Non-homogeneous Pure Birth Process Research Articles

A non-homogeneous Markov early epidemic growth dynamics model. Application to the SARS-CoV-2 pandemic.

This work introduces a new markovian stochastic model that can be described as a non-homogeneous Pure Birth process. We propose a functional form of birth rate that depends on the number of individuals in the population and on the elapsed time, allowing us to model a contagion effect. Thus, we model the early stages of an epidemic. The number of individuals then becomes the infectious cases and the birth rate becomes the incidence rate. We obtain this way a process that depends on two competitive phenomena, infection and immunization. Variations in those rates allow us to monitor how effective the actions taken by government and health organizations are. From our model, three useful indicators for the epidemic evolution over time are obtained: the immunization rate, the infection/immunization ratio and the mean time between infections (MTBI). The proposed model allows either positive or negative concavities for the mean value curve, provided the infection/immunization ratio is either greater or less than one. We apply this model to the present SARS-CoV-2 pandemic still in its early growth stage in Latin American countries. As it is shown, the model accomplishes a good fit for the real number of both positive cases and deaths. We analyze the evolution of the three indicators for several countries and perform a comparative study between them. Important conclusions are obtained from this analysis.

Optimal periodic preventive maintenance policy for a system subject to failures/repairs which follow the non-homogeneous pure birth process

ABSTRACT This paper considers a periodic preventive maintenance policy for a deteriorating and repairable system. The system is subject to random failures and is repaired at each failure. To improve the reliability, at periodic times iT (i = 1, 2, … N–1), the system is preventively maintained (PM). At the time NT the system is replaced. The failure/repair process between two PMs follows the Non-Homogeneous Pure Birth Process (NHPBP) which is a generalized version of the Non-homogeneous Poisson process (NHPP). The reliability improvement made by a PM action is stochastically modeled based on the properties of NHPBP. The cost for each PM depends on the reliability condition of the system at that PM time. The PM policy is characterized by the two parameters (N, T). We develop a two-stage procedure to find the optimal (N*, T*) which minimizes the long-run expected cost rate. Numerical examples are presented to illustrate the properties of the optimal policies. Abbreviations: NHPBP: Non-homogeneous pure birth process; NHPP: Non-homogeneous Poisson process; PM: Preventive maintenance; CM: Corrective maintenance; RM: Replacement maintenance

The optimal preventive-maintenance policy for a NHPBP repairable system under free-repair warranty

This investigation focuses on deriving an optimal preventive maintenance (PM) scheme for a system under the free-repair warranty (FRW). We utilize a Non-Homogeneous Pure Birth Process (NHPBP) to model the process of more general failures and repairs during the coverage period of FRW, propose a type of PM model for such a NHPBP repairable system, and develop the expected cost function. Under the special case of failure intensity, we first prove that there exists a unique optimal PM schedule for FRW with a finite number of PMs and then compute the optimal number of PMs and their implementation times for a given FRW coverage period. Numerical computation and sensitivity analysis are conducted to gain more insights about the NHPBP repairable system under FRW with and without PMs. Our results offer an important analytical tool for marketing managers to design more competitive and cost minimal warranty policies for durable and complex systems with a more general failure process and ‘worse-than-minimal repair’.

Reliability modeling of coherent systems with shared components based on sequential order statistics

Abstract In this paper we are concerned with the reliability properties of two coherent systems having shared components. We assume that the components of the systems are two overlapping subsets of a set of n components with lifetimes X1,...,Xn. Further, we assume that the components of the systems fail according to the model of sequential order statistics (which is equivalent, under some mild conditions, to the failure model corresponding to a nonhomogeneous pure-birth process). The joint reliability function of the system lifetimes is expressed as a mixture of the joint reliability functions of the sequential order statistics, where the mixing probabilities are the bivariate signature matrix associated to the structures of systems. We investigate some stochastic orderings and dependency properties of the system lifetimes. We also study conditions under which the joint reliability function of systems with shared components of order m can be equivalently written as the joint reliability function of systems of order n (n>m). In order to illustrate the results, we provide several examples.

