Showing papers on "Renewal theory published in 2006"
TL;DR: A framework where the ob- served events are modeled as marked point processes, with marks labeling the types of events is presented, where the emphasis is more on modeling than on statistical inference.
Abstract: We review basic modeling approaches for failure and mainte- nance data from repairable systems. In particular we consider imperfect re- pair models, defined in terms of virtual age processes, and the trend-renewal process which extends the nonhomogeneous Poisson process and the renewal process. In the case where several systems of the same kind are observed, we show how observed covariates and unobserved heterogeneity can be included in the models. We also consider various approaches to trend testing. Modern reliability data bases usually contain information on the type of failure, the type of maintenance and so forth in addition to the failure times themselves. Basing our work on recent literature we present a framework where the ob- served events are modeled as marked point processes, with marks labeling the types of events. Throughout the paper the emphasis is more on modeling than on statistical inference.
176 citations
TL;DR: In this article, the authors consider an insurance portfolio situation in which there is possible dependence between the waiting time for a claim and its actual size and obtain explicit exponential estimates for infinite and finite-time ruin probabilities in the case of light-tailed claim sizes.
Abstract: We consider an insurance portfolio situation in which there is possible dependence between the waiting time for a claim and its actual size. By employing the underlying random walk structure we obtain explicit exponential estimates for infinite- and finite-time ruin probabilities in the case of light-tailed claim sizes. The results are illustrated in several examples, worked out for specific dependence structures.
147 citations
TL;DR: This work develops a formal theory for developing RS preventive-maintenance plans using the Markowitz paradigm in which one seeks to optimize a function of the expected cost and its variance.
90 citations
TL;DR: Amathematical model is developed to help analyze the benefit in contact-center performance obtained from increasing employee (agent) retention, which is in turn obtained by increasing agent job satisfaction.
Abstract: Amathematical model is developed to help analyze the benefit in contact-center performance obtained from increasing employee (agent) retention, which is in turn obtained by increasing agent job satisfaction. The contact-center performance may be restricted to a traditional productivity measure such as the number of calls answered per hour, or it may include a broader measure of the quality of service, e.g., revenue earned per hour or the number of problems successfully resolved per hour. The analysis is based on an idealized model of a contact center in which the number of employed agents is constant over time, assuming that a new agent is immediately hired to replace each departing agent. The agent employment periods are assumed to be independent and identically distributed random variables with a general agent-retention probability distribution, which depends on management policy and actions. The steady-state staff-experience distribution is obtained from the agent-retention distribution by applying renewal theory. An increasing real-valued function specifies the average performance as a function of agent experience. Convenient closed-form expressions for the overall performance as a function of model elements are derived when either the agent-retention distribution or the performance function has exponential structure. Management actions may cause the agent-retention distribution to change. The model describes the consequences of such changes on the long-run average staff experience and the long-run average performance.
77 citations
TL;DR: It is shown that there is a close relationship between coverage intensity and the measure of path availability, defined as the probability distribution of durations within which a path (of a fixed number of nodes) remains available.
Abstract: This paper studies wireless sensor networks that operate in low duty cycles, measured by the percentage of time a sensor is on or active. The dynamic change in topology as a result of such duty-cycling has potentially disruptive effect on the performance of the network. We limit our attention to a class of surveillance and monitoring applications and random duty-cycling schemes, and analyze certain coverage property. Specifically, we consider coverage intensity defined as the probability distribution of durations within which a target or an event is uncovered/unmonitored. We derive this distribution using a semi-Markov model, constructed using the superposition of alternating renewal processes. We also present the asymptotic (as the number of sensors approaches infinity) distribution of the target uncovered duration when at least one sensor is required to cover the target, and provide an asymptotic lower bound when multiple sensors are required to cover the target. The analysis using the semi-Markov model serves as a tool with which we can find suitable random duty-cycling schemes satisfying a given performance requirement. Our numerical observations show that the stochastic variation of duty-cycling durations affects performance only when the number of sensors is small, whereas the stochastic mean of duty-cycling durations impacts performance in all cases studied. We also show that there is a close relationship between coverage intensity and the measure of path availability, defined as the probability distribution of durations within which a path (of a fixed number of nodes) remains available. Thus the results presented here are readily applicable to the study of path availability in a low duty-cycled sensor network
72 citations
TL;DR: The interarrival times are characterized as nonnegative random fuzzy variables which is a more reasonable consideration in the real world and Blackwell's theorem in random fuzzy sense is established.
68 citations
TL;DR: In this article, none-exponential asymptotics for solutions of two specific defective renewal equations are obtained, including the special cases of asymPTotics for a compound geometric distribution and the convolution of a compound geometrical distribution with a distribution function.
Abstract: Nonexponential asymptotics for solutions of two specific defective renewal equations are obtained. These include the special cases of asymptotics for a compound geometric distribution and the convolution of a compound geometric distribution with a distribution function. As applications of these results, we study the asymptotic behavior of the demographic birth rate of females, the perpetual put option in mathematics of finance, and the renewal function for terminating renewal processes.
