Generating Function Technique Research Articles

Random Walk on T-Fractal with Stochastic Resetting

In this study, we explore the impact of stochastic resetting on the dynamics of random walks on a T-fractal network. By employing the generating function technique, we establish a recursive relation between the generating function of the first passage time (FPT) and derive the relationship between the mean first passage time (MFPT) with resetting and the generating function of the FPT without resetting. Our analysis covers various scenarios for a random walker reaching a target site from the starting position; for each case, we determine the optimal resetting probability γ* that minimizes the MFPT. We compare the results with the MFPT without resetting and find that the inclusion of resetting significantly enhances the search efficiency, particularly as the size of the network increases. Our findings highlight the potential of stochastic resetting as an effective strategy for the optimization of search processes in complex networks, offering valuable insights for applications in various fields in which efficient search strategies are crucial.

Thermodynamics of a confined macromolecule: An analytical approach

We consider a fully directed self-avoiding walk model on a cubic lattice to mimic the conformations of an infinitely long flexible polymer chain and also to mimic confirmations of a short flexible chain under confined conditions. The confinement conditions is achieved using two parallel impenetrable plates. The confined chain is under good solvent conditions and we revisit this problem to solve the real (self avoiding) polymer's model for any length of the chain and also for any given separation in between the confining plates. The equilibrium statistics of the confined polymer chain are derived using analytical approach of the generating function technique. The force of the confinement, the surface tension and the monomer density profile of the confined chain are obtained analytically. We propose that the methods of calculation are suitable to understand thermodynamics of an arbitrary length confined polymer chain under other possible conditions of the confinement.

Counting Phylogenetic Networks with Few Reticulation Vertices: Galled and Reticulation-Visible Networks.

We give exact and asymptotic counting results for the number of galled networks and reticulation-visible networks with few reticulation vertices. Our results are obtained with the component graph method, which was introduced by L. Zhang and his coauthors, and generating function techniques. For galled networks, we in addition use analytic combinatorics. Moreover, in an appendix, we consider maximally reticulated reticulation-visible networks and derive their number, too.

Analysis of a GEO/G/1 queueing system with repeated attempts pre-emptive priority and replacement in repair times

This paper deals with a discrete-time Geo/G/1 queue with repeated attempts and starting failure. If the server fails to start, it is sent for repair. During a repair process, alterations in the repair times is permitted based on current requirements. Customers are served on priority by the pre-emptive resume queue discipline. The distributions of the various system states when the system is in stable are analysed using the generating function technique. Analytical expressions are supported by numerical illustrations to exhibit the influence of the various parameters of the system on the performancemeasures.

Reliability analysis of interval-valued multi-state sliding window system for sequential tasks

In industrial applications, an interval variable with both an upper and a lower bound is utilized to define the bounded or ranged performance of a system or its elements. This paper proposes a new interval-valued multi-state sliding window system for sequential tasks (IMSWS-ST). The proposed system consists of n circularly or linearly connected multi-state elements (MEs) with their performance states described as interval-valued variables. The system function is determined by the ability of any group of r consecutive MEs for completing predetermined sequential tasks with interval-valued demands. To describe and evaluate system reliability, a universal generating function technique is employed, considering the differences between linear and circular configurations. Additionally, we develop a state aggregation method to reduce the computational burden during reliability evaluation. Numerical experiments are then conducted to demonstrate the proposed system model and validate the suggested algorithms. Finally, we investigate the optimal sequencing problem for MEs within the IMSWS-ST to achieve higher system reliability.

Moran random walk with reset and short memory

<abstract><p>We investigated the statistical properties of the Moran random walk $ (Y_n)_n $ in one dimension, focusing on short memory. Specifically, employing generating function techniques, we determined the cumulative distribution function and the mean of the height $ H_n $. Furthermore, we derived explicit expressions for the distribution, mean, and variance of $ Y_n $, along with its asymptotic distribution. Finally, we provided the distribution of the waiting time $ \tau_h $, which represents the number of steps required to reach a specified level $ h $, as the conclusion of our study.</p></abstract>

Cost optimization of a M/M/1/WV&MAV queueing system using Newton–Raphson and particle swarm optimization techniques

This paper is concerned with the optimal control of a Markovian queueing system subjected to multiple adaptive vacation and working vacation policies. This system is applicable in diverse modern technologies, in particular in call centers. We establish the steady-state solution as well as important system characteristics by means of probability generating functions technique. We also construct the expected total cost for this model and develop a procedure to determine the optimal service rate that yields the minimum cost. Further, we carried out a comparative analysis to obtain the minimum cost using the Newton–Raphson method and particle swarm optimization (PSO) algorithm.

