Polya Process Research Articles

Model for random internalization of nanoparticles by cells

We propose a stochastic model for the internalization of nanoparticles by cells formulating cellular uptake as a compound Poisson process with a random probability of success. This is an alternative approach to the one presented by Rees [] who explained overdispersion in nanoparticle uptake and associated negative binomial distribution by considering a Poisson distribution for particle arrival and a gamma-distributed cell area. In our stochastic model, the formation of new pits is represented by the Poisson process, whereas the capturing process and the population heterogeneity are described by a random Bernoulli process with a beta-distributed probability of success. The random probability of success generates ensemble-averaged conditional transition probabilities that increase with the number of newly formed pits (self-reinforcement). As a result, an ensemble-averaged nanoparticle uptake can be represented as a Polya process. We derive an explicit formula for the distribution of the random number of pits containing nanoparticles. In the limit of the fast nucleation and low probability of nanoparticle capture, we find the negative binomial distribution. Published by the American Physical Society 2024

On the point process with finite memory and its application to optimal age replacement

There has been extensive study of various repair models in the literature, mostly under the assumption that these repairs are minimal or imperfect/better than minimal. Although this is often a realistic assumption, it may not be sufficient to model instances where the repair is worse than minimal. The generalized Polya process (GPP) that has been used to describe this type of repair takes into account all previous events/repairs, which is not often the case in practice. Therefore, in this paper, we define a new process with finite memory that starts as the GPP but, after a certain number of events or elapsed time, becomes the non-homogeneous Poisson process of repairs (minimal repairs). The corresponding age replacement policy is defined and the optimal solutions that minimize the long-run expected cost rate are analyzed. The detailed numerical examples illustrate our findings.

The dynamics of entrepreneurial agglomeration formation: Social selection and simulation.

From the facts that numerous regions with initially similar economic conditions end up with different levels of entrepreneurial agglomeration, this paper constructs a model assuming that the sequential entrants make their career choices based on existing entrepreneurial ratio and describing the dynamics of entrepreneurial agglomeration formation. After mathematical analysis and numerical simulation using NetLogo, it is found that under social selection, a nonlinear Polya process with self-reinforcing and path-dependency characters will emerge, and the repeated entrants' career choices will lead to the agglomeration of entrepreneurship; the agent's risk compensation value, the initial population of agents, the number of role models in the early stage and the initial entrepreneurial ratio are determinants to the formation of entrepreneurial agglomeration. The findings confirm that entrepreneurship has "memory" and the entrepreneurial history could have influence on the future. In order to forge the entrepreneurial agglomeration, our suggestions include exerting influence on the determinants from an early age, and improving the individual's risk-taking abilities.

Incidence rate estimation of SARS-COVID-19 via a Polya process scheme: a comparative analysis in Italy and European countries

During an ongoing epidemic, especially in the case of a new agent, data are partial and sparse, also affected by external factors, as for example climatic effects or preparedness and response capability of healthcare structures. Despite that, we showed how, under some universality assumptions, it is possible to extract strategic insights by modelling the pandemic through a probabilistic Polya urn scheme. Adopting a Polya framework, we provided both the distribution of infected cases and the asymptotic estimation of the incidence rate, showing that data are consistent with a general underlying process at different scales. Using European confirmed cases and diagnostic test data on COVID-19, we also provided an extensive comparison among European countries and between Europe and Italy at regional scale, for both the two big waves of infection. We globally estimated an incidence rate in accordance with previous studies.

Reliability modeling of dependent competing failure processes based on time‐dependent threshold level δ and degradation rate changes

AbstractA new reliability model for dependent competing failure processes is proposed. The reliability model uses the generalized Polya process (GPP) to model the arrival of shocks and considers the degradation rate change. Most systems fail due to performance degradation and external environment shocks. On the one hand, soft failure occurs when the total amount of degradation, including continuous degradation and sudden degenerate increment, exceeds the soft failure threshold level. On the other hand, hard failure occurs when the time interval between two successive shocks is less than the recovery time. As time progresses and natural degradation, the time for a system to recover from damage due to shocks tends to increase gradually. However, conventional models usually regard recovery time as a constant, unrealistic in many practical situations. For this purpose, an increment‐dependent shock process is considered. A hard failure reliability model with recovery time is calculated using the GPP to describe the shock arrival process. On this basis, combined with the deduced soft failure function, the analytical solution of the reliability model is obtained. Finally, a numerical example based on the micro‐electro mechanical systems is conducted to illustrate the effectiveness of the proposed model.

