Generating Function Method Research Articles

Performance analysis of a discrete-time Geo/G/1 retrial queue under <i>J</i>-vacation policy

We have investigated a discrete-time Geo/G/1 retrial queue under J vacation policy and general retrial times, which can be applicable to many digital systems, viz. broadband integrated services digital network (B-ISDN), asynchronous transfer mode (ATM) and circuit-switched time-division multiple access (TDMA) systems. In discrete-time systems, we consider time to be a discrete random variable and measure it in fixed size data units, such as machine cycle time, bits, bytes, packets, etc. The server follows J vacation policy according to which as soon as the retrial orbit becomes empty and no new customer arrives then the server may take at most J number of repeated vacations. The server immediately returns from the vacation if at least one customer arrives in the system. The inter arrival time is assumed to be geometrically distributed whereas retrial time, service time and vacation time are assumed to be generally distributed in the discrete environment. The underlying Markov process is analysed by using generating function method whereby we obtain various performance measures of interest. We provide stochastic decomposition laws for the system size of the developed model.

Performance analysis of a discrete-time Geo/G/1 retrial queue under <i>J</i>-vacation policy

We have investigated a discrete-time Geo/G/1 retrial queue under J vacation policy and general retrial times, which can be applicable to many digital systems, viz. broadband integrated services digital network (B-ISDN), asynchronous transfer mode (ATM) and circuit-switched time-division multiple access (TDMA) systems. In discrete-time systems, we consider time to be a discrete random variable and measure it in fixed size data units, such as machine cycle time, bits, bytes, packets, etc. The server follows J vacation policy according to which as soon as the retrial orbit becomes empty and no new customer arrives then the server may take at most J number of repeated vacations. The server immediately returns from the vacation if at least one customer arrives in the system. The inter arrival time is assumed to be geometrically distributed whereas retrial time, service time and vacation time are assumed to be generally distributed in the discrete environment. The underlying Markov process is analysed by using generating function method whereby we obtain various performance measures of interest. We provide stochastic decomposition laws for the system size of the developed model.

A New Method for the Optimal Control Problem of Path Planning for Unmanned Ground Systems

To model the optimal control problem of path planning for unmanned ground systems (UGSs), the motion and boundary constraints are described first by using the mathematical model proposed in this paper, and the time-energy performance indicators are described by the Bolza cost function. Since the traditional symplectic algorithm hardly solves the problem with uncertain time, the pseudospectral method, almost the only way to solve the optimal control problem of path planning while the rapid path planning is difficult to achieve, is prone to the phenomenon named “Curse of Dimensionality”with increasing the number of discrete points. The symplectic pseudospectral method for improving the efficiency and the precision of the calculation, based on the symplectic theory, third kind of generation function and pseudospectral method is first proposed in this paper. Furthermore, the one-sided approximation is designed, and the one-sided symplectic pseudospectral (OSSP) algorithm is established to solve the model introduced in this paper. Finally, the experiments are conducted using the OSSP method and the pseudospectral method, respectively, to verify the feasibility and the efficiency of the method. The results show that the OSSP is the method with the highest accuracy, efficiency, and good stability to solve the optimal control problem of path planning for UGS, and it has great maneuverability and feasibility for practical application.

Generating Function Approach to the Derivation of Higher-Order Iterative Methods for Solving Nonlinear Equations

In this paper we propose a generating function method for constructing new two and three-point iterations withp(p= 4, 8) order of convergence. This approach allows us to derive a new family of optimal order iterative methods that include well known methods as special cases. Necessary and sufficient conditions forp-th (p= 4, 8) order convergence of the proposed iterations are given in terms of parameters τnand αn. We also propose some generating functions for τnand αn. We develop a unified representation of all optimal eighth-order methods. The order of convergence of the proposed methods is confirmed by numerical experiments.

A higher-order convolution for Bernoulli polynomials of the second kind

In this paper, we perform a further investigation for the Bernoulli polynomials of the second kind. By making use of the generating function methods and summation transform techniques, we establish a higher-order convolution identity for the Bernoulli polynomials of the second kind. We also present some illustrative special cases as well as immediate consequences of the main result.

Developments in quantitative dimensional synthesis (1970–present): four-bar path and function generation

Dimensional synthesis is a type of inverse problem in linkage kinematics where the objective is to calculate the linkage dimensions required to achieve prescribed linkage output motion. It includes three distinct subcategories: motion generation, path generation and function generation. The authors’ first work surveyed developments in quantitative four-bar motion generation. This work concludes the presentation of developments in four-bar dimensional synthesis by surveying the developments made in quantitative four-bar path and function generation since 1970. While early path and function generation methods (like motion generation methods) were primarily qualitative, ongoing advancements in computing hardware and software continue to make quantitative motion generation more practical.