61 citations
26 Jun 2006
TL;DR: An initial analysis of job arrivals in a production data-intensive Grid and several traffic models to characterize the interarrival time processes are investigated and MMPPs are not able to match the autocorrelations for certain VOs, in which strong deterministic semi-periodic patterns are observed.
Abstract: In this paper we present an initial analysis of job arrivals in a production data-intensive Grid and investigate several traffic models to characterize the interarrival time processes. Our analysis focuses on the heavy-tail behavior and autocorrelation structures, and the modeling is carried out at three different levels: Grid, Virtual Organization (VO), and region. A set of m-state Markov modulated Poisson processes (MMPP) is investigated, while Poisson processes and hyperexponential renewal processes are evaluated for comparison studies. We apply the transportation distance metric from dynamical systems theory to further characterize the differences between the data trace and the simulated time series, and estimate errors by bootstrapping. The experimental results show that MMPPs with a certain number of states are successful to a certain extent in simulating the job traffic at different levels, fitting both the interarrival time distribution and the autocorrelation function. However, MMPPs are not able to match the autocorrelations for certain VOs, in which strong deterministic semi-periodic patterns are observed. These patterns are further characterized using different representations. Future work is needed to model both deterministic and stochastic components in order to better capture the correlation structure in the series.
55 citations
TL;DR: In this paper, a single-item inventory system where two demand classes with different service requirements are satisfied from a common inventory is considered, and a critical level, reorder point, order quantity or (s, q, k) policy is in use.
53 citations
TL;DR: In this paper, a new stochastic process, i.e., an extended Poisson process, is introduced for repairable systems where the failure intensity shows a bathtub-type nonmonotonic behaviour.
Abstract: Commonly used repair rate models for repairable systems in the reliability literature are renewal processes, generalised renewal processes or non-homogeneous Poisson processes. In addition to these models, geometric processes (GP) are studied occasionally. The GP, however, can only model systems with monotonously changing (increasing, decreasing or constant) failure intensities. This paper deals with the reliability modelling of failure processes for repairable systems where the failure intensity shows a bathtub-type non-monotonic behaviour. A new stochastic process, i.e. an extended Poisson process, is introduced in this paper. Reliability indices are presented, and the parameters of the new process are estimated. Experimental results on a data set demonstrate the validity of the new process.
49 citations
01 Jan 2006
TL;DR: A multi-objective model for the time–cost trade-off problem in a dynamic PERT network using an interactive approach is developed and extended to the generalized Erlang activity durations.
Abstract: We develop a multi-objective model for the time–cost trade-off problem in a dynamic PERT network using an interactive approach. The activity durations are exponentially distributed random variables and the new projects are generated according to a renewal process and share the same facilities. Thus, these projects cannot be analyzed independently. This dynamic PERT network is represented as a network of queues, where the service times represent the durations of the corresponding activities and the arrival stream to each node follows a renewal process. At the first stage, we transform the dynamic PERT network into a proper stochastic network and then compute the project completion time distribution by constructing a continuous-time Markov chain. At the second stage, the time–cost trade-off problem is formulated as a multi-objective optimal control problem that involves four conflicting objective functions. Then, the STEM method is used to solve a discrete-time approximation of the original problem. Finally, the proposed methodology is extended to the generalized Erlang activity durations. 2006 Elsevier B.V. All rights reserved.
TL;DR: In this article, a simple approach based on Renewal Theory is proposed to determine the precise asymptotic behavior of the partition function, from which the scaling limits of the models and an explicit construction of the infinite volume measure in all regimes, including the critical one are obtained.
Abstract: We consider continuous and discrete (1+1)-dimensional wetting models which undergo a localization/delocalization phase transition. Using a simple approach based on Renewal Theory we determine the precise asymptotic behavior of the partition function, from which we obtain the scaling limits of the models and an explicit construction of the infinite volume measure in all regimes, including the critical one.
TL;DR: This paper addresses some properties of fuzzy random renewal processes generated by a sequence of independent and identically distributed (iid) fuzzy random interarrival times.
Abstract: Fuzzy random variable is a measure function from a probability space to a collection of fuzzy variables. Based on the fuzzy random theory, this paper addresses some properties of fuzzy random renewal processes generated by a sequence of independent and identically distributed (iid) fuzzy random interarrival times. The relationship between the expected value of the fuzzy random renewal variable and the distribution functions of the alpha-pessimistic values and alpha-optimistic values of the interarrival times is discussed. Furthermore, the fuzzy random style of renewal equation is provided. Finally, fuzzy random Blackwell's renewal theorem and Smith's key renewal theorem are also given
TL;DR: In this paper, the renewal risk process with stochastic interest is considered and exact expressions and integral equations for the Gerber-Shiu expected discounted penalty function and the ultimate ruin probability are derived.