A Preventive Replacement Policy for a System Subject to Bivariate Generalized Polya Failure Process

Numerous studies on preventive maintenance of minimally repaired systems with statistically independent components have been reported in reliability literature. However, in practice, the repair can be worse-than-minimal and the components of a system can be statistically dependent. The existing literature does not cover this important in-practice setting. Therefore, our paper is the first to deal with these issues by modeling dependence in the bivariate set up when a system consists of two dependent parts. We employ the bivariate generalized Polya process to model the corresponding failure and repair process. Relevant stochastic properties of this process have been obtained in order to propose and further discuss the new optimal bivariate preventive maintenance policy with two decision parameters: age and operational history. Moreover, introducing these two parameters in the considered context is also a new feature of the study. Under the proposed policy, the long-run average cost rate is derived and the optimal replacement policies are investigated. Detailed numerical examples illustrate our findings and show the potential efficiency of the obtained results in practice.

Periodic maintenance optimization considering the difference of preventive maintenance effect based on generalized Polya process

Previous studies seldom consider the difference of preventive maintenance effect on different failure processes when establishing the preventive maintenance model. Meanwhile, preventive maintenance models usually assume that minimal repair is performed for each failure, but this assumption is not applicable in some systems. Once this assumption is made for the inapplicable system, the maintenance strategy obtained by the established preventive maintenance model will not be optimal. In order to get a better maintenance policy to reduce maintenance cost, this paper divides the system failure processes into two types: wear-out type and damage-type. The model assumes that preventive maintenance can reduce the probability of damage-type failure, but it has no effect on wear-out type failure. At the same time, for the system studied in this paper, it is assumed that after corrective maintenance, the overall state of the system is worse than the moment just prior to the failure. Then, an extended preventive maintenance optimization model based on generalized Polya process is established. Finally, a practical algorithm is designed to solve the model. The analysis of numerical examples shows the potential value of the proposed method, which may provide maintenance guidance for enterprises and save maintenance costs.

On the delayed worse-than-minimal repair model and its application to preventive replacement

Abstract Minimal repair and other imperfect repair models have been intensively studied in the literature. Much less attention has been payed to the ‘worse than minimal’ repair problem, although it often occurs in practice due to the adverse effects of previous repairs, environmental and internal shocks, etc. To model this type of repair, we define a new point process that behaves as the non-homogeneous Poisson process up to a certain event or time (minimal repairs) and then it becomes the generalized Polya process of repairs (worse than minimal repairs). The corresponding replacement policy is defined and the optimal solutions that minimize the long run expected cost rate are analyzed. The replacement can be executed univariately either after the given time T or the given number of repairs (on the k-th failure). Moreover, the system can be also replaced by implementing the bivariate strategy, that is, after the time T or on the k-th failure, whichever comes first. The detailed numerical examples illustrate our findings. It is shown that the k-strategy outperforms the T -strategy (lower cost rates), whereas the bivariate strategy is not worse than the best univariate strategy.

On some general survival models with delayed failures

In practice, the point events (e.g., shocks) affecting a system often result in some consequences only after some random delays. In this paper, we generalize the previous results reported in the literature to a meaningful case of the generalized Polya process of initial shocks, which is characterized by dependent increments. We derive and analyze the distribution of the lifetime and the failure rate of a system. Generalizations to the cases when each initial shock triggers the delay only with a given probability and when it results in the corresponding wear process are also considered.

IS SOVEREIGN DEBT RESISTANCE OVERCOMING A PREMISE OF SOVEREIGN DEFAULT PATH DEPENDENCE OCCURENCE?

