Showing papers on "Renewal theory published in 2019"

Asymptotic Bayesian Theory of Quickest Change Detection for Hidden Markov Models

Abstract: In the 1960s, Shiryaev developed a Bayesian theory of change-point detection in the i.i.d. case, which was generalized in the early 2000s by Tartakovsky and Veeravalli and recently by Tartakovsky (2017) for general stochastic models assuming a certain stability of the log-likelihood ratio process. Hidden Markov models represent a wide class of stochastic processes in a variety of applications. In this paper, we investigate the performance of the Bayesian Shiryaev change-point detection rule for hidden Markov models. We propose a set of regularity conditions under which the Shiryaev procedure is first-order asymptotically optimal in a Bayesian context, minimizing moments of the detection delay up to certain order asymptotically as the probability of false alarm goes to zero. The developed theory for hidden Markov models is based on Markov chain representation for the likelihood ratio and r -quick convergence for Markov random walks. In addition, applying Markov nonlinear renewal theory, we present a high-order asymptotic approximation for the expected delay to detection and a first-order asymptotic approximation for the probability of false alarm of the Shiryaev detection rule. We also study asymptotic properties of another popular change detection rule, the Shiryaev–Roberts rule, and provide some interesting examples.

Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse

Abstract: In this paper, we study the fractional Poisson process (FPP) time-changed by an independent Levy subordinator and the inverse of the Levy subordinator, which we call TCFPP-I and TCFPP-II, respectively. Various distributional properties of these processes are established. We show that, under certain conditions, the TCFPP-I has the long-range dependence property, and also its law of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal process and its waiting time distribution is identified. The bivariate distributions of the TCFPP-II are derived. Some specific examples for both the processes are discussed. Finally, we present simulations of the sample paths of these processes.

Restless bandits with controlled restarts: Indexability and computation of Whittle index

Abstract: Motivated by applications in machine repair, queueing, surveillance, and clinic care, we consider a scheduling problem where a decision maker can reset m out of n Markov processes at each time. Processes that are reset, restart according to a known probability distribution and processes that are not reset, evolve in a Markovian manner. Due to the high complexity of finding an optimal policy, such scheduling problems are often modeled as restless bandits. We show that the model satisfies a technical condition known as indexability. For indexable restless bandits, the Whittle index policy, which computes a function known as Whittle index for each process and resets the m processes with the lowest index, is known to be a good heuristic. The Whittle index is computed by solving an auxiliary Markov decision problem for each arm. When the optimal policy for this auxiliary problem is threshold based, we use ideas from renewal theory to derive closed form expression for the Whittle index. We present detailed numerical experiments which suggest that Whittle index policy performs close to the optimal policy and performs significantly better than myopic policy, which is a commonly used heuristic.

Stochastic life-cycle analysis: renewal-theory life-cycle analysis with state-dependent deterioration stochastic models

Abstract: For the life-cycle analysis (LCA) of deteriorating engineering systems, it is critical to model and incorporate the various deterioration processes and associated uncertainties. This paper proposes...

The dickman subordinator, renewal theorems, and disordered systems

Abstract: We consider the so-called Dickman subordinator, whose Levy measure has density $\frac{1} {x}$ restricted to the interval $(0,1)$. The marginal density of this process, known as the Dickman function, appears in many areas of mathematics, from number theory to combinatorics. In this paper, we study renewal processes in the domain of attraction of the Dickman subordinator, for which we prove local renewal theorems. We then present applications to marginally relevant disordered systems, such as pinning and directed polymer models, and prove sharp second moment estimates on their partition functions.