A Shock-Based Model for the Reliability of Three-State Networks

This paper is a study on the reliability of a three-state network under shocks in which each shock may cause the failure of more than one component at each time instant. The network is assumed to have n binary components and three states: up, partial performance, and down. The components are subject to failure due to the occurrence of shocks appearing based on a counting process, and some of the components may fail as a result of each shock. To give a model for the reliability of the network, a new variant of the notion of two-dimensional signature is introduced, which is called two-dimensional t-signature. Based on this new notion, some mixture representations are given for the joint reliability function of the entrance times into a partial performance state T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and a down state T. Several stochastic orderings and dependence properties regarding T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and T are provided. The results are also explored for the special case when the shocks appear according to a nonhomogeneous pure birth process under different conditions.

A note on a two variable block replacement policy for a system subject to non-homogeneous pure birth shocks

This paper considers an operating system subject to shocks occurring as a non-homogeneous pure birth process (NHPBP). A shock can cause two types of system failures: a type-I failure (minor failure) is rectified by a general repair, whereas a type-II failure (catastrophic failure) is removed by an unplanned replacement or remains inactive until the next planned replacement. The probabilities of these two types of failures depend on the number of shocks since the last replacement. We consider a modified block replacement policy (BRP) policy with a random repair cost as follows: first, an operating system is preventively replaced with a new one at times iT (I = 1,2, …) regardless its failure history. If the system fails in [(i − 1)T, (i − 1)T + T0), it is either replaced by a new one or generally repaired, and if fails in [(i − 1)T + T0, iT), it is either generally repair (minor failure) or remains inactive (catastrophic failure) until the next planned replacement. Thus the policy is called a two variable policy with a fixed replacement interval and a threshold for the system age. The expected cost rate is derived by using the renewal reward theory. The two-variable cost function is transformed into a one variable function and the optimal policy is analyzed.

Optimal number of repairs before replacement for a two-unit system subject to non-homogeneous pure birth process

We consider a discrete replacement model for a two-unit system subject to failure rate interaction and shocks. Two types of shocks occur according to a non-homogeneous pure birth process and can affect the two-unit system. Type I shock causes unit A to fail and can be rectified by a general repair, while type II shock results in a non-repairable failure and must be fixed by a replacement. Two-unit systems also exhibit failure rate interactions between the units: each failure of unit A causes some damage to unit B, while each failure of unit B causes unit A into an instantaneous failure. The occurrence of a particular type of shock is dependent on the number of shocks occurred since the last replacement. The objective of this paper is to determine the optimal number of minor failures before replacement that minimizes the expected cost rate. A numerical example is presented to illustrate application of the model.

Age replacement policy with lead-time for a system subject to non-homogeneous pure birth shocks

A system is subject to shocks that arrive according to a non-homogeneous pure birth process. Whenever a shock occurs, the system enters one of the two types of failure states. Type I failure (minor failure) is fixed by a minimal repair. Type II failure (catastrophic failure) is removed by a replacement. We consider an age replacement policy which replaces the system whenever its age reaches T and a spare for replacement is available. The optimal cost minimization age T∗ is derived under a cost structure. We demonstrate that this model includes more realistic factors and is a generalization of several previous models in the literature.

Optimal Number of Repairs Before Replacement for a System Subject to Shocks of a Non-Homogeneous Pure Birth Process

We consider a system subject to shocks that arrive according to a nonhomogeneous pure birth process (NHPBP). As a shock occurs, the system has two types of failures. Type-I failure (minor failure) is rectified by a general repair, whereas type-II failure (catastrophic failure) is removed by an unplanned replacement. The probabilities of these two types of failures depend on the number of shocks since the last replacement. We consider a policy with which the system is replaced at the n th type-I failure, or at any type-II failure. The aim of this paper is to determine the optimal policy n*, the number of minor failures up to replacement that minimizes the expected cost rate of the system subject to NHPBP shocks. The model is a generalization of the existing models, and is more applicable in practice. We present some numerical examples, and show that several classical models are the special cases of our model.

Age replacement policy for a two-unit system subject to non-homogeneous pure birth shocks

When a two-unit system is subjected to shocks, it may fail due to interactions between the units. Two types of shocks stemming from a non-homogeneous pure birth process can affect a two-unit system. A Type I shock causes a minor failure of Unit A which affects Unit B and can be rectified with minimal repair. The Type II shock causes total (catastrophic) system failure, at which point an unplanned maintenance is required to replace the failed system. Two-unit systems also exhibit failure rate interactions between units: failure of Unit A causes an internal shock that increases the failure rate of Unit B, while failure of Unit B causes instantaneous failure of Unit A. The occurrence of a particular type of shock is based on a random mechanism dependent on the number of shocks that have occurred since the last replacement. The goal of this study is to derive the expected replacement cost rate by introducing relative costs as a factor, and determining the optimal replacement period T∗ which minimizes cost. A numerical example is presented to illustrate application of the method.