Generating function method for constructing new iterations

In this paper we propose a generating function method for constructing new two- and three-point iterations with p (3 ≤ p ≤ 8) order of convergence. This approach allows us to derive a new family of the optimal order iterative methods that include well known methods as special cases. The necessary and sufficient conditions for pth order convergence of the proposed iterations are given in terms of parameters τn and αn. We also propose some generating functions for τn and αn. We give the extension of a class of optimal fourth-order Jarratt’s type iterations with a≠23. We develop a unified representation of all optimal eighth-order methods. Several numerical results are given to demonstrate the efficiency and the performance of the presented methods and compare them with some other existing methods.

Solution to urn models of pairwise interaction with application to social, physical, and biological sciences.

We investigate a family of urn models that correspond to one-dimensional random walks with quadratic transition probabilities that have highly diverse applications. Well-known instances of these two-urn models are the Ehrenfest model of molecular diffusion, the voter model of social influence, and the Moran model of population genetics. We also provide a generating function method for diagonalizing the corresponding transition matrix that is valid if and only if the underlying mean density satisfies a linear differential equation and express the eigenvector components as terms of ordinary hypergeometric functions. The nature of the models lead to a natural extension to interaction between agents in a general network topology. We analyze the dynamics on uncorrelated heterogeneous degree sequence networks and relate the convergence times to the moments of the degree sequences for various pairwise interaction mechanisms.

An exactly solvable model of polymerization

This paper considers the evolution of a polydisperse polymerizing system comprising g1,g2… – mers carrying ϕ1,ϕ2… functional groups reacting with one another and binding the g-mers together. In addition, the g-mers are assumed to be added at random by one at a time with a known rate depending on their mass g and functionality ϕ. Assuming that the rate of binding of two g-mers is proportional to the product of the numbers of nonreacted functional groups the kinetic equation for the distribution of clusters (g-mers) over their mass and functionalities is formulated and then solved by applying the generating function method. In contrast to existing approaches this kinetic equation operates with the efficiencies proportional to the product of the numbers of active functional groups in the clusters rather than to the product of their masses. The evolution process is shown to reveal a phase transition: the emergence of a giant linked cluster (the gel) whose mass is comparable to the total mass of the whole polymerizing system. The time dependence of the moments of the distribution of linked components over their masses and functionalities is investigated. The polymerization process terminates by forming a residual spectrum of sol particles in addition to the gel.

Statistical moments in superposition models and strongly intensive measures

First, we present a concise glossary of formulas for composition of standard, cumulant, factorial, and factorial cumulant moments in superposition (compound) models, where final particles are created via independent emission from a collection of sources. Explicit mathematical formulas for the composed moments are given to all orders. We discuss the composition laws for various types of moments via the generating-function methods and list the formulas for the unfolding of the unwanted fluctuations. Second, the technique is applied to the difference of the scaled multiplicities of two particle types. This allows us for a systematic derivation and a simple algebraic interpretation of the so-called strongly intensive fluctuation measures. With the help of the formalism we obtain several new strongly intensive measures involving higher-rank moments. The reviewed as well as the new results may be useful in investigations of mechanisms of particle production and event-by-event fluctuations in high-energy nuclear and hadronic collisions, and in particular in the search for signatures of the QCD phase transition at a finite baryon density.

Some new formulas for the products of the Frobenius-Euler polynomials

The main purpose of this paper is, using the generating function methods and summation transform techniques, to establish some new formulas for the products of an arbitrary number of the Frobenius-Euler polynomials and give some illustrative special cases.

A relatively simple model for percolation properties of real networks

Analyzing percolation rules of real networks has some great realistic significance. In this paper, we develop a relatively simple model based on generating function method to study percolation properties of real networks. We construct our model for both site and bond percolation, compare its estimates with those of the message passing algorithm and simulation results on computer-generated networks as well as practical networks, and discuss causes of the inaccuracy. The conclusions show that the accuracy of our model could be accepted though it is lower than that of the message passing algorithm and the discrepancies between the estimates of our model and the simulation values mainly come from the disagreement of those real networks with the model hypotheses.

Weighted Voting System With Unreliable Links

In a weighted voting system (WVS), outputs of voting units are sometimes not properly transferred to a sink unit because of the unreliability of links between the voting units and the sink unit. In this study, a WVS model with unreliable links is established, and two different kinds of unreliable links are introduced. Type I links have two states, which are connection and disconnection, and type II links have one more state (error state) than type I links. An approach based on a universal generating function method is suggested for evaluating a WVS with unreliable links. Two definitions of sensitivity are presented, which reflect the effect of a link's reliability or a system parameter on system reliability. Reliability evaluation and sensitivity analysis are demonstrated by some examples, and a conclusion is given.

