Recursive Formula Research Articles

Equilibrium Strategies in an Mn/M/1 Queue with Server Breakdowns and Delayed Repairs

Ophthalmic units use sophisticated equipment to enable accurate diagnosis of refractive errors. This equipment is subject to two types of breakdowns. One is simple breakdown which can be repaired in-house, and the other is complex breakdown which requires repair by the original equipment manufacturer (OEM) and results in a delay time between the server breakdown and the start of the repair. In this paper, we model this scenario as an Mn/M/1 queuing system with two types of breakdowns and delayed repairs due to complex breakdowns, where the delay time and repair times for simple and complex breakdowns are generally distributed. We obtain the steady-state probabilities and provide the recursive formulas for the Laplace–Stieltjes transforms (LSTs) of conditional residual delay time and repair times given the system state. For the fully observable case, we derive the equilibrium joining strategies of customers who decide to join or balk based on their observation of the system state. Moreover, two numerical experiments are conducted to explore the equilibrium joining probabilities.

X-vine models for multivariate extremes

Abstract Regular vine sequences permit the organization of variables in a random vector along a sequence of trees. Vine-based dependence models have become greatly popular as a way to combine arbitrary bivariate copulas into higher-dimensional ones, offering flexibility, parsimony, and tractability. In this project, we use regular vine sequences to decompose and construct the exponent measure density of a multivariate extreme value distribution, or, equivalently, the tail copula density. Although these densities pose theoretical challenges due to their infinite mass, their homogeneity property offers simplifications. The theory sheds new light on existing parametric families and facilitates the construction of new ones, called X-vines. Computations proceed via recursive formulas in terms of bivariate model components. We develop simulation algorithms for X-vine multivariate Pareto distributions as well as methods for parameter estimation and model selection on the basis of threshold exceedances. The methods are illustrated by Monte Carlo experiments and a case study on US flight delay data.

Polynomial Invariants for Rooted Trees Related to Their Random Destruction

We consider three bivariate polynomial invariants $P$, $A$, and $S$ for rooted trees, as well as a trivariate polynomial invariant $M$. These invariants are motivated by random destruction processes such as the random cutting model or site percolation on rooted trees. We exhibit recursion formulas for the invariants and identities relating $P$, $S$, and $M$. The main result states that the invariants $P$ and $S$ are complete, that is they distinguish rooted trees (in fact, even rooted forests) up to isomorphism. The proof method relies on the obtained recursion formulas and on irreducibility of the polynomials in suitable unique factorization domains. For $A$, we provide counterexamples showing that it is not complete, although that question remains open for the trivariate invariant $M$.

Recursive and Explicit Formulas for Expansion and Connection Coefficients in Series of Classical Orthogonal Polynomial Products

Recursive and Explicit Formulas for Expansion and Connection Coefficients in Series of Classical Orthogonal Polynomial Products

Interval and $\ell$-interval Rational Parking Functions

Interval parking functions are a generalization of parking functions in which cars have an interval preference for their parking. We generalize this definition to parking functions with $n$ cars and $m\geq n$ parking spots, which we call interval rational parking functions and provide a formula for their enumeration. By specifying an integer parameter $\ell\geq 0$, we then consider the subset of interval rational parking functions in which each car parks at most $\ell$ spots away from their initial preference. We call these $\ell$-interval rational parking functions and provide recursive formulas to enumerate this set for all positive integers $m\geq n$ and $\ell$. We also establish formulas for the number of nondecreasing $\ell$-interval rational parking functions via the outcome map on rational parking functions. We also consider the intersection between $\ell$-interval parking functions and Fubini rankings and show the enumeration of these sets is given by generalized Fibonacci numbers. We conclude by specializing $\ell=1$, and establish that the set of $1$-interval rational parking functions with $n$ cars and $m$ spots are in bijection with the set of barred preferential arrangements of $[n]$ with $m-n$ bars. This readily implies enumerative formulas. Further, in the case where $\ell=1$, we recover the results of Hadaway and Harris that unit interval parking functions are in bijection with the set of Fubini rankings, which are enumerated by the Fubini numbers.

Finite-region dissipative control for 2-D switched systems via mode-dependent average dwell time

Dissipativity is a significant concern for 2-D switched systems, but existing research often overlooks the transient dynamics occurring within a finite region. This paper aims to address finite-region dissipativity and dissipative control for a class of Fornasini-Marchesini local state-space (FMLSS) type 2-D switched systems. The analysis employs the mode-dependent average dwell time approach. Firstly, the concept of dissipativity on finite-region for 2-D switched systems is introduced. Then, a sufficient condition under which the considered systems can demonstrate the finite-region boundedness with the external disturbances is established by the special recursive formulas. Additionally, a sufficient condition for finite-region dissipativity of the considered systems is provided. Furthermore, the dissipative state feedback controller is derived to ensure the finite-region dissipativity of the closed-loop systems. To illustrate the effectiveness of the proposed controller design approach, an example is presented.

