Renewal Function Research Articles

Exponential and Pareto-type bounds for the renewal function and the excess lifetime of a renewal process

In this paper, our objective is to derive two-sided bounds for the renewal function even if the interarrival times have no finite moment. By proving a relationship between the renewal function and a suitable ‘truncated' compound geometric distribution, under a truncated generalized Lundberg condition for proper renewal equations, we derive upper and lower bounds for this truncated compound geometric distribution. Using this relationship, we obtain bounds for the renewal function. Under specific truncated generalized Lundberg conditions, we obtain two-sided exponential-type bounds and Pareto-type lower bounds for the renewal function. Also, by proving a relationship between the excess lifetime of a renewal process and the df of a suitable ‘truncated deficit at ruin' of a renewal risk (or Sparre Andersen) model, we obtain two-sided exponential bounds for the df of the excess lifetime.

Advances in Cell Transplantation Therapy for Limbal Stem Cell Deficiency.

Limbal stem cells (LSCs) are essential for maintaining corneal transparency and ocular surface integrity. Many external factors or genetic diseases can lead to corneal limbal stem cell deficiency (LSCD), resulting in the loss of barrier and corneal epithelial cell renewal functions. Stem cell transplantation is one of the primary treatments for LSCD, including limbal transplantation and cultivated limbal epithelial transplantation. In addition, a variety of non-limbal stem cell lines have been experimented with for LSCD treatment. Biological scaffolds are also used to support in vitro stem cell culture and transplantation. Here, we review the mechanisms of corneal maintenance by LSCs, the clinical stage and surgical treatment of LSCD, the source of stem cells, and the biological scaffolds required for in vitro culture. This study is a narrative retrospective study aimed at collecting available information on various aspects of surgical treatments for LSCD. Relevant literature was searched in a range of online databases, including Web of Science, Scopus, and PubMed from 2005 to March, 2023. A total of 397 relevant articles were found, and 49 articles with strong relevance to the studies in this paper were obtained and analyzed. Moreover, 11 of these articles were on the concept of LSCD and the mechanism of LESCs maintaining the corneal epithelium, 3 articles on the staging and grading of LSCD, 17 articles on cell transplantation methods and donor cell sources, and 18 articles on scaffolds for delivering stem cells. We also summarized the advantages and disadvantages of different cell transplantation methods and the benefits and limitations of scaffolds based on the above literature. The treatment of LSCD is determined by the clinical stage and whether it involves monocular or binocular eyes. Appropriate surgical techniques should be taken for LSCD patients in order to reconstruct the ocular surface, relieve symptoms, and restore visual function. Meanwhile, biological scaffolds assist in the ex vivo culture and implantation of stem cells.

Non-Parametric Estimation of the Renewal Function for Multidimensional Random Fields

This paper addresses the almost sure convergence and the asymptotic normality of an estimator of the multidimensional renewal function based on random fields. The estimator is based on a sequence of non-negative independent and identically distributed (i.i.d.) multidimensional random fields and is expressed as infinite sums of k-folds convolutions of the empirical distribution function. It is an extension of the work from the case of the two-dimensional random fields to the case of the d-dimensional random fields where d>2. This is established by the definition of a “strict order relation”. Concrete applications are given.

Semaglutide 6-months therapy of type 2 diabetes mellitus restores adipose progenitors potential to develop metabolically active adipocytes

BackgroundNowadays type 2 diabetes mellitus (T2DM) leads to population mortality growth. Today glucagon-like peptide type 1 receptor agonists (GLP-1 RA) are one of the most promising glucose-lowered drugs with anorexigenic and cardioprotective effects. The present study aims to determine the effects of GLP-1 RA semaglutide 6-month therapy on T2DM patient metabolic parameters and adipose progenitor cell health. MethodsT2DM patients (N = 8) underwent clinical characterization and subcutaneous fat biopsy at start point and after semaglutide 6-month therapy. Adipose-derived stem cells (ADSC) were isolated by enzymatic method. Cell proliferation analysis was performed by MTT and immunocytochemistry. White and beige adipogenesis was analyzed by BODIPY493/503 staining and confocal microscopy. Adipocyte’s metabolic properties were estimated by 3H- and 14C-based metabolic assays. Thermogenesis analysis was performed by ERthermAC staining and confocal microscopy. Protein markers were assessed by Western blotting. ResultsSemaglutide 6-month therapy demonstrated significant anorexigenic and glucose-lowering effects. However, insulin sensitivity (HOMA-IR and M-index) was unchanged after therapy. Semaglutide 6-month therapy increased ADSC proliferation and white and beige adipogenesis. Moreover, lipid droplets fragmentation was observed in beige adipocytes. Both white and beige adipocytes after semaglutide therapy demonstrated 2–3 fold growth of glucose uptake without changes in insulin sensitivity. Newly formed white adipocytes demonstrated glucose utilization for active ATP synthesis, whereas beige adipocytes for canonical thermogenesis. ConclusionsOur study has revealed that semaglutide 6-month therapy has not only systemic anorexigenic effects, but can markedly improve adipose tissue health. We have demonstrated critical restoration of ADSC renewal functions, which potentially can be involved in semaglutide based weight loss.