Reliability analysis for complex electromechanical multi-state systems utilizing universal generating function techniques

The large-scale adoption of complex electromechanical systems (CMESs) renders it challenging to perform reliability assessments for such systems. In terms of the reliability evaluation of CMESs, previous studies generally assume that only binary states exist within the systems. However, owing to the stochastic failures and various operating characteristics of the system components, not only the good/bad states but also the performance levels within the system should be taken into account when analyzing the reliability of CMESs. Hence, we abstract the CMES as a multi-layer and multi-state network in which minimum maintenance units (MMUs) and three types of connections constitute nodes and edges in different layers. Furthermore, multi-state models for different MMUs in the CMES are developed based on universal generating functions (UGFs). Aggregation operators are then utilized to aggregate these UGFs to obtain the multi-state models. Subsequently, a cascading failure model is introduced to incorporate MMUs into the system reliability assessment. Thus, the effects of the operating mechanism and topological attributes can be considered in the reliability evaluation of the CEMS. Finally, the CRHX high-speed train system is used as an example to demonstrate the applicability and effectiveness of the proposed method in the reliability assessment of the CMES.

Relevance of the calculation of the diffusion coefficient in a capillary electrophoresis experiment.

This paper analyzes the role of the diffusion coefficient in the movement of analytes that can reversibly react with a selector given a product in the presence of drift. The problem mimics the movement of enantiomers in a capillary electrophoresis experiment. As is well known, the signal in the capillary must be sharp enough to make a good determination of the effective mobility of the analytes being analyzed. The essence of the technique is based on fast interconversion rates. Therefore, the effective diffusion coefficient must be negligible during the experiment. In the present work, an exact expression for both the apparent mobility and the diffusion coefficient is obtained. This is done by writing the rate equations governing the process and solving them using the generating function technique. The effective mobility coincides with the Wren and Rowe equation, whereas the diffusion coefficient allows us to determine the values of the parameters to be taken into account so that this quantity is minimal or close to zero. On the other hand, the numerical solution of the kinetic equations and Monte Carlo simulations allow us to follow the signal in the capillary and to determine its space-time evolution.

Full calculation of the long-time-interval limit of noise in reactors

The analysis of the stochastic behavior of detected signals is useful in multiplicative systems such as nuclear reactors. In this work, we derive a simple, exact expression for the long-time limit of the variance-to-mean ratio of neutron detection near a reactor using the probability generating function technique. The obtained result incorporates continuous energy and spatial effects and depends only on the steady-state flux and its adjoint. We then perform MCNP simulations to calculate the adjoint flux in the Israeli Research Reactor no. 1 (IRR-1). This allows us to explicitly calculate the said values for different configurations of varying homogeneity and criticality, and to compare the results of the simulations to the Feynman α point model results. In the comparison we find significant deviations of 10%-80%, with strong dependence on the core's multiplication factor, accentuating the need for spatial corrections.

Signature reliability scrutiny of domestic refrigerator

PurposeThe purpose of this paper is to evaluate the reliability and structure function of refrigeration complex system consisted of four components in complex manner.Design/methodology/approachAlthough, a variety of methodologies have been used to assess the refrigeration system's reliability function that has proven to be effective, the universal generating function approach is the basis of this research study, which is used in the calculation of a domestic refrigeration system with four separate components that are related in series and parallel with a corresponding sample to form a complex machine.FindingsIn this paper, signature reliability of the refrigeration system has been evaluated with the universal generating function technique. There are four components present in the proposed system in complex (series and parallel) manner. The tail signature, signature, Barlow–Proschan index, expected lifetime and expected cost of independent identically distributed are all computed.Originality/valueThis is the first study of domestic refrigeration system to examine the signature reliability with the help of universal generating function techniques with various measures. Refrigeration systems are an essential process in industries and home applications as they perform cooling or the maintain temperature at the desired value. A cycle of refrigeration consists of four main components such as, heat exchange, compression and expansion with a refrigerant flowing through the units within the cycle.

