Study of average losses caused by ill processing in a production line with immediate feedback and multi server facility at each of the processing units
This academic article analyzes average losses caused by ill-processing in a production line with immediate feedback and multi-server facilities at each processing unit. The authors model the production line as a queuing network with an arbitrary number of processing units in series, where each unit has multi-server capacity. They analyze the stationary behavior and find the solution in product form. Considering processing costs, the average loss to the system due to rejection of items from ill-processing is obtained.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
The document summarizes a queueing model with two component mixture of doubly truncated exponential service times. The service time distribution is a two component mixture of doubly truncated exponential distributions, which can characterize heterogeneous and finite range service times. Assuming Poisson arrivals, the embedded Markov chain technique is used to analyze the system. Explicit expressions are derived for performance measures like average number of customers, average waiting time, throughput, and probability of idleness. Numerical analysis studies the sensitivity of performance measures to parameter changes. The model includes two component mixture of exponential, doubly truncated exponential, and exponential service time models as special cases.
Random Matrix Theory and Machine Learning - Part 4Fabian Pedregosa
Deep learning models with millions or billions of parameters should overfit according to classical theory, but they do not. The emerging theory of double descent seeks to explain why larger neural networks can generalize well. Random matrix theory provides a tractable framework to model double descent through random feature models, where the number of random features controls model capacity. In the high-dimensional limit, the test error of random feature regression exhibits a double descent shape that can be computed analytically.
The document discusses several brute-force algorithms including bubble sort, selection sort, string matching, closest pair of points, convex hulls, traveling salesman problem, knapsack problem, and assignment problem. It analyzes the runtime of each algorithm, which are often quadratic or exponential time due to considering all possible combinations in a systematic way.
Random Matrix Theory and Machine Learning - Part 1Fabian Pedregosa
This document provides an introduction to random matrix theory and its applications in machine learning. It discusses several classical random matrix ensembles like the Gaussian Orthogonal Ensemble (GOE) and Wishart ensemble. These ensembles are used to model phenomena in fields like number theory, physics, and machine learning. Specifically, the GOE is used to model Hamiltonians of heavy nuclei, while the Wishart ensemble relates to the Hessian of least squares problems. The tutorial will cover applications of random matrix theory to analyzing loss landscapes, numerical algorithms, and the generalization properties of machine learning models.
The document analyzes the analytic solution of Burger's equations using the variational iteration method. It begins by introducing the variational iteration method and how it can be used to solve differential equations. It then applies the method to obtain exact solutions for the (1+1), (1+2), and (1+3) dimensional Burger equations. Lengthy iterative solutions are presented for each case. The variational iteration method is shown to provide exact solutions to these Burger equations without requiring linearization.
Nonlinear Stochastic Optimization by the Monte-Carlo MethodSSA KPI
This document describes a method for solving stochastic optimization problems using Monte Carlo simulation. It introduces Monte Carlo estimators for the objective function, its gradient, and the covariance matrix that can be computed using a random sample. It then presents an iterative stochastic gradient descent procedure where the sample size is adjusted at each iteration inversely proportional to the square of the gradient estimate. Two theorems prove that this approach ensures convergence to the optimal solution and provides accuracy bounds on the estimate of the distance to the optimal point. The method offers a way to efficiently solve stochastic optimization problems using adaptive sample sizes.
Mixed Spectra for Stable Signals from Discrete Observationssipij
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial kernel to build a periodogram which we then smooth by two spectral windows taking into account the width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing often encountered in the case of estimation from discrete observations of a continuous time process.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
The document summarizes a queueing model with two component mixture of doubly truncated exponential service times. The service time distribution is a two component mixture of doubly truncated exponential distributions, which can characterize heterogeneous and finite range service times. Assuming Poisson arrivals, the embedded Markov chain technique is used to analyze the system. Explicit expressions are derived for performance measures like average number of customers, average waiting time, throughput, and probability of idleness. Numerical analysis studies the sensitivity of performance measures to parameter changes. The model includes two component mixture of exponential, doubly truncated exponential, and exponential service time models as special cases.
Random Matrix Theory and Machine Learning - Part 4Fabian Pedregosa
Deep learning models with millions or billions of parameters should overfit according to classical theory, but they do not. The emerging theory of double descent seeks to explain why larger neural networks can generalize well. Random matrix theory provides a tractable framework to model double descent through random feature models, where the number of random features controls model capacity. In the high-dimensional limit, the test error of random feature regression exhibits a double descent shape that can be computed analytically.
The document discusses several brute-force algorithms including bubble sort, selection sort, string matching, closest pair of points, convex hulls, traveling salesman problem, knapsack problem, and assignment problem. It analyzes the runtime of each algorithm, which are often quadratic or exponential time due to considering all possible combinations in a systematic way.
Random Matrix Theory and Machine Learning - Part 1Fabian Pedregosa
This document provides an introduction to random matrix theory and its applications in machine learning. It discusses several classical random matrix ensembles like the Gaussian Orthogonal Ensemble (GOE) and Wishart ensemble. These ensembles are used to model phenomena in fields like number theory, physics, and machine learning. Specifically, the GOE is used to model Hamiltonians of heavy nuclei, while the Wishart ensemble relates to the Hessian of least squares problems. The tutorial will cover applications of random matrix theory to analyzing loss landscapes, numerical algorithms, and the generalization properties of machine learning models.