Extended optimal replacement policy for a system subject to non-homogeneous pure birth shocks

This article studies the optimal replacement policy with general repairs for an operating system subject to shocks occurring to a non-homogeneous pure birth process (NHPBP). A shock causes that the system experiences one of two types of failures: type-I failure (minor failure) is rectified by a general repair, or type-II failure (catastrophic failure) is removed by an unplanned replacement. The probabilities of these two types of failures depend on the number of shocks since the last replacement. We consider a bivariate replacement policy (n,T) under which the system is replaced at planned life age T, or at the nth type-I failure, or at any type-II failure, whichever occurs first. The optimal replacement schedule which minimizes the expected cost rate model is derived analytically and discussed numerically.

A Block Replacement Policy for Systems Subject to Non-homogeneous Pure Birth Shocks

This note studies the block replacement policy with general repairs for an operating system subject to shocks occurring according to a non-homogeneous pure birth process. A shock causes the system to fail. There are two types of failures: a type-I failure (minor failure) is fixed by a general repair, whereas a type-II failure (catastrophic failure) is removed by an unplanned (or unscheduled) replacement. The failure type probabilities depend on the number of type-I failure shocks that occurred since the last replacement. Under the block replacement policy, the operating system is replaced every time units to reduce the chances of more expensive unplanned replacements due to type-II failures. The aim of this note is to determine the optimal block interval T*, which minimizes the expected cost rate and the expected total discounted cost rate of the proposed policy. As the shock process is a more general non-homogeneous pure birth process, several previous models become the special cases of our model.

A note on replacement policy for a system subject to non-homogeneous pure birth shocks

A system is subject to shocks that arrive according to a non-homogeneous pure birth process. As shocks occur, the system has two types of failures. Type-I failure (minor failure) is removed by a general repair, whereas type-II failure (catastrophic failure) is removed by an unplanned replacement. The occurrence of the failure type is based on some random mechanism which depends on the number of shocks occurred since the last replacement. Under an age replacement policy, a planned (or scheduled) replacement happens whenever an operating system reaches age T. The aim of this note is to derive the expected cost functions and characterize the structure of the optimal replacement policy for such a general setting. We show that many previous models are special cases of our general model. A numerical example is presented to show the application of the algorithm and several useful insights.

Inference and model comparison for species accumulation functions using approximating pure birth processes

Soberon and Llorente proposed using the mean S(t; ϕ) of a pure birth process as a model for species accumulation functions and obtained ordinary least squares point estimates for the vector of parameters ϕ. They provided useful biological interpretations for S(t; ϕ) as well as for the birth rates of the process; they also expressed the strong need for an objective tool to compare different models for a given species accumulation data set. We show that the functions S(t; ϕ) proposed by Soberon and Llorente are generally not means of pure birth processes but that they can be well approximated by the mean of a suitable nonhomogeneous pure birth process B(t). We suggest using S(t; ϕ) and an upper bound for the (unknown) variance of the approximating pure birth process to make statistical inferences about ϕ. A nonlinear regression model with mean S(t; ϕ) and normal errors, possibly correlated, is proposed using a likelihood approach. This model can be compared with other relevant models under consideration by means of their corresponding likelihood ratio, providing a useful, objective, and quantitative tool for comparing different models in the light of the observed data. The suggested approach is exemplified with two species accumulation data sets.

Species accumulation functions and pure birth processes

Many authors in Ecology, following Soberón and Llorente (Conserv. Biol. 7(3) (1993) 480), model species accumulation functions as means of pure birth processes. Their approach has inconsistencies, but we show that it can be justified as an approximation, in the sense that given a species accumulation function S( t) of a proposed type, there exists a non-homogeneous pure birth process (PBP) whose mean is close to S( t). The variance of the PBP is basic for statistical inference and model assessment, but it is not known. We provide an upper bound for the variance that can be used in its place.