Discrete-time queue with batch renewal input and random serving capacity rule: $$GI^X/ Geo^Y/1$$ G I X / G e o Y / 1

Abstract: In this paper, we provide a complete analysis of a discrete-time infinite buffer queue in which customers arrive in batches of random size such that the inter-arrival times are arbitrarily distributed. The customers are served in batches by a single server according to the random serving capacity rule, and the service times are geometrically distributed. We model the system via the supplementary variable technique and further use the displacement operator method to solve the non-homogeneous difference equation. The analysis done using these methods results in an explicit expression for the steady-state queue-length distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the underlying characteristic equation. Our approach enables one to estimate the asymptotic distribution at a pre-arrival epoch by a unique largest root of the characteristic equation lying inside the unit circle. With the help of few numerical results, we demonstrate that the methodology developed throughout the work is computationally tractable and is suitable for light-tailed inter-arrival distributions and can also be extended to heavy-tailed inter-arrival distributions. The model considered in this paper generalizes the previous work done in the literature in many ways.

Intermittent Waiting-Time Noises Through Embedding Processes

Abstract: An intermittent two-state noise can be modelled through a renewal process characterized by two different time scales. A (four state) Markovian embedding of this non-Markovian process is presented. The equivalence between the renewal approach and the enlarged master equation is shown. Analytical results for n-time moments of the intermittent dichotomic noise are obtained. The Monte Carlo simulations supports our analytical results. The advantage of using the enlarged master equation for calculating higher order moments is established.

Extending Discourse of Renewal to Preparedness: Construct and Scale Development of Readiness for Renewal:

Abstract: This study developed the construct of readiness for renewal in organizations and evaluated its underlying psychometric properties. We drew on Discourse of Renewal theory to develop, pilot, and refi...

Large deviations in renewal models of statistical mechanics

Abstract: In Ref. [1] the author has recently established sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a separable Banach space. The renewal model has been there identified with constrained and non-constrained pinning models of polymers, which amount to Gibbs changes of measure of a classical renewal process. In this paper we show that the constrained pinning model is the common mathematical structure to the Poland-Scheraga model of DNA denaturation and to some relevant one-dimensional lattice models of Statistical Mechanics, such as the Fisher-Felderhof model of fluids, the Wako-Saito-Munoz-Eaton model of protein folding, and the Tokar-Dreysse model of strained epitaxy. Then, in the framework of the constrained pinning model, we develop an analytical characterization of the large deviation principles for cumulative rewards corresponding to multivariate deterministic rewards that are uniquely determined by, and at most of the order of magnitude of, the time elapsed between consecutive renewals. In particular, we outline the explicit calculation of the rate functions and successively we identify the conditions that prevent them from being analytic and that underlie affine stretches in their graphs. Finally, we apply the general theory to the number of renewals. From the point of view of Equilibrium Statistical Physics and Statistical Mechanics, cumulative rewards of the above type are the extensive observables that enter the thermodynamic description of the system. The number of renewals, which turns out to be the commonly adopted order parameter for the Poland-Scheraga model and for also the renewal models of Statistical Mechanics, is one of these observables.

Moments of Random Variables: A Systems-Theoretic Interpretation

Abstract: Moments of continuous random variables admitting a probability density function are studied. We show that, under certain assumptions, the moments of a random variable can be characterized in terms of a Sylvester equation and of the steady-state output response of a specific interconnected system. This allows to interpret well-known notions and results of probability theory and statistics in the language of systems theory, including the sum of independent random variables, the notion of mixture distribution and results from renewal theory. The theory developed is based on tools from center manifold theory, the theory of the steady-state response of nonlinear systems, and the theory of output regulation. Our formalism is illustrated by means of several examples and can be easily adapted to the case of discrete and multivariate random variables.

Rademacher complexity for Markov chains: Applications to kernel smoothing and Metropolis–Hastings

Abstract: The concept of Rademacher complexity for independent sequences of random variables is extended to Markov chains. The proposed notion of “regenerative block Rademacher complexity” (of a class of functions) follows from renewal theory and allows to control the expected values of suprema (over the class of functions) of empirical processes based on Harris Markov chains as well as the excess probability. For classes of Vapnik–Chervonenkis type, bounds on the “regenerative block Rademacher complexity” are established. These bounds depend essentially on the sample size and the probability tails of the regeneration times. The proposed approach is employed to obtain convergence rates for the kernel density estimator of the stationary measure and to derive concentration inequalities for the Metropolis–Hastings algorithm.