Reliability and importance measures for m-consecutive-k, l-out-of-n system with non-homogeneous Markov-dependent components

We study an m-consecutive-k, l-out-of-n system with non-homogenous Markov-dependent components. The m-consecutive-k, l-out-of-n:F system fails if and only if there are at least m runs of k consecutive failed components and each of the runs may have at most l components overlapping with the previous run of k consecutive failed components. Using probability generating function method, we derive closed-form formulas for the reliability of the m-consecutive-k, l-out-of-n:F system, the marginal reliability importance measure of a single component, and the joint reliability importance measure of two or more components when components are non-homogenous Markov-dependent. We also extend these results into an analogous m-consecutive-k, l-out-of-n:G system, which is developed by considering consecutive working components. The results can be simplified to the situations of the homogenous Markov-dependent components and the independent components. We present a practical application in quality control and related numerical examples that demonstrate the use of derived formulas and provide the insights on the m-consecutive-k, l-out-of-n system and the importance measures.

Inhomogeneous Site Percolation on an Irregular Bethe Lattice with Random Site Distribution

In this paper, we study inhomogeneous site percolation on an irregular Bethe lattice, for considering that percolation often occurs on irregular grids or lattices with variable site neighbours in real-world problems. The explicit expression for cluster-size distribution of this percolation is derived based on probability theory. Moreover, the exact formulas for critical occupation probability, mean cluster size, and percolation probability are obtained using generating function method and generalised recursive approach. In addition, sensitivity analysis and numerical simulation are given to deepen and illustrate the results.

Summation formulae of products of the Apostol–Bernoulli and Apostol–Euler polynomials

In this paper, a further investigation for the Apostol–Bernoulli and Apostol–Euler polynomials is performed, and a new formula of products of the Apostol–Bernoulli and Apostol–Euler polynomials is established by applying the generating function methods and some summation transform techniques. It turns out that some known results are obtained as special cases.

A linear weighted (n, f, k) system for non-homogeneous Markov-dependent components

ABSTRACTWe study a linear weighted (n, f, k) system, denoted by L(n, f, k, w) system and consider the situation where components are non-homogeneous Markov-dependent. An L(n, f, k, w) system consists of n components ordered in a line, and each component u has a positive integer weight wu for u = 1, 2, …, n and w = (w1, w2, …, wn). The L(n, f, k, w):F (G) system fails (works) if the total weight of failed (working) components is at least f or the total weight of consecutive failed (working) components is at least k. For the L(n, f, k, w):F system with non-homogeneous Markov-dependent components, we derive closed-form formulas for the system reliability, the marginal reliability importance measure of a single component, and the joint reliability importance measure of multiple components using a conditional probability generating function method. We extend these results to the L(n, f, k, w):G systems, the weighted consecutive-k-out-of-n systems, and the weighted f-out-of-n systems. Our numerical examples and a case study on a bridge system demonstrate the use of derived formulas and provide the insights on the L(n, f, k, w) systems and the importance measures. In addition, the two failure modes associated with the L(n, f, k, w):F systems are analyzed by comparing to the single failure mode associated with the weighted consecutive-k-out-of-n:F systems and the single failure mode associated with the weighted f-out-of-n:F systems.

Effect of feed rate of end-capping molecules on structure of hyperbranched polymers formed from monomers A2 and B4 in semibatch process

Hyperbranched polymers that were prepared by the polymerization of monomers A2 and B4 with the gradual addition of the end-capping molecules AR to the reactor were studied using a kinetic model. The degree of polymerization (DP) and fraction of branch points (FB) under various compositions and feed rate parameters (ϕ) were calculated using a generating function method. The growing profile of the reduced weight-average DP (DPw∗) dependent on the conversion of group B, α(B), in a semibatch process was different from that in a batch reactor. At the same composition of the reactants, the DPw∗ of HBPs prepared under a lower addition rate of AR was higher than that at a quicker feed rate or in the batch process. Either a lower ratio of AR (λ1) or a higher ratio of A2 (λ2) could increase the DPw∗. According to the proposed model, the HBPs without the gel fraction could be obtained at high conversion via the continuous addition of AR to the reactor. For example, the HBPs with a high weight-average DP approximately 478, could be obtained with ϕ=0.1; λ1=2.68; and λ2=0.66 at α(B)=α(A)=0.95 in the semi-batch process, and the DB and FB of the HBP were 0.987 and 0.238, respectively. On the contrary, the DPw∗ was only 45 in a batch system at the same composition of reactants.

Some convolution identities for Frobenius-Euler polynomials

In this paper, by applying the generating function methods and summation transform techniques, we establish some new convolution identities for the Frobenius-Euler polynomials. It turns out that some well-known results are obtained as special cases.

Evaluation of Hessenberg determinants via generating function approach

In this paper, we will present various results on computing of wide classes of Hessenberg matrices whose entries are the terms of any sequence. We present many new results on the subject as well as our results will cover and generalize earlier many results by using generating function method. Moreover, we will present a new approach on computing Hessenberg determinants, whose entries are general higher order linear recursions with arbitrary constant coefficients, based on finding an adjacency-factor matrix. We will give some interesting showcases to show how to use our new method.