Refinement bounds for the expected number of renewal epochs over a finite interval

ABSTRACT Primary quantities of interest in renewal theory are the expected number of renewals in ( 0 , t ] , known as renewal function (denoted by U ( t ) ), and the expected number of renewals in ( t , t + h ] , given by the difference U ( t + h ) − U ( t ) .One of the most known bounds for the expected number of renewals over a finite interval is the one delivered by Barlow and Proschan [Statistical theory of reliability and life testing. MD: Silver Spring; 1981. Reprint Edition. To begin with]. The essential quantity of this bound is the difference U ( t + h ) − U ( t ) − U ( h ) . In this paper, initially, we offer bounds for the conditional tails of the backward and forward recurrence times, and we give a renewal-type equation for the difference U ( t + h ) − U ( t ) − U ( h ) . Employing those results, we refine the provided bound by Barlow and Proschan [Statistical theory of reliability and life testing. MD: Silver Spring; 1981. Reprint Edition. To begin with]. The results are illustrated with numerical examples.

Moment-based approximations for stochastic control model of type (s, S)

In this study, we propose an approximation for a renewal reward process that describes a stochastic control model of type (s, S) based on the first three moments of demand random variables. Various asymptotic expansions for this model exist in the literature. All these studies rely on the condition of knowing the distribution function of demand random variables and require obtaining the asymptotic expansion of the renewal function produced by them. However, obtaining a renewal function can be challenging for certain distribution families, and in some cases, the mathematical structure of the renewal function is difficult to apply. Therefore, in this study, simple and compact approximations are presented for the stochastic control model of type (s, S). The findings of this study rely on Kambo’s method, through which we obtain approximations for the ergodic distribution, and the n t h order ergodic moments of this process. To conclude the study, the accuracy of the proposed approximate formulas are examined through a specialized illustrative example. Moreover, it has been noted that the proposed approximation is more accurate than the approximations existing in the literature.

Asymptotic Behavior of a Nonparametric Estimator of the Renewal Function for Random Fields

In this paper, we study the asymptotic normality of a nonparametric estimator of the renewal function associated with a sequence of absolutely continuous nonnegative two-dimensional random fields. We prove that this estimator is asymptotically unbiased. The asymptotic normality of this estimator is established.

Investigation the ergodic distribution of a semi-Markovian inventory model of type (s,S) with intuitive approximation approach

This paper concerns a stochastic process expressing (s,S) type inventory system with intuitive approximation approach. The stock level in the system is modeled as a semi-Markovian renewal reward process X(t). Therefore, the ergodic distributions of this process can be analyzed with the help of the renewal function. Obtaining explicit formula for renewal function U(x) is difficult from a practical standpoint. Mitov and Omey recently present some intuitive approximations in literature for renewal function which cover a large number of existing results. Using their approach we were able to establish asymptotic approximations for ergodic distribution of a stochastic process X(t). Obtained results can be used in many situations where demand random variables have different distributions from different classes such as Γ(g) class.

Residual Lifetime of Repairable Systems and Phase type Approximation

ABSTRACT This paper considers a repairable system whose periods of operations and repairs alternate during the run time. The residual lifetime of the system and its age, when the system operates at time t , are proposed, and their properties are investigated. Under the condition that the system is under repair, the remaining time until the end of system repair time and the elapsed time of a repair are studied. Since computing a renewal function is not straightforward, we use phase-type approximation to find the best-fitted distribution of corresponding random variables. We provide some examples and utilize a real data set to demonstrate the results.

On a parametric model for the mean number of system repairs with applications

In reliability engineering and other disciplines, there are many situations, in which one deals with the number of events in a time period that happen as an outcome of a stochastic process. Examples are the number of repairs of a system in an age-based block replacement and the number of repairs for a product sold under warranty in a specific period. In these circumstances, the mean number of failures (repair) plays a crucial role in assessing reliability characteristics and the optimal time of maintenance of the system. In this paper, we study a parametric function for the mean number of repairs of the system that covers a wide range of repair degrees, such as better than minimal repair, minimal repair and worse than minimal repair. Using the suggested model, we propose an approximation for the renewal function in the case that the repairs are complete. We also give applications of the proposed function in age-based block replacement and optimal warranty policy.