Maintenance management of load haul dumper using reliability analysis

Abstract: Load haul dumper (LHD) is one of the main ore transporting machineries used in underground mining industry. Reliability of LHD is very significant to achieve the expected targets of production. The performance of the equipment should be maintained at its highest level to fulfill the targets. This can be accomplished only by reducing the sudden breakdowns of component/subsystems in a complex system. The identification of defective component/subsystems can be possible by performing the downtime analysis. Hence, it is very important to develop the proper maintenance strategies for replacement or repair actions of the defective ones. Suitable maintenance management actions improve the performance of the equipment. This paper aims to discuss this issue.,Reliability analysis (renewal approach) has been used to analyze the performance of LHD machine. Allocations of best-fit distribution of data sets were made by the utilization of Kolmogorov–Smirnov (K–S) test. Parametric estimation of theoretical probability distributions was made by utilizing the maximum likelihood estimate (MLE) method.,Independent and identical distribution (IID) assumption of data sets was validated through trend and serial correlation tests. On the basis of test results, the data sets are in accordance with IID assumption. Therefore, renewal process approach has been utilized for further investigation. Allocations of best-fit distribution of data sets were made by the utilization of Kolmogorov–Smirnov (K–S) test. Parametric estimation of theoretical probability distributions was made by utilizing the MLE method. Reliability of each individual subsystem has been computed according to the best-fit distribution. In respect of obtained reliability results, the reliability-based preventive maintenance (PM) time schedules were calculated for the expected 90 percent reliability level.,As the reliability analysis is one of the complex techniques, it requires strategic decision making knowledge for the selection of methodology to be used. As the present case study was from a public sector company, operating under financial constraints the conclusions/findings may not be universally applicable.,The present study throws light on this equipment that need a tailored maintenance schedule, partly due to the peculiar mining conditions, under which they operate. This study mainly focuses on estimating the performance of four numbers of well-mechanized LHD systems with reliability, availability and maintainability (RAM) modeling. Based on the drawn results, reasons for performance drop of each machine were identified. Suitable recommendations were suggested for the enhancement of performance of capital intensive production equipment. As the maintenance management is only the means for performance improvement of the machinery, PM time intervals were estimated with respect to the expected rate of reliability level.

Flexible Regression Models for Count Data Based on Renewal Processes: The Countr Package

Abstract: A new alternative to the standard Poisson regression model for count data is suggested. This new family of models is based on discrete distributions derived from renewal processes, i.e., distributions of the number of events by some time t. Unlike the Poisson model, these models have, in general, time-dependent hazard functions. Any survival distribution can be used to describe the inter-arrival times between events, which gives a rich class of count processes with great flexibility for modelling both underdispersed and overdispersed data. The R package Countr provides a function, renewalCount(), for fitting renewal count regression models and methods for working with the fitted models. The interface is designed to mimic the glm() interface and standard methods for model exploration, diagnosis and prediction are implemented. Package Countr implements stateof-the-art recently developed methods for fast computation of the count probabilities. The package functionalities are illustrated using several datasets.

Large Deviations in Discrete-Time Renewal Theory

Abstract: We establish sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. The framework we consider is the pinning model of polymers, which amounts to a Gibbs change of measure of a classical renewal process and includes it as a special case. We first tackle the problem in a constrained pinning model, where one of the renewals occurs at a given time, by an argument based on convexity and super-additivity. We then transfer the results to the original pinning model by resorting to conditioning.