The document analyzes the analytic solution of Burger's equations using the variational iteration method. It begins by introducing the variational iteration method and how it can be used to solve differential equations. It then applies the method to obtain exact solutions for the (1+1), (1+2), and (1+3) dimensional Burger equations. Lengthy iterative solutions are presented for each case. The variational iteration method is shown to provide exact solutions to these Burger equations without requiring linearization.
Nonlinear Stochastic Optimization by the Monte-Carlo MethodSSA KPI
This document describes a method for solving stochastic optimization problems using Monte Carlo simulation. It introduces Monte Carlo estimators for the objective function, its gradient, and the covariance matrix that can be computed using a random sample. It then presents an iterative stochastic gradient descent procedure where the sample size is adjusted at each iteration inversely proportional to the square of the gradient estimate. Two theorems prove that this approach ensures convergence to the optimal solution and provides accuracy bounds on the estimate of the distance to the optimal point. The method offers a way to efficiently solve stochastic optimization problems using adaptive sample sizes.
Mixed Spectra for Stable Signals from Discrete Observationssipij
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial kernel to build a periodogram which we then smooth by two spectral windows taking into account the width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing often encountered in the case of estimation from discrete observations of a continuous time process.
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
This document discusses developing near-optimal state feedback controllers for nonlinear discrete-time systems using iterative approximate dynamic programming (ADP) algorithms. Specifically:
1) An infinite-horizon optimal state feedback controller is developed for discrete-time systems based on the dual heuristic programming (DHP) algorithm.
2) A new optimal control scheme is developed using the generalized DHP (GDHP) algorithm and a discounted cost functional.
3) An infinite-horizon optimal stabilizing state feedback controller is designed based on the globalized dual heuristic programming (GHJB) algorithm.
4) Finite-horizon optimal controllers with an ε-error bound are proposed, where the number of optimal control steps can be determined
This paper studies an approximate dynamic programming (ADP) strategy of a group of nonlinear switched systems, where the external disturbances are considered. The neural network (NN) technique is regarded to estimate the unknown part of actor as well as critic to deal with the corresponding nominal system. The training technique is simul-taneously carried out based on the solution of minimizing the square error Hamilton function. The closed system’s tracking error is analyzed to converge to an attraction region of origin point with the uniformly ultimately bounded (UUB) description. The simulation results are implemented to determine the effectiveness of the ADP based controller.
Nonlinear Stochastic Programming by the Monte-Carlo methodSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 4.
More info at http://summerschool.ssa.org.ua
This document summarizes a research paper on the existence of periodic solutions of certain fourth order differential equations with delay. It begins by introducing the general form of the differential equation being studied and defining the relevant terms and functions. It then examines the linear case where some coefficients are constants. Two lemmas are presented, the first addressing when the linear equation has no non-trivial periodic solutions, and the second addressing when the linear equation with a variable coefficient admits only the trivial solution. This sets the groundwork for analyzing the non-linear case.
The document describes using threshold-based agent models to optimize plant placement in a landscape. It proposes an agent-based algorithm where individual "plants" search the landscape for optimal locations based on their light and water requirements. A genetic algorithm approach is also mentioned. The goal is to maximize overall plant growth by finding placements where each plant meets a 70% threshold of its ideal growth conditions. Future work could include formal analysis and comparisons to determine how well the approach works at finding the optimal plant collection for a given landscape.
NIPS2010: optimization algorithms in machine learningzukun
The document summarizes optimization algorithms for machine learning applications. It discusses first-order methods like gradient descent, accelerated methods like Nesterov's algorithm, and non-monotone methods like Barzilai-Borwein. Gradient descent converges at a rate of 1/k, while methods like heavy-ball, conjugate gradient, and Nesterov's algorithm can achieve faster linear or 1/k^2 convergence rates depending on the problem structure. The document provides convergence analysis and rate results for various first-order optimization algorithms applied to machine learning problems.
This document summarizes optimization techniques for matrix factorization and completion problems. Section 8.1 introduces the matrix factorization problem and considers minimizing reconstruction error subject to a nuclear norm penalty. Section 8.2 discusses properties of the nuclear norm, including relationships to the trace norm and Frobenius norm. Section 8.3 provides performance guarantees for matrix completion when the underlying matrix is exactly low-rank. Section 8.4 describes proximal gradient methods for optimization, including updates that involve singular value thresholding. The document concludes by discussing an extension of these methods to dictionary learning and alignment problems.
The document discusses digital image processing and two-dimensional transforms. It provides an agenda that covers two-dimensional mathematical preliminaries and two transforms: the discrete Fourier transform (DFT) and discrete cosine transform (DCT). It then discusses the DFT and DCT in more detail over several pages, covering properties, examples, and applications such as image compression.
Fixed Point Theorem of Compatible of Type (R) Using Implicit Relation in Fuzz...IRJET Journal
This document presents a theorem on the existence of a common fixed point for compatible mappings of type (R) in a fuzzy metric space.