시뮬레이션 방법에 의한 보조재생함수의 근사

It is known that it is very difficult to calculate the auxiliary renewal function in most cases. We propose three methods of approximating the auxiliary renewal function by simulaion. At first, we review some theoretical results on M/G/1 queueing model with impatient customers, and observe that the auxiliary renewal function is included in the results. Then we propose three ideas by which we can approximate the auxiliary renewal function by simulation method, and give some approximated value of the function when the service time of customers are ditributed to Weibull and uniform distribution and constant vaule. It is found that three methods by simulation give almost same estimate for auxliliary renewal function with very small standard errors in one million repetition of simulation and the graph of the auxiliary renewal function can be showed. Finally, we conclude that any method given in this paper is acceptable and reliable if the simulation is carried with sufficiently many repetition.

Sequences of Improved Two-Sided Bounds for the Renewal Function and the Solutions of Renewal-Type Equations

Sequences of Improved Two-Sided Bounds for the Renewal Function and the Solutions of Renewal-Type Equations

An Age of Infection Kernel, an R Formula, and Further Results for Arino–Brauer A, B Matrix Epidemic Models with Varying Populations, Waning Immunity, and Disease and Vaccination Fatalities

In this work, we first introduce a class of deterministic epidemic models with varying populations inspired by Arino et al. (2007), the parameterization of two matrices, demography, the waning of immunity, and vaccination parameters. Similar models have been focused on by Julien Arino, Fred Brauer, Odo Diekmann, and their coauthors, but mostly in the case of “closed populations” (models with varying populations have been studied in the past only in particular cases, due to the difficulty of this endeavor). Our Arino–Brauer models contain SIR–PH models of Riano (2020), which are characterized by the phase-type distribution (α→,A), modeling transitions in “disease/infectious compartments”. The A matrix is simply the Metzler/sub-generator matrix intervening in the linear system obtained by making all new infectious terms 0. The simplest way to define the probability row vector α→ is to restrict it to the case where there is only one susceptible class s, and when matrix B (given by the part of the new infection matrix, with respect to s) is of rank one, with B=bα→. For this case, the first result we obtained was an explicit formula (12) for the replacement number (not surprisingly, accounting for varying demography, waning immunity and vaccinations led to several nontrivial modifications of the Arino et al. (2007) formula). The analysis of (A,B) Arino–Brauer models is very challenging. As obtaining further general results seems very hard, we propose studying them at three levels: (A) the exact model, where only a few results are available—see Proposition 2; and (B) a “first approximation” (FA) of our model, which is related to the usually closed population model often studied in the literature. Notably, for this approximation, an associated renewal function is obtained in (7); this is related to the previous works of Breda, Diekmann, Graaf, Pugliese, Vermiglio, Champredon, Dushoff, and Earn. (C) Finally, we propose studying a second heuristic “intermediate approximation” (IA). Perhaps our main contribution is to draw attention to the importance of (A,B) Arino–Brauer models and that the FA approximation is not the only way to tackle them. As for the practical importance of our results, this is evident, once we observe that the (A,B) Arino–Brauer models include a large number of epidemic models (COVID, ILI, influenza, illnesses, etc.).

Modelling of Loan Non-Payments with Count Distributions Arising from Non-Exponential Inter-Arrival Times

The number of non-payments is an indicator of delinquent behaviour in credit scoring, hence its estimation and prediction are of interest. The modelling of the number of non-payments, as count data, can be examined as a renewal process. In a renewal process, the number of events (such as non-payments) which has occurred up to a fixed time t is intimately connected with the inter-arrival times between the events. In the context of non-payments, the inter-arrival times correspond to the time between two subsequent non-payments. The probability mass function and the renewal function of the count distribution are often complicated, with terms involving factorial and gamma functions, and thus their computation may encounter numerical difficulties. In this paper, with the motivation of modelling the number of non-payments through a renewal process, a general method for computing the probabilities and the renewal function based on numerical Laplace transform inversion is discussed. This method is applied to some count distributions which are derived given the distributions of the inter-arrival times. Parameter estimation with maximum likelihood estimation is considered, with an application to a data set on number of non-payments from the literature.

