In this present work A single server fixed batch queuing method is considered with fixed arrival and service occurs according to poisson process, under multiple vacation and system reset. Further, It is supposed that the vacation has an exponential distribution and that the system reset happens as a Poisson process. In this model, the system starts with k customers, and the server immediately begins service for a batch of size k. When a service is finished and there are fewer than k customers in line, the server takes a vacation. When the vacation ends, If the count of customers is still insufficient, the server continues taking multiple vacations of average length 1/α. If there are k or more customers, the server selects the first k customers and starts the next batch service. The system also considers the possibility of a system reset. Whenever a reset occurs, all customers in the system are removed, and the server goes on a vacation. Numerical experiments and graphical representations are used to study the mean and variance change with different Different rates for service rate, batch size, vacation rate, and reset rate have been considered in the present study. It has been observed that the developed model is helpful in the analysis of the performance of automatic service systems facing periodic disruptions.
The Heat Equation is a well-known partial differential equation that can be solved numerically by finite difference methods in time coupled with finite element methods in space. Crank-Nicolson methods can therefore be applied as finite difference methods and coupled with linear Lagrange finite element methods in order to solve the Heat equation and obtain an efficient convergence rate, the Heat equation is said to be solved by a Crank-Nicolson finite element method. However, the convergence rate of the Crank-Nicolson finite element method for the Heat equation can be affected if the exact solution entails time singularities; in that case the lack of smoothness of the solution though it is local in time, affects the convergence of the finite element method in the whole domain. This paper presents a Crank-Nicolson finite element method coupled to a Predictor-corrector algorithm to recover the optimal convergence rate when the solution has time singularities. The finite element method presented is based on the approximation of the time singular functions using a Fourier decomposition of the exact solution that leads to computable formulas of the time dependent coefficients of singularities that reduce the smoothness of the solution; so removing those coefficients ameliorate the efficiency of the Crank-Nicolson finite element method. Numerical experiments are presented to show the efficiency of the method.
The root-finding problem is one of the most important problems in Numerical Analyis. It arises in a wide variety of practical applications in physics, chemistry, biosciences, engineering, etc. As a matter of fact, determination of any unknown appearing implicitly led to the evolution of root-finding problem. Therefore, this paper focuses on the modification and analysis of a high order derivative-free iteration methods for finding roots of nonlinear algebraic equations of the form f (x) = 0. The methods require only one initial approximation. The proposed method is seen as an extension of the second-order Steffensen’s scheme, which is an Iterative method for approximating roots of non-linear equations which often breaks down when the derivative of the function value is zero or near zero at the point of iteration. This work therefore, seeks to introduce a method that overcomes such breakdown. The method herein is a combination of forward difference formula with Simpson’s quadrature in spirit of Steffensen. The idea is to modify the Steffensen’s method, which were recently developed to obtain derivative-free methods. The modified methods are shown to converge. We also describe how to obtain derivative-free methods to find solutions to multiple roots. Several numerical examples are provided to validate the theoretical order of convergence for nonlinear functions with simple roots and results obtained show the comparative advantage the proposed method has over well-known methods.
Abstract: A dominating set of a connected graph G = (V, E) is a subset D of V (G) such that every vertex of G is either in D or adjacent to a vertex in D . A resolving set of a connected graph G is a subset D of V (G) such that each vertex v of G has a unique representation with respect to D . A dominant resolving set of a graph G is a subset D of V (G) that resolves all vertices of G and dominates G . The dominant metric dimension, Ddim(G) , is the cardinality of the smallest such set. The dominant metric dimension of a graph combines metric dimension (location identification) with domination (coverage), making it ideal for sensor networks that require both unique positioning and full coverage. Similarly, a dominant edge resolving set of a connected graph G is a vertex subset D of V that resolves all edges and is a covering of G. Star fan graphs are composite graph structures formed by attaching fan graphs to the pendant vertices of a star graph, creating hierarchical networks that are useful for modeling hub-and-spoke systems with additional path structures. This paper computes the dominant metric dimension and dominant edge metric dimension of star fan graphs.
In the context of the ongoing digital transformation of governmental and corporate information systems, the development of intelligent document management solutions capable of efficient processing, structuring, and analysis of textual data has become increasingly important. Particular challenges arise in the processing of multilingual data and low-resource languages, such as Tajik, due to the limited availability of annotated corpora. The aim of this study is to develop and formalize a mathematical model of an intelligent document management system based on microservice architecture and transformer-based natural language processing techniques. The proposed approach integrates a distributed microservice architecture using gRPC with a named entity recognition (NER) model based on multilingual BERT. To address data scarcity, a synthetic data generation mechanism is introduced to augment the training corpus. The NER task is formulated as a probabilistic sequence labeling problem, and the training procedure includes fine-tuning of the transformer model and comparison with baseline approaches, including rule-based methods, Conditional Random Fields (CRF), and BiLSTM-CRF models. Experimental evaluation is conducted on a curated corpus of Tajik-language documents, divided into training, validation, and test subsets. The results demonstrate that the proposed model achieves an F1-score of 0.93, outperforming all baseline methods. In addition, the system exhibits near-linear scalability under horizontal scaling conditions and ensures fault tolerance through a hybrid mechanism that switches to a rule-based extractor in case of service unavailability. The proposed model provides a scalable and robust framework for intelligent document processing systems and can be effectively applied in governmental and corporate environments undergoing digital transformation.