Ergodicity of an SPDE associated with a many-server queue

Abstract: We consider the so-called GI/GI/N queueing network in which a stream of jobs with independent and identically distributed service times arrive according to a renewal process to a common queue served by $N$ identical servers in a first-come-first-serve manner. We introduce a two-component infinite-dimensional Markov process that serves as a diffusion model for this network, in the regime where the number of servers goes to infinity and the load on the network scales as $1-\beta N^{-1/2}+o(N^{-1/2})$ for some $\beta>0$. Under suitable assumptions, we characterize this process as the unique solution to a pair of stochastic evolution equations comprised of a real-valued Ito equation and a stochastic partial differential equation on the positive half line, which are coupled together by a nonlinear boundary condition. We construct an asymptotic (equivalent) coupling to show that this Markov process has a unique invariant distribution. This invariant distribution is shown in a companion paper [Aghajani and Ramanan (2016)] to be the limit of the sequence of suitably scaled and centered stationary distributions of the GI/GI/N network, thus resolving (for a large class service distributions) an open problem raised by Halfin and Whitt in [Oper. Res. 29 (1981) 567–588]. The methods introduced here are more generally applicable for the analysis of a broader class of networks.

Generalized fractional Poisson process and related stochastic dynamics

Abstract: We survey the 'generalized fractional Poisson process' (GFPP). The GFPP is a renewal process generalizing Laskin's fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges $0 0$ and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in $d$ dimensions the 'well-scaled' diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.

Long Term Behaviour of a Reversible System of Interacting Random Walks

Abstract: This paper studies the long-term behaviour of a system of interacting random walks labelled by vertices of a finite graph. We show that the system undergoes phase transitions, with different behaviour in various regions, depending on model parameters and properties of the underlying graph. We provide the complete classification of the long-term behaviour of the corresponding continuous time Markov chain, identifying whether it is null recurrent, positive recurrent, or transient. The proofs are partially based on the reversibility of the model, which allows us to use the method of electric networks. We also provide some alternative proofs (based on the Lyapunov function method and the renewal theory), which are of interest in their own right, since they do not require reversibility and can be applied to more general situations.

First hitting time of uncertain random renewal reward process and its application in insurance risk process

Abstract: The renewal reward process is used to record the cumulative rewards of a system, which is widely applied in the queuing problems and insurance pricing problems. This paper studies a type of renewal reward processes with random inter-arrival times and uncertain rewards from the point of view of first hitting time. The analytic expressions of the chance distribution and the expected value of the first hitting time are derived, and a numerical method for calculating the chance distribution is designed based on the Monte-Carlo simulation. Besides, the concept of first hitting time is applied to the insurance risk process and is employed to model the ruin index of an insurance company.

Superimposed Renewal Processes in Reliability

Abstract: This paper reviews the existing literature on the superimposed renewal process, with its foci on probabilistic and statistical properties, statistical inference, and applications in reliability analysis and maintenance policy optimisation. It then proposes future research topics.

Large Deviations in Renewal Models of Statistical Mechanics

Abstract: In Ref. [1] the author has recently established sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a separable Banach space. The renewal model has been there identified with constrained and non-constrained pinning models of polymers, which amount to Gibbs changes of measure of a classical renewal process. In this paper we show that the constrained pinning model is the common mathematical structure to the Poland-Scheraga model of DNA denaturation and to some relevant one-dimensional lattice models of Statistical Mechanics, such as the Fisher-Felderhof model of fluids, the Wako-Saito-Munoz-Eaton model of protein folding, and the Tokar-Dreysse model of strained epitaxy. Then, in the framework of the constrained pinning model, we develop an analytical characterization of the large deviation principles for cumulative rewards corresponding to multivariate deterministic rewards that are uniquely determined by, and at most of the order of magnitude of, the time elapsed between consecutive renewals. In particular, we outline the explicit calculation of the rate functions and successively we identify the conditions that prevent them from being analytic and that underlie affine stretches in their graphs. Finally, we apply the general theory to the number of renewals. From the point of view of Equilibrium Statistical Physics and Statistical Mechanics, cumulative rewards of the above type are the extensive observables that enter the thermodynamic description of the system. The number of renewals, which turns out to be the commonly adopted order parameter for the Poland-Scheraga model and for also the renewal models of Statistical Mechanics, is one of these observables.