The document begins with definitions of key concepts such as fuzzy metric spaces, Cauchy sequences, limits, compatibility, and compatibility of type (R). It then states a theorem that if mappings A, B, S, and T satisfy certain conditions, including being compatible of type (R) and satisfying an implicit relation, then they have a unique common fixed point.
The conditions and proof of the theorem are then provided. The proof constructs a Cauchy sequence and uses properties of the mappings and space like completeness to show the sequence converges to a common fixed point of the mappings.
(日本語説明下記)Here I try to derive the Y Combinator as simple as possible.
Yコンビネータは、変数を使わずに再帰関数をつくるための便利な考え方です。 Y Combinatorという名前は、同名のシリコンバレーの起業家支援プログラムが登場したため、一躍有名になりました(Google検索が難しくなりました)。
こっちの Y Combinatorを検索したいときは、
「Y Combinator 再帰」「Y Combinator ラムダ」などで検索してみてください。
This document provides a course calendar and lecture plans for topics related to Bayesian estimation methods. The course calendar lists 12 class dates from September to December covering topics like Bayes estimation, Kalman filters, particle filters, hidden Markov models, supervised learning, and clustering algorithms. One lecture plan provides details on the hidden Markov model, including the introduction, definition of HMMs, and problems of evaluation, decoding, and learning. Another lecture plan covers particle filters, including the sequential importance sampling algorithm, choice of proposal density, and the particle filter algorithm of sampling, weight update, resampling, and state estimation.
JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron modelshirokazutanaka
This document provides an overview of topics to be covered in a lecture on single neuron models. It will discuss:
1) The basic anatomy and physiology of neurons including their morphology and membrane properties.
2) Phenomenological models of subthreshold dynamics like the integrate-and-fire, quadratic-and-fire, and resonate-and-fire models.
3) Biophysical models of spiking mechanisms including the Hodgkin-Huxley model and its use of ion channels and master equations.
4) Analysis techniques like phase plots and bifurcation analysis applied to models like FitzHugh-Nagumo and Hindmarsh-Rose.
5) Modern single neuron models such
Stochastic Alternating Direction Method of MultipliersTaiji Suzuki
This document discusses stochastic optimization methods for solving regularized learning problems with structured regularization and large datasets. It proposes applying the alternating direction method of multipliers (ADMM) in a stochastic manner. Specifically, it introduces two stochastic ADMM methods for online data: RDA-ADMM, which extends regularized dual averaging with ADMM updates; and OPG-ADMM, which extends online proximal gradient descent with ADMM updates. These methods allow the regularization term to be optimized in batches, resolving computational difficulties, while the loss is optimized online using only a small number of samples per iteration.
Seminar Talk: Multilevel Hybrid Split Step Implicit Tau-Leap for Stochastic R...Chiheb Ben Hammouda
The document describes a multilevel hybrid split-step implicit tau-leap method for simulating stochastic reaction networks. It begins with background on modeling biochemical reaction networks stochastically. It then discusses challenges with existing simulation methods like the chemical master equation and stochastic simulation algorithm. The document introduces the split-step implicit tau-leap method as an improvement over explicit tau-leap for stiff systems. It proposes a multilevel Monte Carlo estimator using this method to efficiently estimate expectations of observables with near-optimal computational work.
This document discusses algorithms complexity and data structures efficiency. It covers topics like time and memory complexity, asymptotic notation, fundamental data structures like arrays, lists, trees and hash tables, and choosing proper data structures. Computational complexity is important for algorithm design and efficient programming. The document provides examples of analyzing complexity for different algorithms.
The document discusses algorithms and their analysis. It defines an algorithm as a well-defined computational procedure that takes inputs and produces outputs. It discusses analyzing algorithms based on their time complexity, space complexity, and correctness. It provides examples of analyzing simple algorithms and calculating their complexity based on the number of elementary operations.
Introducción al Análisis y diseño de algoritmosluzenith_g
The document discusses algorithms and their analysis. It defines an algorithm as a well-defined computational procedure that takes inputs and produces outputs. It discusses analyzing algorithms to determine their time and space complexity, and how this involves determining how the resources required grow with the size of the problem. It provides examples of analyzing simple algorithms and determining whether they have linear, quadratic, or other complexity.
This paper proposes modeling and identification of dynamical systems in delta
domain using neural network. The properties of delta operator are used such as greater
numerical robustness in computation and superior coefficients representation in finite word
length in implementation and well ensured numerical conditioning at high sampling
frequency. To formulate the identification scheme delta operator model is recasted into a
realizable neural network structure using the properties of inverse delta operator.
Random Matrix Theory and Machine Learning - Part 3Fabian Pedregosa
ICML 2021 tutorial on random matrix theory and machine learning.
Part 3 covers: 1. Motivation: Average-case versus worst-case in high dimensions 2. Algorithm halting times (runtimes) 3. Outlook
This document discusses developing near-optimal state feedback controllers for nonlinear discrete-time systems using iterative approximate dynamic programming (ADP) algorithms. Specifically:
1) An infinite-horizon optimal state feedback controller is developed for discrete-time systems based on the dual heuristic programming (DHP) algorithm.
2) A new optimal control scheme is developed using the generalized DHP (GDHP) algorithm and a discounted cost functional.
3) An infinite-horizon optimal stabilizing state feedback controller is designed based on the globalized dual heuristic programming (GHJB) algorithm.