The Reliability of a Class of Two-Layer Networks with Unreliable Edges

It is well known that networks are dynamic graphs, and the topology of a network can be described by a graph. Thus, the reliability of a network under edge failure is defined as the probability that its corresponding topological graph remains connected under the condition that the edges fail with independent probabilities. In this paper, the reliability of a class of two-layer networks is considered, where each layer is a complete graph and the edges joining different layers are one-to-one correspondingly connected. The edge failure probability is uniform in the same layer and distinct in different layers and between layers. The recursive formula for the reliability (reliability polynomial) of the two-layer network is obtained, and the corresponding algorithm is also given. Furthermore, the reliability of several networks is computed by Python, which verifies the correctness of the algorithm.

Local invariants of conformally deformed non-commutative tori II: Multiple operator integrals

We explicitly compute the local invariants (heat kernel coefficients) of a conformally deformed non-commutative d-torus using multiple operator integrals. We derive a recursive formula that easily produces an explicit expression for the local invariants of any order k and in any dimension d. Our recursive formula can conveniently produce all formulas related to the modular operator, which before were obtained in incremental steps for d∈{2,3,4} and k∈{0,2,4}. We exemplify this by writing down some known (k=2, d=2) and some novel (k=2, d≥3) formulas in the modular operator.

Transient analysis of a SIQS model with state capacities using a non-homogeneous Markov system

In this paper, a novel stochastic model is proposed to model the spread of a virus in epidemics. The model is based on a discrete-time non-homogeneous Markov system with state capacities. In order to study the distributions of the state sizes, recursive formulae for their factorial and mixed factorial moments were derived in matrix form. As a consequence, the probability mass function of each state size can be evaluated in the transient period. To avoid the computational complexity of the proposed algorithm, an alternative method for the computation of the state size distributions was recommended. The proposed Markovian approach was then tailored to the characteristics of a SIQS (susceptible-infected-quarantined-susceptible) epidemic scheme, which took into account external infections and the potential for secondary infections. This epidemic model is well-suited for describing infections in computer networks, where the quarantine capacity can be likened to the number of working people (IT professionals) available to restore an infected computer. We presented numerical examples and sensitivity analysis to illustrate the behavior and performance of the system under different scenarios and parameter values. We show that the state capacities and the infection rates have significant effects on the evolution and extinction of the epidemic. We note that the optimal number of employed technicians can be identified, aiming to keep the computer network functional. Higher internal infection rates significantly affect the sustainability of the computer network, while controlling external infections is not always feasible. On the other hand, faster detection rates and higher malware elimination rates will considerably increase the number of computers that remain operational in the long term. Consequently, the quality of services provided by IT technicians plays a crucial role in the system’s viability.

DUAL RECURSIVE FORMULAS FOR THE SUMS OF POWERS OF INTEGERS

In this note, we introduce the concept of a dual pair of recursive formulas for the sums of powers of integers Central to this concept is the symmetry property exhibited by the power sum polynomial . We illustrate the concept by some examples taken from the literature, and derive our own pair of dual recursive formulas for .

Leibniz rule for the high q$$ q $$‐derivatives of a quotient of two functions

In this paper, we introduce some explicit and recursive formulas for the th ‐derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between commutative algebras and the properties of ‐differential operator. One of the formulas involves the determinant of the functions and their ‐derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, we used formulas to obtain some interesting ‐identities.

In-profile monitoring on univariate profile with application to industrial busbar

In the current research on profile monitoring, most studies treat each profile as a whole to design monitoring statistic. These monitoring methods can only detect whether there exist some anomalies in the process after a complete profile sample is collected. This leads to a lag between the occurrence of a shift and the signaling of an alarm, which hinders engineers from promptly intervening in the out-of-control process. To address this limitation, an in-profile monitoring scheme is proposed in this paper, in which the dynamic influence mechanism of covariates on the response variable is considered. In phase-I, a random varying-coefficient model is utilized to model the dynamic time-varying relationship between covariates and the response variable, and the model parameters are estimated. In Phase-II, for the sequential observations generated by the monitored process, a new monitoring scheme based on the generalized likelihood ratio test is designed. This scheme can adapt to within-profile autocorrelation and arbitrary design points. To enhance online computational efficiency, recursive formulas for calculating the charting statistic are developed. Numerical studies demonstrate that the proposed scheme exhibits satisfactory and robust monitoring performance. Finally, an application to industrial busbar running process monitoring is given to demonstrate the implementation of the scheme.

