Reliability Test Plan Based on Logistic-Exponential Distribution and Its Application

In this article, a reliability test plan is developed for Logistic-exponential distribution (LoED) under time truncated life test scheme. The distribution has been chosen because it can used to model lifetime of several reliability phenomenon and it performs better than many well known existing distributions. With the discussions of statistical properties of the aforesaid model, the reliability test plan has been established under the assumption of median quality characteristics when minimum confidence level P* is given. To quench the objective of the paper i.e; to serve as a guiding aid to the emerging practitioners, minimum sample sizes have been obtained by using binomial approximation and Poisson approximation for the proposed plan. Further, operating characteristic (OC) values for the various choices of quality level are placed. Also, minimum ratio of true median life to specified life has been presented for specified producer’s risk. Important findings of the proposed reliability test plan are given for considered value of k=0.75,1,2. To demonstrate the appropriateness of suggested reliability test plan is achieved using four real life situation.

Periodic Solutions of One-Dimensional Cellular Automata with Uniformly Chosen Random Rules

We study cellular automata whose rules are selected uniformly at random. Our setting are two-neighbor one-dimensional rules with a large number $n$ of states. The main quantity we analyze is the asymptotic distribution, as $n \to \infty$, of the number of different periodic solutions with given spatial and temporal periods. The main tool we use is the Chen-Stein method for Poisson approximation, which establishes that the number of periodic solutions, with their spatial and temporal periods confined to a finite range, converges to a Poisson random variable with an explicitly given parameter. The limiting probability distribution of the smallest temporal period for a given spatial period is deduced as a corollary and relevant empirical simulations are presented.

Managing Advance Admission Requests for Obstetric Care

This paper is devoted to the management of advance admission requests for obstetric care. Pregnant women in China select one hospital and request admission for both antenatal and postnatal care after nine weeks of pregnancy. Schedulers must make the admission decision instantly based on the availability of the most critical resource, that is, hospital beds for postnatal care. The random delay between admission requests and postnatal care has created a distinct advance admission control problem. To address this issue, we propose a basic model that assumes a unit bed requirement for one day. Each admission generates a unit of revenue and each unit of overcapacity use incurs an overcapacity cost. With the objective of maximizing the expected net revenue, we establish an optimal policy for unlimited requests, that is, an expected arrival time quota (EATQ) policy that accepts a fixed quota of advance admission requests with the same expected date of confinement. We then propose an extended model for general capacity requirements. Using the Poisson approximation, we establish the optimality of the EATQ policy, which is shown to be solvable by a simple linear programming model. We compare the numerical results from the different policies and conduct a sensitivity analysis. The EATQ policy is demonstrated to be the best option in all test instances and notably outperforms the current admission rules used in hospitals, which usually accept admission requests according to some empirical monthly quota of the expected delivery month. The Poisson approximation is shown to be effective for determining the optimal EATQ policy for both stationary and nonstationary arrivals. Summary of Contribution: First, this paper investigates the advance admission control problem for obstetric care. Pregnant women in China choose one hospital and request admission for both antenatal and postnatal care after nine weeks of pregnancy but the most critical resource is hospitalization beds needed for postnatal care. The random delay between admission request and postnatal care makes the problem unique and challenging to solve. It belongs to the scope of computing and operations research. Second, this paper formulates a dynamic programming model, analyzes the structural properties of the optimal control policy, and finally proposes a mathematical programming model to determine the optimal quota. Numerical experiments show the validity of the proposed approach. It covers the research contents of theories on dynamic stochastic control, mathematic programming model, and experiments. Moreover, this paper is motivated by the practical problem (advance admission control) in obstetric units of Shanghai. Using these optimality properties, solution approaches, and numerical results, this paper provides guidance on how to manage advance obstetric admission requests.

Extremal lifetimes of persistent cycles

Persistent homology captures the appearances and disappearances of topological features such as loops and cavities when growing disks centered at a Poisson point process. We study extreme values for the lifetimes of features dying in bounded components and with birth resp. death time bounded away from the threshold for continuum percolation and the coexistence region. First, we describe the scaling of the minimal lifetimes for general feature dimensions, and of the maximal lifetimes for cavities in the Čech filtration. Then, we proceed to a more refined analysis and establish Poisson approximation for large lifetimes of cavities and for small lifetimes of loops. Finally, we also study the scaling of minimal lifetimes in the Vietoris-Rips setting and point to a surprising difference to the Čech filtration.

On Approximation of the Tails of the Binomial Distribution with These of the Poisson Law

A subject of this study is the behavior of the tail of the binomial distribution in the case of the Poisson approximation. The deviation from unit of the ratio of the tail of the binomial distribution and that of the Poisson distribution, multiplied by the correction factor, is estimated. A new type of approximation is introduced when the parameter of the approximating Poisson law depends on the point at which the approximation is performed. Then the transition to the approximation by the Poisson law with the parameter equal to the mathematical expectation of the approximated binomial law is carried out. In both cases error estimates are obtained. A number of conjectures are made about the refinement of the known estimates for the Kolmogorov distance between binomial and Poisson distributions.

Joint Distribution of the Number of Vertices and the Area of Convex Hulls Generated by a Uniform Distribution in a Convex Polygon

A convex hull generated by a sample uniformly distributed on the plane is considered in the case when the support of a distribution is a convex polygon. A central limit theorem is proved for the joint distribution of the number of vertices and the area of a convex hull using the Poisson approximation of binomial point processes near the boundary of the support of distribution. Here we apply the results on the joint distribution of the number of vertices and the area of convex hulls generated by the Poisson distribution given in [6]. From the result obtained in the present paper, in particular, follow the results given in [3, 7], when the support is a convex polygon and the convex hull is generated by a homogeneous Poisson point process