4) Finite-horizon optimal controllers with an ε-error bound are proposed, where the number of optimal control steps can be determined
This paper studies an approximate dynamic programming (ADP) strategy of a group of nonlinear switched systems, where the external disturbances are considered. The neural network (NN) technique is regarded to estimate the unknown part of actor as well as critic to deal with the corresponding nominal system. The training technique is simul-taneously carried out based on the solution of minimizing the square error Hamilton function. The closed system’s tracking error is analyzed to converge to an attraction region of origin point with the uniformly ultimately bounded (UUB) description. The simulation results are implemented to determine the effectiveness of the ADP based controller.
Nonlinear Stochastic Programming by the Monte-Carlo methodSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 4.
More info at http://summerschool.ssa.org.ua
This document summarizes a research paper on the existence of periodic solutions of certain fourth order differential equations with delay. It begins by introducing the general form of the differential equation being studied and defining the relevant terms and functions. It then examines the linear case where some coefficients are constants. Two lemmas are presented, the first addressing when the linear equation has no non-trivial periodic solutions, and the second addressing when the linear equation with a variable coefficient admits only the trivial solution. This sets the groundwork for analyzing the non-linear case.
The document describes using threshold-based agent models to optimize plant placement in a landscape. It proposes an agent-based algorithm where individual "plants" search the landscape for optimal locations based on their light and water requirements. A genetic algorithm approach is also mentioned. The goal is to maximize overall plant growth by finding placements where each plant meets a 70% threshold of its ideal growth conditions. Future work could include formal analysis and comparisons to determine how well the approach works at finding the optimal plant collection for a given landscape.
NIPS2010: optimization algorithms in machine learningzukun
The document summarizes optimization algorithms for machine learning applications. It discusses first-order methods like gradient descent, accelerated methods like Nesterov's algorithm, and non-monotone methods like Barzilai-Borwein. Gradient descent converges at a rate of 1/k, while methods like heavy-ball, conjugate gradient, and Nesterov's algorithm can achieve faster linear or 1/k^2 convergence rates depending on the problem structure. The document provides convergence analysis and rate results for various first-order optimization algorithms applied to machine learning problems.
This document summarizes optimization techniques for matrix factorization and completion problems. Section 8.1 introduces the matrix factorization problem and considers minimizing reconstruction error subject to a nuclear norm penalty. Section 8.2 discusses properties of the nuclear norm, including relationships to the trace norm and Frobenius norm. Section 8.3 provides performance guarantees for matrix completion when the underlying matrix is exactly low-rank. Section 8.4 describes proximal gradient methods for optimization, including updates that involve singular value thresholding. The document concludes by discussing an extension of these methods to dictionary learning and alignment problems.
The document discusses digital image processing and two-dimensional transforms. It provides an agenda that covers two-dimensional mathematical preliminaries and two transforms: the discrete Fourier transform (DFT) and discrete cosine transform (DCT). It then discusses the DFT and DCT in more detail over several pages, covering properties, examples, and applications such as image compression.
Fixed Point Theorem of Compatible of Type (R) Using Implicit Relation in Fuzz...IRJET Journal
This document presents a theorem on the existence of a common fixed point for compatible mappings of type (R) in a fuzzy metric space.
The document begins with definitions of key concepts such as fuzzy metric spaces, Cauchy sequences, limits, compatibility, and compatibility of type (R). It then states a theorem that if mappings A, B, S, and T satisfy certain conditions, including being compatible of type (R) and satisfying an implicit relation, then they have a unique common fixed point.
The conditions and proof of the theorem are then provided. The proof constructs a Cauchy sequence and uses properties of the mappings and space like completeness to show the sequence converges to a common fixed point of the mappings.
(日本語説明下記)Here I try to derive the Y Combinator as simple as possible.
Yコンビネータは、変数を使わずに再帰関数をつくるための便利な考え方です。 Y Combinatorという名前は、同名のシリコンバレーの起業家支援プログラムが登場したため、一躍有名になりました(Google検索が難しくなりました)。
こっちの Y Combinatorを検索したいときは、
「Y Combinator 再帰」「Y Combinator ラムダ」などで検索してみてください。
This document provides a course calendar and lecture plans for topics related to Bayesian estimation methods. The course calendar lists 12 class dates from September to December covering topics like Bayes estimation, Kalman filters, particle filters, hidden Markov models, supervised learning, and clustering algorithms. One lecture plan provides details on the hidden Markov model, including the introduction, definition of HMMs, and problems of evaluation, decoding, and learning. Another lecture plan covers particle filters, including the sequential importance sampling algorithm, choice of proposal density, and the particle filter algorithm of sampling, weight update, resampling, and state estimation.
JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron modelshirokazutanaka
This document provides an overview of topics to be covered in a lecture on single neuron models. It will discuss:
1) The basic anatomy and physiology of neurons including their morphology and membrane properties.
2) Phenomenological models of subthreshold dynamics like the integrate-and-fire, quadratic-and-fire, and resonate-and-fire models.
3) Biophysical models of spiking mechanisms including the Hodgkin-Huxley model and its use of ion channels and master equations.
4) Analysis techniques like phase plots and bifurcation analysis applied to models like FitzHugh-Nagumo and Hindmarsh-Rose.