Recursive Formula for Sum of Powers of Natural Numbers and its Generalization to Arithmetic Progression

Recursive Formula for Sum of Powers of Natural Numbers and its Generalization to Arithmetic Progression

Efficient Modeling of Skin and Proximity Effects Over Ultrawide Frequency Range, Part II: Multi-Conductor Arrays and Recursive Formulas

Efficient Modeling of Skin and Proximity Effects Over Ultrawide Frequency Range, Part II: Multi-Conductor Arrays and Recursive Formulas

Finite-frequency model order reduction of linear and bilinear systems via low-rank approximation

In this paper, we first investigate the finite-frequency model order reduction for linear systems based on low-rank Gramian approximations. An efficient algorithm for computing low-rank approximations of the finite-frequency and frequency-dependent Gramians based on Laguerre functions is proposed. The approach constructs the low-rank decomposition factors of the finite-frequency Gramians or frequency-dependent Gramians through a recursive formula of Laguerre functions expansion coefficient vectors and then combines the low-rank square root method and frequency-dependent balanced truncation method to obtain the reduced-order models. In this process, it avoids dealing with the matrix-valued functions and solving the related (generalized) Lyapunov matrix equations directly, making them computationally efficient. Furthermore, the above method is successfully extended to bilinear systems, and a corresponding efficient computation method for low-rank approximations of the finite-frequency Gramians of bilinear systems is derived. Finally, some numerical simulations are provided to illustrate the effectiveness of our proposed algorithms.

Generalized Poisson random variable: its distributional properties and actuarial applications

Abstract Generalized Poisson (GP) distribution was introduced in Consul & Jain ((1973). Technometrics, 15(4), 791–799.). Since then it has found various applications in actuarial science and other areas. In this paper, we focus on the distributional properties of GP and its related distributions. In particular, we study the distributional properties of distributions in the $\mathcal{H}$ family, which includes GP and generalized negative binomial distributions as special cases. We demonstrate that the moment and size-biased transformations of distributions within the $\mathcal{H}$ family remain in the same family, which significantly extends the results presented in Ambagaspitiya & Balakrishnan ((1994). ASTINBulletin: the Journal of the IAA, 24(2), 255–263.) and Ambagaspitiya ((1995). Insurance Mathematics and Economics, 2(16), 107–127.). Such findings enable us to provide recursive formulas for evaluating risk measures, such as Value-at-Risk and conditional tail expectation of the compound GP distributions. In addition, we show that the risk measures can be calculated by making use of transform methods, such as fast Fourier transform. In fact, the transformation method showed a remarkable time advantage over the recursive method. We numerically compare the risk measures of the compound sums when the primary distributions are Poisson and GP. The results illustrate the model risk for the loss frequency distribution.

Equivariant Divergence Formula for Hyperbolic Chaotic Flows

We prove the equivariant divergence formula for axiom A flow attractors. It is a pointwisely-defined and recursive formula for perturbation of SRB measures along center-unstable manifolds. It depends on only the zeroth and first order derivatives of the map, the observable, and the perturbation. Hence, the linear response acquires an ‘ergodic theorem’, which means that it can be sampled by recursively computing a few vectors on one orbit.

A Stein characterisation of the distribution of the product of correlated normal random variables

We obtain a Stein characterisation of the distribution of the product of two correlated normal random variables with non-zero means, and more generally the distribution of the sum of independent copies of such random variables. Our Stein characterisation is shown to naturally generalise a number of other Stein characterisations in the literature. From our Stein characterisation we derive recursive formulas for the moments of the product of two correlated normal random variables, and more generally the sum of independent copies of such random variables, which allows for efficient computation of higher order moments.

A Fusion Tracking Algorithm for Electro-Optical Theodolite Based on the Three-State Transition Model.

This study presents a novel approach to address the autonomous stable tracking issue in electro-optical theodolite operating in closed-loop mode. The proposed methodology includes a multi-sensor adaptive weighted fusion algorithm and a fusion tracking algorithm based on a three-state transition model. A refined recursive formula for error covariance estimation is developed by integrating attenuation factors and least squares extrapolation. This formula is employed to formulate a multi-sensor weighted fusion algorithm that utilizes error covariance estimation. By assigning weighted coefficients to calculate the residual of the newly introduced error term and defining the sensor's unique states based on these coefficients, a fusion tracking algorithm grounded on the three-state transition model is introduced. In cases of interference or sensor failure, the algorithm either computes the weighted fusion value of the multi-sensor measurement or triggers autonomous sensor switching to ensure the autonomous and stable measurement of the theodolite. Experimental results indicate that when a specific sensor is affected by interference or the off-target amount cannot be extracted, the algorithm can swiftly switch to an alternative sensor. This capability facilitates the precise and consistent generation of data, thereby ensuring the stable operation of the tracking system. Furthermore, the algorithm demonstrates robustness across various measurement scenarios.

Spin‐s$s$ Dicke States and Their Preparation

AbstractThe notion of spin‐ Dicke states is introduced, which are higher‐spin generalizations of usual (spin‐1/2) Dicke states. These multi‐qudit states can be expressed as superpositions of qudit Dicke states. They satisfy a recursion formula, which is used to formulate an efficient quantum circuit for their preparation, whose size scales as , where is the number of qudits and is the number of times the total spin‐lowering operator is applied to the highest‐weight state. The algorithm is deterministic and does not require ancillary qudits.