UGF-based signature reliability for solar panel k-out-of-n- multiplex systems

PurposeThe purpose of this paper is to evaluate the reliability and signature reliability of solar panel k-out-of-n-multiplex system with the help of universal generating function.Design/methodology/approachEnergy scarcity and global warming issues have become important concerns for humanity in recent decades. To solve these problems, various nations work for renewable energy sources (RESs), including sun, breeze, geothermal, wave, radioactive and biofuels. Solar energy is absorbed by solar panels, referred to as photovoltaic panels, which then transform it into electricity that can be used to power buildings or residences. Remote places can be supplied with electricity using these panels. Solar energy is often generated using a solar panel that is connected to an inverter for power supply. As a result, a converter reliability evaluation is frequently required. This paper presents a study on the reliability analysis of k-out-of-n systems with heterogeneous components. In this research, the universal generating function methodology is used to identify the reliability function and signature reliability of the solar array components. This method is commonly used to assess the tail signature and Barlow-Proschan index with independent and identically distributed components.FindingsThe Barlow-Proschan index, tail signature, signature, expected lifetime, expected cost and minimal signature of independent identically distributed are all computed.Originality/valueThis is the first study of solar panel k-out-of-n-multiplex systems to examine the signature reliability with the help of universal generating function techniques with various measures.

Stationary Analysis Fluid Model Driven by an Queue

A Fluid queue is a mathematical model used to describe fluid level in a reservoir subject to randomly determined periods of filling and emptying the system without interruption called a buffer, according to a randomly varying rate regulated by an external stochastic environment. Such fluid queues are used as a mathematical tool for modeling, for example, to approximate discrete models, model the spread of wildfires in ruin theory and to model high speed data networks, a router, computer networks including call admission control, traffic shaping and modeling of TCP and production and inventory systems. The fluid from the first phase (i.e, fluid output of the queue) goes to the second phase represents a buffer with a constant leak rate c. We always assume that service rate is greater than buffer, μ ˃ c. In this paper, a fluid queue driven by infinite queue with fixed –size batch arrivals. The generating function technique is used to obtain the expressions for steady-state distribution of both the buffer content and stationary state probabilities of background birth-death process. Hence, the performance measures are computed whereas the server utilization analysis and mean buffer content are investigated. . In addition, some numerical results are provided to illustrate the effect of various parameters on the distribution of the buffer content.

Efficient numerical evaluation of weak restricted compositions

We propose an algorithm to calculate the number of weak compositions, wherein each part is restricted to a different range of integers. This algorithm performs different orders of approximation up to the exact solution by using the Inclusion-Exclusion Principle. The great advantage of it with respect to the classical generating function technique is that the calculation is exponentially faster as the size of the numbers involved increases.

Modeling of the Activators Regenerated by Electron Transfer Copolymerization of Styrene-Acrylonitrile with Prediction of the Bivariate Molecular Weight Distribution─Copolymer Composition Distribution Using Parallel Computing and Probability Generating Functions

A mathematical model of the activators regenerated by electron transfer atom transfer radical polymerization copolymerization of styrene-acrylonitrile (SAN) is developed for the first time that calculates the bivariate molecular weight distribution (MWD)-copolymer composition distribution (CCD) of the copolymer. In addition, the model calculates the overall MWD, the overall CCD, average molecular weights and composition, and conversion. The model is implemented in the open-source programming language Julia, a high-level language designed for scientific computing, which is easy to use and that performs very fast calculations. The probability generating function technique is used to process the infinite population balances of the system. The mathematical model, composed of a very large system of equations, is solved by applying parallel computing to speed up its execution. In this way, the distributions of polymer properties are calculated in a very efficient manner. The evolution of the joint MWD–CCD and other molecular properties along the reaction path is analyzed under different operating conditions. The model allows obtaining very detailed insights into the copolymer microstructure during the polymerization reaction.