5) Modern single neuron models such
Stochastic Alternating Direction Method of MultipliersTaiji Suzuki
This document discusses stochastic optimization methods for solving regularized learning problems with structured regularization and large datasets. It proposes applying the alternating direction method of multipliers (ADMM) in a stochastic manner. Specifically, it introduces two stochastic ADMM methods for online data: RDA-ADMM, which extends regularized dual averaging with ADMM updates; and OPG-ADMM, which extends online proximal gradient descent with ADMM updates. These methods allow the regularization term to be optimized in batches, resolving computational difficulties, while the loss is optimized online using only a small number of samples per iteration.
Seminar Talk: Multilevel Hybrid Split Step Implicit Tau-Leap for Stochastic R...Chiheb Ben Hammouda
The document describes a multilevel hybrid split-step implicit tau-leap method for simulating stochastic reaction networks. It begins with background on modeling biochemical reaction networks stochastically. It then discusses challenges with existing simulation methods like the chemical master equation and stochastic simulation algorithm. The document introduces the split-step implicit tau-leap method as an improvement over explicit tau-leap for stiff systems. It proposes a multilevel Monte Carlo estimator using this method to efficiently estimate expectations of observables with near-optimal computational work.
Similar to Study of average losses caused by ill processing in a production line with immediate feedback and multi server facility at each of the processing units
This document discusses algorithms complexity and data structures efficiency. It covers topics like time and memory complexity, asymptotic notation, fundamental data structures like arrays, lists, trees and hash tables, and choosing proper data structures. Computational complexity is important for algorithm design and efficient programming. The document provides examples of analyzing complexity for different algorithms.
The document discusses algorithms and their analysis. It defines an algorithm as a well-defined computational procedure that takes inputs and produces outputs. It discusses analyzing algorithms based on their time complexity, space complexity, and correctness. It provides examples of analyzing simple algorithms and calculating their complexity based on the number of elementary operations.
Introducción al Análisis y diseño de algoritmosluzenith_g
The document discusses algorithms and their analysis. It defines an algorithm as a well-defined computational procedure that takes inputs and produces outputs. It discusses analyzing algorithms to determine their time and space complexity, and how this involves determining how the resources required grow with the size of the problem. It provides examples of analyzing simple algorithms and determining whether they have linear, quadratic, or other complexity.
This paper proposes modeling and identification of dynamical systems in delta
domain using neural network. The properties of delta operator are used such as greater
numerical robustness in computation and superior coefficients representation in finite word
length in implementation and well ensured numerical conditioning at high sampling
frequency. To formulate the identification scheme delta operator model is recasted into a
realizable neural network structure using the properties of inverse delta operator.
The document discusses algorithms complexity and data structures efficiency, explaining that algorithm complexity can be measured using asymptotic notation like O(n) or O(n^2) to represent operations scaling linearly or quadratically with input size, and different data structures have varying time efficiency for operations like add, find, and delete.
A priori and a posteriori analysis are two methods for analyzing algorithms. A priori analysis involves determining the time and space complexity of an algorithm without running it on a specific system, while a posteriori analysis involves analyzing an algorithm after running it on a system. Big-O notation is commonly used to describe an algorithm's time complexity as the input size increases. Common time complexities include constant, logarithmic, linear, quadratic, and exponential time.
Glowworm swarm optimization (GSO) is a swarm intelligence based algorithm, introduced by K.N. Krishnanand and D. Ghose in 2005, for simultaneous computation of multiple optima of multimodal functions
This document summarizes an academic research paper that analyzes an optimal N-policy for a Bernoulli feedback Mx/G/1 machining system with general setup times. The paper develops a mathematical model of the system using supplementary variable technique to obtain the probability generating function of the system queue size distribution and mean number of failed units. It also derives the Laplace-Stieltjes transform of the waiting time and evaluates the mean waiting time. Finally, it formulates the total operational cost function to determine the optimal value of N that minimizes costs.
The document discusses algorithms, including their definition, properties, analysis of time and space complexity, and examples of recursion and iteration. It defines an algorithm as a finite set of instructions to accomplish a task. Properties include inputs, outputs, finiteness, definiteness, and effectiveness. Time complexity is analyzed using big-O notation, while space complexity considers static and variable parts. Recursion uses function calls to solve sub-problems, while iteration uses loops. Examples include factorial calculation, GCD, and Towers of Hanoi solved recursively.
The document discusses algorithms and their analysis. It begins by defining an algorithm and key aspects like correctness, input, and output. It then discusses two aspects of algorithm performance - time and space. Examples are provided to illustrate how to analyze the time complexity of different structures like if/else statements, simple loops, and nested loops. Big O notation is introduced to describe an algorithm's growth rate. Common time complexities like constant, linear, quadratic, and cubic functions are defined. Specific sorting algorithms like insertion sort, selection sort, bubble sort, merge sort, and quicksort are then covered in detail with examples of how they work and their time complexities.
This document summarizes a research paper that develops a mathematical model of a queueing system with batch arrivals, two optional phases of service, optional re-service, and Bernoulli vacations. The server provides an essential first phase of service, after which customers may optionally repeat that service or proceed to an optional second phase of service. After each service, the server may take a vacation. Differential equations describing the time-dependent behavior of the system are presented. The long-term properties of the queue are also analyzed.
