakly asymmetric simple exclusion process on the discrete space {1,⋯,n-1}(n∈ℕ), in contact with stochastic reservoirs, both with density ρ∈(0,1) at the extremity points, and starting from the invariant state, namely the Bernoulli product measure of parameter ρ. Under time diffusive scaling tn 2 and for ρ=1 2, when the asymmetry parameter is taken of order 1/n, we prove that the density fluctuations at stationarity are macroscopically governed by the energy solution of the stochastic Burgers equation with Dirichlet boundary conditions, which is shown to be unique and to exhibit different boundary behavior than the Cole–Hopf solution.

BBS invariant measures with independent soliton components

The Box-Ball System (BBS) is a one-dimensional cellular automaton in the configuration space $\{0,1\}^{\mathbb{Z} }$ introduced by Takahashi and Satsuma [8], who identified conserved quantities called solitons. Ferrari, Nguyen, Rolla and Wang [4] map a configuration to a family of soliton components, indexed by the soliton sizes $k\ge 1$. Building over this decomposition, we give an explicit construction of a large family of invariant measures for the BBS that are also shift invariant, including Ising-like Markov and Bernoulli product measures. The construction is based on the concatenation of iid excursions of the associated walk trajectory. Each excursion has the property that the law of its $k$ component given the larger components is product of a finite number of geometric distributions with a parameter depending on $k$. As a consequence, the law of each component of the resulting ball configuration is product of identically distributed geometric random variables, and the components are independent. This last property implies invariance for BBS, as shown by [4].

Exact solution of the facilitated totally asymmetric simple exclusion process

We obtain the exact solution of the facilitated totally asymmetric simple exclusion process (F-TASEP) in 1D. The model is closely related to the conserved lattice gas (CLG) model and to some cellular automaton traffic models. In the F-TASEP a particle at site j in jumps, at integer times, to site j + 1, provided site j − 1 is occupied and site j + 1 is empty. When started with a Bernoulli product measure at density , the system approaches a stationary state. This non-equilibrium steady state (NESS) has phase transitions at and . The different density regimes , , and exhibit many surprising properties; for example, the pair correlation satisfies, for all , , with k = 2 when , k = 6 when , and k = 3 when . The quantity , where is the variance in the number of particles in an interval of length L, jumps discontinuously from to 0 when and when .

The invariant measures and the limiting behaviors of the facilitated TASEP

We study the facilitated totally asymmetric exclusion process on the one dimensional integer lattice. A particle jumps to right at rate one provided that the target site is empty and that the left neighboring site of the particle is occupied. We investigate the invariant measures and the limiting behaviors of the process. We mainly prove the non-existence of spatial ergodic non-degenerate invariant measures having particle density less than or equal to 1∕2, and derive the limiting distribution of the process when the initial distribution is the Bernoulli product measure with density less than or equal to 1∕2. We also prove that in the low density regime, the system finally converges to an absorbing state.

A Good-Turing estimator for feature allocation models

Feature allocation models generalize classical species sampling models by allowing every observation to belong to more than one species, now called features. Under the popular Bernoulli product model for feature allocation, we assume $n$ observable samples and we consider the problem of estimating the expected number $M_{n}$ of hitherto unseen features that would be observed if one additional individual was sampled. The interest in estimating $M_{n}$ is motivated by numerous applied problems where the sampling procedure is expensive, in terms of time and/or financial resources allocated, and further samples can be only motivated by the possibility of recording new unobserved features. We consider a nonparametric estimator $\hat{M}_{n}$ of $M_{n}$ which has the same analytic form of the popular Good-Turing estimator of the missing mass in the context of species sampling models. We show that $\hat{M}_{n}$ admits a natural interpretation both as a jackknife estimator and as a nonparametric empirical Bayes estimator. Furthermore, we give provable guarantees for the performance of $\hat{M}_{n}$ in terms of minimax rate optimality, and we provide with an interesting connection between $\hat{M}_{n}$ and the Good-Turing estimator for species sampling. Finally, we derive non-asymptotic confidence intervals for $\hat{M}_{n}$, which are easily computable and do not rely on any asymptotic approximation. Our approach is illustrated with synthetic data and SNP data from the ENCODE sequencing genome project.

Limit theorems for the tagged particle in exclusion processes on regular trees

We consider exclusion processes on a rooted $d$-regular tree. We start from a Bernoulli product measure conditioned on having a particle at the root, which we call the tagged particle. For $d\geq 3$, we show that the tagged particle has positive linear speed and satisfies a central limit theorem. We give an explicit formula for the speed. As a key step in the proof, we first show that the exclusion process "seen from the tagged particle" has an ergodic invariant measure.