Recurrence times and the expected number of renewal epochs over a finite interval

This paper initially presents bounds for the joint tail and the conditional tails of the backward and forward recurrence times. Using these bounds, we deliver an improvement to the lower bound for the renewal function given by Brown [Inequalities for distributions with increasing failure rate. In: Gelfand AE, editors. Contributions to the theory and applications of statistics, a volume in honour of Herbert Solomon. Orlando, FL: Academic; 1987. p. 3–17] for IFR inter-arrival times. Finally, this paper proposes a renewal type equation for the expected number of renewals in and, under certain conditions, improves Lorden's [On excess over the boundary. Ann Math Stat. 1970;41(2):520–527] well-known general upper bound for the expected number of renewals in .

Creeping of Lévy processes through curves

A Lévy process is said to creep through a curve if, at its first passage time across this curve, the process reaches it with positive probability. We first study this property for bivariate subordinators. Given the graph {(t,f(t)):t≥0} of any continuous, non increasing function f such that f(0)>0, we give an expression of the probability that a bivariate subordinator (Y,Z) issued from 0 creeps through this graph in terms of its renewal function and the drifts of the components Y and Z. We apply this result to the creeping probability of any real Lévy process through the graph of any continuous, non increasing function at a time where the process also reaches its past supremum. This probability involves the density of the renewal function of the bivariate upward ladder process as well as its drift coefficients. We also investigate the case of Lévy processes conditioned to stay positive creeping at their last passage time below the graph of a function. Then we provide some examples and we give an application to the probability of creeping through fixed levels by stable Ornstein-Uhlenbeck processes. We also raise a couple of open questions along the text.

Renewals in Systems with Two Serially Connected Components

In this paper, we study renewals in a system with two components connected in series. Both components can undergo corrective maintenance (i.e., either full replacement/perfect repair or minimal repair) when a failure occurs. When one of the units fails and is correctively maintained, the other one is either preventively replaced (if cost-feasible) or simply left working as is. When a component is minimally repaired or left working as is, its remaining lifetime is reevaluated (“memory effect”) and is taken into consideration in calculating the next renewal. A “coupled” lifetime that combines the lifetimes of the two serially connected components is proposed to represent the joint lifetime of the system. We develop renewal functions based on the coupled lifetimes and show that they follow the classical or generalized renewal theory depending on whether the components work without memory or not. Approximation formulas for the new renewal functions are also obtained and validated by Monte Carlo simulations for various combinations of distributions, and a comparative cost analysis is conducted.

Some bounds for the renewal function and the variance of the renewal process

Renewal equations are frequently encountered in several applications when regenerative arguments are used in modelling. Since, these equations usually do not have analytical solutions, bounds have a great practical importance. The aim of this paper is to present some new bounds for the renewal function and the variance of the renewal process. A general lower bound for the renewal function is given, which is a refinement of all known lower bounds and attains the well-known asymptotic result of Asmussen (2003). Also, some new tighter upper and lower bounds for the renewal function, which are based on several reliability classifications of the distribution of the inter-arrival times, are given. Improved upper and lower bounds for the variance of the renewal process are also presented. Finally, several numerical examples are given to illustrate the effectiveness of the proposed new bounds.

On a Dirichlet Process Mixture Representation of Phase-Type Distributions

An explicit representation of phase-type distributions as an infinite mixture of Erlang distributions is introduced. The representation unveils a novel and useful connection between a class of Bayesian nonparametric mixture models and phase-type distributions. In particular, this sheds some light on two hot topics, estimation techniques for phase-type distributions, and the availability of closed-form expressions for some functionals related to Dirichlet process mixture models. The power of this connection is illustrated via a posterior inference algorithm to estimate phase-type distributions, avoiding some difficulties with the simulation of latent Markov jump processes, commonly encountered in phase-type Bayesian inference. On the other hand, closed-form expressions for functionals of Dirichlet process mixture models are illustrated with density and renewal function estimation, related to the optimal salmon weight distribution of an aquaculture study.

Improved bounds for the solutions of renewal equations

Sequences of non-decreasing (non-increasing) lower (upper) bounds for the renewal-type equation as well as for the renewal function which are improvements of the famous corresponding bounds of Marshal [(1973). Linear bounds on the renewal function. SIAM Journal on Applied Mathematics 24(2): 245–250] are given. Also, sequences such bounds converging to the ordinary renewal function are obtained for several reliability classes of the lifetime distributions of the inter-arrival times, which are refinements of all of the existing known corresponding bounds. For the first time, a lower bound for the renewal function with DMRL lifetimes is given. Finally, sequences of such improved bounds are given for the ordinary renewal density as well as for the right-tail of the distribution of the forward recurrence time.