11.generalized and subset integrated autoregressive moving average bilinear t...Alexander Decker
This document proposes generalized integrated autoregressive moving average bilinear (GBL) time series models and subset generalized integrated autoregressive moving average bilinear (GSBL) models to achieve stationary for all nonlinear time series. It presents the models' formulations and discusses their properties including stationary, convergence, and parameter estimation. An algorithm is provided to fit the one-dimensional models. The generalized models are applied to Wolfer sunspot numbers and the GBL model is found to perform better than the GSBL model.
The document discusses algorithm analysis and asymptotic notation. It begins by explaining how to analyze algorithms to predict resource requirements like time and space. It defines asymptotic notation like Big-O, which describes an upper bound on the growth rate of an algorithm's running time. The document then provides examples of analyzing simple algorithms and classifying functions based on their asymptotic growth rates. It also introduces common time functions like constant, logarithmic, linear, quadratic, and exponential time and compares their growth.
The document discusses fundamentals of analyzing algorithm efficiency, including:
- Measuring an algorithm's time efficiency based on input size and number of basic operations.
- Using asymptotic notations like O, Ω, Θ to classify algorithms by order of growth.
- Analyzing worst-case, best-case, and average-case efficiencies.
- Setting up recurrence relations to analyze recursive algorithms like merge sort.
This document discusses analyzing recursive algorithms and forming recurrence relations. It provides examples of writing recurrence relations for recursive functions. The key steps are:
1) Identify the base case(s) where recursive calls stop.
2) Express the work done and size of subproblems at each recursive call.
3) Derive the recurrence relation relating the function at different inputs sizes.
The recurrence relation captures the work at each level of recursion and sums the costs to determine overall runtime. Analyzing recurrences helps understand the asymptotic complexity of recursive algorithms.
NumPy is a Python package that provides multidimensional array and matrix objects as well as tools to work with these objects. It was created to handle large, multi-dimensional arrays and matrices efficiently. NumPy arrays enable fast operations on large datasets and facilitate scientific computing using Python. NumPy also contains functions for Fourier transforms, random number generation and linear algebra operations.
Data structures and algorithms involve organizing data to solve problems efficiently. An algorithm describes computational steps, while a program implements an algorithm. Key aspects of algorithms include efficiency as input size increases. Experimental studies measure running time but have limitations. Pseudocode describes algorithms at a high level. Analysis counts primitive operations to determine asymptotic running time, ignoring constant factors. The best, worst, and average cases analyze efficiency. Asymptotic notation like Big-O simplifies analysis by focusing on how time increases with input size.
Second Genetic algorithm and Job-shop scheduling presentationAccenture
This document describes a genetic algorithm approach for solving job shop scheduling problems. It proposes new crossover and mutation operators designed based on the characteristics of job shop problems. The crossover operator combines operation orders from different machines in the parents. The mutation operator permutes successive operations on the same machine if they are on the critical path, to potentially reduce makespan. Experimental results using the new operators show improved convergence speed over a simple genetic algorithm.
Analysis and design of algorithms part2Deepak John
Analysis of searching and sorting. Insertion sort, Quick sort, Merge sort and Heap sort. Binomial Heaps and Fibonacci Heaps, Lower bounds for sorting by comparison of keys. Comparison of sorting algorithms. Amortized Time Analysis. Red-Black Trees – Insertion & Deletion.
Similar to Study of average losses caused by ill processing in a production line with immediate feedback and multi server facility at each of the processing units (20)
Abnormalities of hormones and inflammatory cytokines in women affected with p...Alexander Decker
Women with polycystic ovary syndrome (PCOS) have elevated levels of hormones like luteinizing hormone and testosterone, as well as higher levels of insulin and insulin resistance compared to healthy women. They also have increased levels of inflammatory markers like C-reactive protein, interleukin-6, and leptin. This study found these abnormalities in the hormones and inflammatory cytokines of women with PCOS ages 23-40, indicating that hormone imbalances associated with insulin resistance and elevated inflammatory markers may worsen infertility in women with PCOS.
A usability evaluation framework for b2 c e commerce websitesAlexander Decker
This document presents a framework for evaluating the usability of B2C e-commerce websites. It involves user testing methods like usability testing and interviews to identify usability problems in areas like navigation, design, purchasing processes, and customer service. The framework specifies goals for the evaluation, determines which website aspects to evaluate, and identifies target users. It then describes collecting data through user testing and analyzing the results to identify usability problems and suggest improvements.
A universal model for managing the marketing executives in nigerian banksAlexander Decker
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Study of average losses caused by ill processing in a production line with immediate feedback and multi server facility at each of the processing units
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.8, 2012
Study of Average Losses Caused by Ill-Processing in a Production
Line with Immediate Feedback and Multi Server Facility at Each
of the Processing Units
Abhimanu Singh 1 * Prof. C. K. Datta2 Dr. S. R. Singh3
1. Faculty of Technology, University of Delhi, Delhi, India
2. PDM College of Engineering, Bahadurgarh, Haryana, India
3. DN College, Meerut, India
*E-.mail address of corresponding author: asingh19669@yahoo.co.in
Abstract
In this paper, we have modeled a production line consisting of an arbitrary number of processing units arranged
in a series. Each of the processing units has multi-server facility. Arrivals at the first processing unit are
according to Poisson distribution and service times at each of the processing units are exponentially distributed.