Modeling extreme negative returns using marked renewal Hawkes processes

Abstract: Extreme return financial time series are often challenging to model due to the presence of heavy temporal clustering of extremes and strong bursts of return volatility. One approach to model both these phenomena in extreme financial returns is the marked Hawkes self-exciting process. However, the Hawkes process restricts the arrival times of exogenously driven returns to follow a Poisson process and may fail to provide an adequate fit to data. In this work, we introduce a model for extreme financial returns, which provides added flexibility in the specification of the background arrival rate. Our model is a marked version of the recently proposed renewal Hawkes process, in which exogenously driven extreme returns arrive according to a renewal process rather than a Poisson process. We develop a procedure to evaluate the likelihood of the model, which can be optimized to obtain estimates of model parameters and their standard errors. We provide a method to assess the goodness-of-fit of the model based on the Rosenblatt residuals, as well as a procedure to simulate the model. We apply the proposed model to extreme negative returns for five stocks traded on the Australian Stock Exchange. The models identified for the stocks using in-sample data were found to be able to successfully forecast the out-of-sample risk measures such as the value at risk and provide a better quality of fit than the competing Hawkes model.

A Renewal Shot Noise Process with Subexponential Shot Marks

Abstract: We investigate a shot noise process with subexponential shot marks occurring at renewal epochs. Our main result is a precise asymptotic formula for its tail probability. In doing so, some recent results regarding sums of randomly weighted subexponential random variables play a crucial role.

Reliability analysis of underground rock bolters using the renewal process, the non-homogeneous Poisson process and the Bayesian approach

Abstract: Reliability plays an important role in the execution of the maintenance improvement and the understanding of its concepts is essential to predict the type of maintenance according to the equipment state. Thereby, a computational tool was developed and programming with VBA in Excel® for reliability and failure analysis in a mining context. The paper aims to discuss these issues.,The developed approach use the modeling of stochastic processes, such as the renewal process, the non-homogeneous Poisson process and less conventional method as the Bayesian approach, by considering Jeffreys non-informative prior. The resolution gives the best associated model, the parameters estimation, the mean time between failure and the reliability estimate. This approach is validated with the reliability analysis of inter-failure times from underground rock bolters subsystems, over a two-year period.,Results show that Weibull and lognormal probability distribution fit to the most subsystems inter-failure times. The study revealed that the bolting head, the rock drill, the screen handler, the electric/electronic system, the hydraulic system, the drilling feeder and the structural consume the most repair frequency. The hydraulic and electric/electronic subsystems represent the lowest reliability after 50 operation hours.,For the first time, this case study defines practical failures and reliability information for rock bolter subsystems based on real operation data. This paper is useful to the comparative evaluation of rock bolter by detecting the weakest elements and understanding failure patterns in the individual observation subsystems on the overall machine performance.

The Mittag-Leffler function in the thinning theory for renewal processes

Abstract: The main purpose of this note is to point out the relevance of the Mittag-Leffler probability distribution in the so-called thinning theory for a renewal process with a queue of power law type. This theory, formerly considered by Gnedenko and Kovalenko in 1968 without the explicit reference to the Mittag-Leffler function, was used by the authors in the theory of continuous random walk and consequently of fractional diffusion in a plenary lecture by the late Prof Gorenflo at a Seminar on Anomalous Transport held in Bad-Honnef in July 2006, published in a 2008 book. After recalling the basic theory of renewal processes including the standard and the fractional Poisson processes, here we have revised the original approach by Gnedenko and Kovalenko for convenience of the experts of stochastic processes who are not aware of the relevance of the Mittag-Leffler function

On some generalization of Lorden's inequality for renewal processes

Abstract: In queueing theory, Lorden's inequality can be used for bounds estimation of the moments of backward and forward renewal times. Two random variables called backwards renewal time and forward renewal time for this process are defined. Lorden's inequality it's true for the renewal process, so expectations of backward and forward renewal times are bounded by the relation of expectation of moment of the random variable for any fixed moment of time, where random variables are i.i.d. We generalised and proved a similar result for dependent random variables with finite expectations, some constant C and integrable function Q(s): if X is not independent and have absolutely continuous distribution function which satisfies some boundary conditions, then the analogue of Lorden's inequality for renewal process is true. In August 2021 reviewed version is uploaded.