Analytical Solution of the serial Bernoulli production line steady-state performance and its application in the shipbuilding process

The modern shipbuilding industry is faced with numerous challenges urging abiding improvement of the shipbuilding process and its management. Such a demanding problem is usually approached using complex production management models involving large data basis handling. The same problem can be solved using simpler, intuitive and mathematically transparent models developed within production system engineering. For these purposes, the analytical solution of the serial Bernoulli production line steady-state performance involving an arbitrary number of machines and buffers of arbitrary occupancy is developed based on Markov chain approach and eigenvalue problem including formulation of the generalised transition matrix as a key novelty. Performance measures in case of general serial Bernoulli production line are developed and the theory is validated using serial Bernoulli lines composed of two, three, four and five machines. The obtained results are compared to those determined using a semi-analytical approach. In order to further demonstrate the applicability of the developed theory, performance measure analysis is performed in cases of longer lines composed of 6, 7, 8, 9 and 10 machines with non-equal buffer capacities. Application of the developed procedure in the analysis of the shipbuilding process is illustrated in a case of the plate prefabrication line usually placed in each shipyard.

Stochastic domination in space‐time for the contact process

AbstractLiggett and Steif (2006) proved that, for the supercritical contact process on certain graphs, the upper invariant measure stochastically dominates an i.i.d. Bernoulli product measure. In particular, they proved this for and (for infection rate sufficiently large) d‐ary homogeneous trees Td. In this paper, we prove some space‐time versions of their results. We do this by combining their methods with specific properties of the contact process and general correlation inequalities. One of our main results concerns the contact process on Td with . We show that, for large infection rate, there exists a subset Δ of the vertices of Td, containing a “positive fraction” of all the vertices of Td, such that the following holds: The contact process on Td observed on Δ stochastically dominates an independent spin‐flip process. (This is known to be false for the contact process on graphs having subexponential growth.) We further prove that the supercritical contact process on observed on certain d‐dimensional space‐time slabs stochastically dominates an i.i.d. Bernoulli product measure, from which we conclude strong mixing properties important in the study of certain random walks in random environment.

Analysis, design, and control of Bernoulli production lines with waiting time constraints

Waiting time constraints between two processes are one of common scheduling requirements in many production systems, such as semiconductor, automotive, food, and battery manufacturing. When the time constraints are introduced, quality inspection should be carried out on parts that have exceeded a given time limit and such parts are subject to be re-processed or scrapped, which incurs additional expenses and efforts. Therefore, the foremost goal is to maximize yield, which is defined by the production rate of parts that do not violate the time constraints. In this paper, we present a mathematical model of Bernoulli production lines to evaluate yield and examine its properties with respect to time constraints, buffer capacity, and machine reliability. System properties such as monotonicity and asymptotic characteristics are analyzed. With the analysis results, an efficient algorithm to design an optimal buffer capacity is developed. It is shown that, in contrast to traditional production lines, there is no monotonicity of yield on the reliability of an upstream machine; that is, an upstream machine becoming more reliable does not always contribute to increasing yield. Therefore, optimal control policies are presented to control the upstream machine for maximal yield. Finally, a case study is introduced to illustrate the applicability of the method.

Projections of planar Mandelbrot random measures

Let μ be a planar Mandelbrot measure and π⁎μ its orthogonal projection on one of the principal axes. We study the thermodynamic and geometric properties of π⁎μ. We first show that π⁎μ is exact dimensional, with dim⁡(π⁎μ)=min⁡(dim⁡(μ),dim⁡(ν)), where ν is the Bernoulli product measure obtained as the expectation of π⁎μ. We also prove that π⁎μ is absolutely continuous with respect to ν if and only if dim⁡(μ)>dim⁡(ν). Our results provides a new proof of Dekking–Grimmett–Falconer formula for the Hausdorff and box dimension of the topological support of π⁎μ, as well as a new variational interpretation. We obtain the free energy function τπ⁎μ of π⁎μ on a wide subinterval [0,qc) of R+. For q∈[0,1], it is given by a variational formula which sometimes yields phase transitions of order larger than 1. For q>1, it is given by min⁡(τν,τμ), which can exhibit first order phase transitions. This is in contrast with the analyticity of τμ over [0,qc). Also, we prove the validity of the multifractal formalism for π⁎μ at each α∈(τπ⁎μ′(qc−),τπ⁎μ′(0+)].