At each of the processing units, the authors have taken into account immediate feedback and the rejection
possibility. Taking into account the stationary behavior of queues in series, the solution for infinite queuing space
have been found in the product form. Considering the processing cost at each of the processing units, the average
loss to the system due to rejection, caused by ill processing at various processing units, is obtained.
Keywords: Queuing Network, Processing Units, Production Line, Multi-Server, Immediate Feedback,
Stationary behavior.
1. Introduction
A production line is a sequence of a finite number of processing units arranged in a specific order. At each of the
processing units, service may be provided by one person or one machine that is called single- server facility, or it
can be provided by more than one persons or more than one machines that is called multi-server facility at the
respective processing unit. In this paper we have considered multi-server facility at each of the processing units.
At each of the processing units a specific type processing is performed i.e. at different processing units material
is processed differently. At a processing unit the processing times of different jobs or materials are independent
and are exponentially distributed around a certain value, called mean processing time. To estimate the required
measures, we represent a production line by a serial network of queues with multi- server facility at each of the
node.
Several researches have been considered the queues in series having infinite queuing space before each
servicing unit. Specifically, Jackson had considered finite and infinite queuing space with phase type service
taking two queues in series. In [7] has found that the steady state distribution of queue length taking two queues
in the system, where each of the two non-serial servers is separately in service. O.P. Sharma [1973] studied the
stationary behavior of a finite space queuing model consisting of queues in series with multi-server service
facility at each node.
In an production line the processing of raw material starts at the first processing unit. It is processed for a
certain time interval at the first processing unit and then it is transferred to the second processing unit for other
type of processing, if its processing is done correctly at the first processing unit. This sequence is followed til the
processing at the last processing unit is over.
End of processing at each of the processing units give rise to the following three possibilities:
(a) Processing at a unit is done correctly and the job or material is transferred to the next processing unit for
other type of processing.
(b) Processing at a unit is not done correctly but can be reprocessed once more at the same processing unit.
(c) Processing at a unit is neither done correctly nor it can be reprocessed at the same processing unit i.e. this
job or material is lost, in this situation the job or material is rejected and put into the scrap.
2. Modeling
Let us consider an assembl line consisting of an arbitrary number(r) processing units arranged in a series in a
specific order. Each of the processing units has multi- server facility.
26
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.8, 2012
Let λ = Mean arrival rate to the first processing unit from an infinite source, following Poisson’s rule.
µi = Mean service rate of an individual server at the i th processing unit having exponentially
distributed service times.
= Number of servers at the processing unit.
ni = Number of unprocessed jobs before the i th processing unit waiting for service, including one in
service, if any, at any time t.
pi , i +1 = Probability that the processing of a job or material at the i th processing unit is done correctly and it is
transferred to the (i + 1) st processing unit.
pi ,i = Probability that the processing of a job or material at the i th processing unit is not done correctly but it
can be reprocessed once more, so, it is transferred to the same processing unit for processing once more.
pi ,o = Probability that the processing of a job or material at the i th processing unit is neither
done correctly nor it remains suitable for reprocessing.
Ci= Processing cost per unit at ith processing unit.
L = Average loss to the system due to rejection of items at various processing units.
P ( n1 , n2 ,...nr , t ) = Probability that there are n1 jobs for processing before the first processing unit, n2
jobs before the second processing unit, and so on, nr jobs before the r th processing unit at time t, with
ni ≥ 0(1 ≤ i ≤ r ) and P (n1 , n2 ,...n r , t ) =0, if some ni <0 (because number of jobs cannot be negative).
The above production line can be represented by a serial network of queues in which each processing unit is
equivalent to a queue with the same number of similar servers and the same numbers of jobs waiting for service.
In the above serial network of queues, each queue has immediate feedback. To analyze this serial network of
queues firstly we remove the immediate feedback. After the removal of immediate feedback the above serial
network of queues is replaced by one as follows:
27
3. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.8, 2012
Here µi =
'
µi (1− pi ,i ) , where µ i ' is the effective service at the i th processing unit after the
removal of the immediate feedback as given by J.Warland (1988). We define the respective probabilities as
follows
pio pii +1
qio = , qii +1 = (1)
(1 − pii ) (1 − pii )
3. Equations Governing the Queuing System
Under the steady state conditions, we have,
[λ + n µ 1
'
1 + n 2 µ 2 + ... + n r µ r' .P (n1 , n2 ,.....nr )
'
]
r
= λ P (n1 − 1, n2 n3 ,......nr ) + ∑ n µ .q i
'
i i ,i +1 .P(n1 , n2 ,....ni + 1, ni +1 − 1,........nr ) +
i =1
r
∑ n µ .q i i
'
i ,o .P(n1 , n2 ,.....ni + 1.ni +1 ,....nr ) (2)
i =1
Dividing the above steady state equation by the factor λ + µ1' + µ 2 + ..... + µ r' the above equation is
'
[ ]
reduced to P.Q=P, where P is the row vector of the steady state probability matrix and Q is the stochastic
transition matrix.