On the Hofer–Zehnder conjecture on ℂPd via generating functions

We use generating function techniques developed by Givental, Théret and ourselves to deduce a proof in [Formula: see text] of the homological generalization of Franks theorem due to Shelukhin. This result proves in particular the Hofer–Zehnder conjecture in the nondegenerate case: every Hamiltonian diffeomorphism of [Formula: see text] that has at least [Formula: see text] nondegenerate periodic points has infinitely many periodic points. Our proof does not appeal to Floer homology or the theory of [Formula: see text]-holomorphic curves. An appendix written by Shelukhin contains a new proof of the Smith-type inequality for barcodes of Hamiltonian diffeomorphisms that arise from Floer theory, which lends itself to adaptation to the setting of generating functions.

Analysis of Hetrogeneous Feedback Queue Model in Stochastic and in Fuzzy Environment Using L-R Method

In this paper, we analyse a feedback queue network in stochastic and in fuzzy environment. We consider a model with three heterogeneous servers which are commonly attached to a server in starting. At the initial stage, all queue performance measures are obtained in steady-state that is in stochastic environment. After that, work is extended to fuzzy environment because practically all characteristics of the system are not exact, they are uncertain in nature. In the present work we use probability generating function technique, triangular fuzzy numbers, classical formulae for the calculation of all queue characteristics and L-R method to calculate queue characteristics in fuzzy environment.

Reliability analysis and optimal generator allocation and protection strategy of a non-repairable power grid system

In a power grid system, generators may fail due to the internal structure and material degradation caused by stress levels or work conditions. However, power grid systems may also be subject to external impacts such as landslides and earthquakes. Protecting these systems can mitigate the external negative effects. In this research, we extend the general network reliability model with consideration of the individual generator protection and the overarching protection. The generators can be destroyed if and only if both the individual protection and the overarching protection are destroyed. A joint generator allocation and protection optimization problem is formulated to increase the system reliability. We propose an algorithm to evaluate the system reliability by extending the universal generating function technique. Numerical experiments show that determining the generator allocation and the protection strategy optimally can improve the system reliability.

Computing Optimized Path Integrals for Knapsack Feasibility

A generating function technique for solving integer programs via the evaluation of complex path integrals is discussed from a theoretical and computational perspective. Applying the method to knapsack feasibility problems, it is demonstrated how the presented numerical integration algorithm benefits from a preoptimized path of integration. After discussing the algorithmic setup in detail, a numerical study is implemented to evaluate the computational performance of the preoptimized integration method, and the algorithmic parameters are tuned to a set of knapsack instances. The goal is to highlight the method’s computational advantage for hard knapsack instances. Summary of Contribution: A method for evaluating the feasibility of knapsack problems is discussed and connected to the existing theory on generating function techniques for computational integer optimization. Specifically, the number of solutions to knapsack instances is computed using numerical quadrature of complex path integrals. The choice of the path of integration is identified as an important degree of freedom, and it is shown that preoptimizing the path improves the computational performance for hard instances significantly. The scope of this work is to give a self-contained presentation of the mathematical theory, introduce and discuss path optimization as a presolving routine, and demonstrate the computational performance of the overall approach with the help of a detailed numerical study. The main goal is to highlight the computational advantage of the new algorithm for hard knapsack instances.

Orthogonal polynomials associated with a continued fraction of Hirschhorn

We study orthogonal polynomials associated with a continued fraction due to Hirschhorn. Hirschhorn’s continued fraction contains as special cases the famous Rogers–Ramanujan continued fraction and two of Ramanujan’s generalizations. The orthogonality measure of the set of polynomials obtained has an absolutely continuous component. We find generating functions, asymptotic formulas, orthogonality relations, and the Stieltjes transform of the measure. Using standard generating function techniques, we show how to obtain formulas for the convergents of Ramanujan’s continued fractions, including a formula that Ramanujan recorded himself as Entry 16 in Chapter 16 of his second notebook.