The main aim of the article is to analyze the interconnection between sovereign debt resilience and sovereign default path dependence with its application for Ukrainian economy. There were system dynamics methods used in this research with the simulation using Polya process probability (Polya urn). We used the statistical data base of the Ukrainian economy and fundamental works in system dynamics and sovereign debt analysis. We examine in the article the problem of overcoming existing sovereign debt mental model resilience to external and internal shocks in a situation of uncertainty and the issue of the appropriate instruments for sovereign debt resilience conceptual analysis application. The causal loops diagrams are used to demonstrate the main effects arising under the policy dilemma conducting of sovereign debt default or its repayment. Among the main consequences of not overcoming the existing mental model is 1) the obsolete sovereign debt management financial structure of the society management method (The Deming cycle is used according to the scheme: "Plan - work - check - activity", or "Plan - execution - check - adjustment" to explain the existing principle of sovereign debt service.) 2) foreign direct investment decreasing versus external debt decreasing, which slows down the process of economic development 3) The disproportion between national savings and investment 4) The absence of the resistance in sovereign debt policy mechanism. The author highlights, that under Pola process the system does not lock in to possible balance - we are not observing the uniform distribution of the proportion of sovereign defaults. Positive feedback loops are overlapped by undetermined negative one. Polya process in a case of the sovereign default versus debt repayment with the appropriate density function for sovereign default probability after 29 years (trials) has been used to demonstrate the nonlinearity processes of the appropriate policy choice. The main conclusion is that the equilibrium distribution of the Polya process is not quite uniform. We are observing a locally unstable disequilibrium with the positive feedback but impossibility to lock in in a certain point.

A general multivariate new better than used (MNBU) distribution and its properties

In this paper, we develop a general multivariate new better than used (MNBU) distribution based on a multivariate common shock model. Assuming that the external shock process follows the generalized Polya process and a shock can destroy each component with some given probability, the multivariate survival distribution is obtained. The dependence structure of the multivariate distribution is analyzed. The properties on the dependence order and the multivariate ageing are also studied. Finally, as a special case, we consider the distribution of the minimum random variable and briefly discuss its properties.

A new repair model and its optimization for cold standby system

In this paper, we propose a new repair model for a cold standby system, which consists of two components and one repairman. It is assumed that the consecutive working time follows decreasing geometric process after repair, and the repair time interval is a constant for component 1. For component 2 (standby component), the failure process during working time follows Generalized Polya Process, which is a generalized version of the nonhomogeneous Poisson process. Component 2 is rectified by Generalized Polya Process repair when it fails. The repair time of component 2 is assumed to be negligible. Component 1 is assumed to have priority in use. The long-run average cost rate function of the system is deduced based on the failure number of component 1. Moreover, the optimal replacement policy of model is established by minimizing the long-run average cost rate function theoretically, which proves the existence and uniqueness of the optimal replacement policy. Numerical examples are provided to verify the effectiveness of the proposed approaches. Sensitivity analysis are conducted to illustrate the influence of parameters under the optimal replacement policy.

Two-dimensional extended warranty strategy including maintenance level and purchase time: A win-win perspective

In practice, consumers need to decide whether and when to purchase extended warranty (EW), and there are many factors that influence this decision, such as product reliability and preventive maintenance (PM) conditions, etc. Under the consideration of consumer PM options, purchase time of EW and consumer usage rate diversity, this research first determines the product failure processes based on Generalised Polya Process (GPP) failure mode. Moreover, considering the impact of consumers purchasing decision of EW on warranty costs, we establish warranty cost models based on the product failure processes. Finally, given the warranty cost model, win-win EW price decision models are proposed in the product life cycle. A real case from a leading automobile manufacturer of China is presented to illustrate the application of the proposed model. The isoline of win-win regions and win-win EW intervals are obtained. Also, two management recommendations are given to help manufacturers make relevant decisions.

A martingale approach for Pólya urn processes

This paper is devoted to a direct martingale approach for Polya urn models asymptotic behaviour. A Polya process is said to be small when the ratio of its remplacement matrix eigenvalues is less than or equal to 1/2, otherwise it is called large. We find again some well-known results on the asymptotic behaviour for small and large urns processes. We also provide new almost sure properties for small urns processes.

On optimal preventive-maintenance policy for generalized Polya process repairable products under free-repair warranty

This paper focuses on analyzing the cost behavior of a free-repair warranty (FRW) policy with a preventive maintenance (PM) schedule for a complex product. We utilize a generalized Polya process to model the more general failure and repairs during the FRW coverage period. The expected cost function of the FRW with PMs is derived. We firstly prove that a unique optimal PM schedule exists for a FRW with a finite number of PMs, and then propose a search algorithm for computing the optimal number of PMs and their implementation times for a given FRW coverage period. A certain amount of numerical analysis is conducted to gain further insights into the FRW with PMs. Owing to the significant cost-saving advantages of scheduling PMs for the FRW, our study offers an important analytical tool for marketing managers in the design of more competitive and cost-effective warranty policies for complex and durable products.