4. Solution for Infinite Queuing System
Under the steady state conditions all the queues behave independently and thus the solution of steady state
equation in product form is given by
r
P (n1 , n2 ,.......nr ) = ∏ (1 − ρ )ρ
i =1
i i
ni
, (3) where
ni ≥ 0(1 ≤ i ≤ r ) and ρ i < 1 (1 ≤ i ≤ r )
If any ρ i (1 ≤ i ≤ r ) >1 then the stability is disturbed and the behavior of the system will not remain
stationary consequently solution will not remain valid
Here, we have
λi
ρi = ,
ni µi '
i pk −1, k
Where λ i = λ ∏ (1 − p
k =1 )
, p0, 0 = 0
k −1, k −1
Thus
λ i p k −1,k
ρi =
ni µ i
∏ (1 − p ),
k =1
With p0,1 = 1 (4)
k ,k
r
It can be seen that ∑ λ .q
i =1
i i ,o + λ r .q r , f =λ (5)
5. Evaluation of Average Loss
28
4. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.8, 2012
Let c1 be the processing cost at the first processing unit, c 2 the processing cost at the second processing unit
and so on … cr , the processing cost at the r th processing unit.
If an item is rejected just after its processing at the first processing unit is over, then it causes a loss c1 to the
system. If an item is rejected just after its processing at the second processing unit is over, then it causes a loss
(c1+c2 ) to the system. Thus, in general if an item is rejected just after its processing at the rth processing unit is
over, then it causes a loss (c1+c2+c3+…+cr ) to the system.
L, the average loss per unit time to the system due to rejection of items just after the processing at various
processing units, due to ill-processing (processing of an item is neither done correctly nor it can be reprocessed)
is
L= c1 λ q1, o + (c1 + c2 ) λ q1, 2 q2, o +… + (c1 + c2 + ...cr ) λ q1, 2 q2,3 … qr −1, r qr , o
r
= ∑ (c
i =1
1 + c2 + ...ci ) λ q1, 2 .q2,3...qi −1,i q i , o ,
With q0,1 = 1
r
∑ (c + c2 + ...ci ) λ
p1, 2 p 2,3 p i −1,i p i ,o
= … ,
1 − p1,1 1 − p 2 , 2 1 − p i −1,i −1 1 − pi ,i
1
i =1
r i p k −1,k pi,o
=λ ∑(c1 +c2 +...ci ) ∏
i=1
k =1 1 − p k −1, k −1 (
. 1− p ) (6)
i,i
With p0,0 = 0, and p0,1 = 1
6. Conclusion
The work can be used to find the approximate loss in a manufacturing system and can be extended to make
decision policies.
7. Acknowledgements
Author is thankful to Siddhartha Sirohi, Assistant Professor, Delhi University, Delhi, and Achal Kaushik,
Assistant Professor, BPIT, Rohini, Delhi, for their continuous encouragement and support. I am thankful to
management of BPIT also, for providing research oriented environment in the institute.
8. Biography
Abhimanu Singh born on 4th may 1969, got his M.Sc. Degree in mathematics from Ch.
Charan Singh University, Meerut, U. P., India, in 1996.
He started teaching Mathematics to B. Sc. Students in 1996. He has been teaching
Engineering Mathematics for the last fifteen years at Delhi Technological University
(formerly Delhi College of Engineering), Delhi, and GGSIP University, Delhi, affiliated
institutions. He has authored three books on Engineering Mathematics.
1. Applied Mathematics-I, Delhi, Delhi, Ane books Pvt. Ltd., 2010.
2. Engineering Mathematics-I, Delhi, Delhi, Ane books Pvt. Ltd., 2011.
3. Applied Mathematics-II Delhi, Delhi, Ane books Pvt. Ltd., 2011.
His current area of research is modeling with applications of Queuing Theory.
29
5. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.8, 2012
References
1) T.L.Saaty, Elements of Queueing Theory, New York, McGraw-Hill, 1961, ch. 12, pp. 260.
2) U. Narayan Bhat, An Introduction to Queueing Theory, Birkhäuser Boston, 2008, ch. 7, pp 144-147.
3) D.Gross and C.M.Harris, Fundamentals of Queueing Theory, John-Wiley New York, 1985, ch. 4, pp.
220-226
4) Guy L. Curry. Richard M. Feldman, Manufacturing System, Springer-Verlag Berlin Heidelberg, 2011, ch. 3,
pp. 77-80.
5) Kishor.S.Trivedi., Probability & Statistics with Reliability, Queuing and Computer Science Applications.
Wiley India (P.) Ltd., 4435/7, Ansari Road, Daryaganj, New Delhi, (2002), ch. 9, pp. 564
6) J. R. Jackson (1957), “Networks of waiting lines”, Oper. Res.,5, pp. 518-521.
http://or.journal.informs.org/content/5/4/518.full.pdf
7) K.L.Arya (1972), Study of a Network of Serial and Non-serial Servers with Phase Type Service and Finite
Queueing Space, Journal of Applied Probability [online] 9(1), pp198-201.
http://www.jstor.org/stable/3212649
8) O. P. Sharma (1973), A Model for Queues in Series. Journal of Applied Probability [Online] 10(3), pp.
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