A simulation design for the regional lock-in effect of urbanisation in Taiwan

This study was divided into two phases. The first phase is theoretical discussion, including new economic geography, theory of increasing returns, lock-in effect, and the Polya process. The second phase is simulation design for the regional lock-in effect of Taiwan's urbanisation, that is, constructing a regional development system to simulate manufacturers' location selection (the dynamic process), where the rules of system design are based on the theoretical discussion in the first phase, and the world view will be through the simulation of four regions according to the Taiwan Regional Planning Act, then scenario analysis with or without the development strategies will be adopted to compare the effect, before and after the restructure of the five counties and cities, on the behaviour of firms' location selection, and further on the change and balance of Taiwan's regional development (that is, to examine whether any region would be locked in).

A simulation design for the regional lock-in effect of urbanisation in Taiwan

This study was divided into two phases. The first phase is theoretical discussion, including new economic geography, theory of increasing returns, lock-in effect, and the Polya process. The second phase is simulation design for the regional lock-in effect of Taiwan's urbanisation, that is, constructing a regional development system to simulate manufacturers' location selection (the dynamic process), where the rules of system design are based on the theoretical discussion in the first phase, and the world view will be through the simulation of four regions according to the Taiwan Regional Planning Act, then scenario analysis with or without the development strategies will be adopted to compare the effect, before and after the restructure of the five counties and cities, on the behaviour of firms' location selection, and further on the change and balance of Taiwan's regional development (that is, to examine whether any region would be locked in).

A Polya Contagion Model for Networks

A network epidemics model based on the classical Polya urn scheme is investigated. Temporal contagion processes are generated on the network nodes using a modified Polya sampling scheme that accounts for spatial infection among neighbouring nodes. The stochastic properties and the asymptotic behaviour of the resulting network contagion process are analyzed. Unlike the classical Polya process, the network process is noted to be non-stationary in general, although it is shown to be time-invariant in its first and some of its second-order statistics and to satisfy martingale convergence properties under certain conditions. Three classical Polya processes, one computational and two analytical, are proposed to statistically approximate the contagion process of each node, showing a good fit for a range of system parameters. Finally, empirical results compare and contrast our model with the well-known discrete time SIS model.

A new class of multivariate counting processes and its characterization

ABSTRACTIn this paper, we suggest a new class of multivariate counting processes which generalizes and extends the multivariate generalized Polya process recently studied in Cha and Giorgio [On a class of multivariate counting processes, Adv. Appl. Probab. 48 (2016), pp. 443–462]. Initially, we define this multivariate counting process by means of mixing. For further characterization of it, we suggest an alternative definition, which facilitates convenient characterization of the proposed process. We also discuss the dependence structure of the proposed multivariate counting process and other stochastic properties such as the joint distributions of the number of events in an arbitrary interval or disjoint intervals and the conditional joint distribution of the arrival times of different types of events given the number of events. The corresponding marginal processes are also characterized.

Extensions of the Generalized Pólya Process

A new self-exciting counting process is here considered, which extends the generalized Polya process introduced by Cha (Adv Appl Probab 46:1148–1171, 2014). Contrary to Cha’s original model, where the intensity of the process (linearly) increases at each jump time, the extended version allows for more flexibility in the dependence between the point-wise intensity of the process at some time t and the number of already observed jumps. This allows the “extended Polya process” to be appropriate, e.g., for describing successive failures of a system subject to imperfect but effective repairs, where the repair can lower the intensity of the underlying counting process. Probabilistic properties of the new process are studied (construction from a homogeneous pure-birth process, conditions of non explosion, computation of distributions, convergence of a sequence of such processes, ...) and its connection with Generalized Order Statistics is highlighted. Positive dependence properties are next explored. Finally, the maximum likelihood method is considered in a parametric setting and tested on a few simulated data sets, to highlight the practical use of the new process in an application context.