TL;DR: It is shown that the dynamic approach to fractional Brownian motion establishes a link between a non-Poisson renewal process with abrupt jumps resetting to zero the system's memory and correlated dynamic processes, whose individual trajectories keep a nonvanishing memory of their past time evolution.
Abstract: We show that the dynamic approach to fractional Brownian motion (FBM) establishes a link between a non-Poisson renewal process with abrupt jumps resetting to zero the system's memory and correlated dynamic processes, whose individual trajectories keep a nonvanishing memory of their past time evolution. It is well known that the recrossings of the origin by an ordinary one-dimensional diffusion trajectory generates a L\'evy (and thus renewal) process of index $\ensuremath{\theta}=1∕2$. We prove with theoretical and numerical arguments that this is the special case of a more general condition, insofar as the recrossings produced by the dynamic FBM generates a L\'evy process with $0l\ensuremath{\theta}l1$. This result is extended to produce a satisfactory model for the fluorescent signal of blinking quantum dots.
TL;DR: A T-related fuzzy renewal theorem and a fuzzy renewal reward theorem are proved using a law of large numbers for fuzzy variables under t-norm-based fuzzy operations.
TL;DR: In this article, the authors consider the transformation T that takes a distribution F into the distribution of the length of the interval covering a fixed point in the stationary renewal process corresponding to F. This transformation has been referred to as size-biasing, length biasing, the renewal length transformation, and the stationary lifetime operator.
Abstract: We consider the transformation T that takes a distribution F into the distribution of the length of the interval covering a fixed point in the stationary renewal process corresponding to F. This transformation has been referred to as size-biasing, length-biasing, the renewal length transformation, and the stationary lifetime operator. We review and develop properties of this transformation and apply it to diverse areas.
TL;DR: A general class of stochastic processes forced by instantaneous random fires that reset the state variable x to a given value that is based on the deterministic drift of the fire activity is introduced.
Abstract: We introduce a general class of stochastic processes forced by instantaneous random fires (i.e., jumps) that reset the state variable $x$ to a given value. Since in many physical systems the fire activity is often dependent on the actual value of the state variable, as in the case of natural fires in ecosystems and firing dynamics in neuronal activity, the frequency of fire occurrence is assumed to be state dependent. Such dynamics leads to independent interfire statistics---i.e., to renewal point processes. Various functions relating the frequency of fire occurrence to $x(t)$ are analyzed and compared. The relation between the probabilistic dynamics of $x(t)$ and the interfire statistics is derived and some exact probability distribution of both $x(t)$ and the interfire times are obtained for systems with different degrees of complexity. After studying processes in which the fire activity is coupled only to a deterministic drift, we also analyze processes forced by either additive or multiplicative Gaussian white noise.
TL;DR: In this article, the authors considered a data network model in which sources begin to transmit at renewal time points fSng and transmissions are assumed to proceed at xed rate unity.
TL;DR: In this article, a general approach to anomalous diffusion is provided by the integral equation for the so-called continuous time random walk (CTRW), which can be understood as a random walk subordinated to a renewal process.
Abstract: A mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. A more general approach is however provided by the integral equation for the so-called continuous time random walk (CTRW), which can be understood as a random walk subordinated to a renewal process. We show how this integral equation reduces to our fractional diffusion equations by a properly scaled passage to the limit of compressed waiting times and jumps. The essential assumption is that the probabilities for waiting times and jumps behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. Illustrating examples are given, numerical results and plots of simulations are displayed.
TL;DR: In this article, a defective renewal equation for the conditional Gerber-Shiu expected discounted penalty function is obtained based on the associated ordinary renewal process, and the relationship between the conditional expected discount function in the ordinary renewal case and that in the delayed renewal case is established.
Abstract: This paper considers the compound Markov binomial risk model proposed by Cossette et al. (2003 2004). Two discrete-time renewal (ordinary renewal and delayed renewal) risk processes associated with the compound Markov binomial risk model are analyzed. Based on the associated ordinary renewal process, a defective renewal equation for the conditional Gerber–Shiu expected discounted penalty function is obtained. The relationship between the conditional expected discounted penalty function in the ordinary renewal case and that in the delayed renewal case is then established. From these results, the conditional ultimate probability of ruin as well as the conditional joint distribution of the surplus just prior to ruin and the deficit at ruin are studied. Finally, it is shown that a modified version of the compound Markov binomial risk model is a special case of the discrete-time semi-Markov risk model introduced by Reinhard and Snoussi (2001 2002).
TL;DR: The Context Tree Weighting method is shown to be almost adaptive on the classes of renewal and Markov renewal processes and it is shown that almost minimax redundancy can be achieved on massive classes of sources (classes that cannot be smoothly parameterized by subsets of finite-dimensional spaces).
Abstract: The Context Tree Weighting method (CTW) is shown to be almost adaptive on the classes of renewal and Markov renewal processes. Up to logarithmic factor, ctw achieves the minimax pointwise redundancy described by I. Csiszaacuter and P. Shields in IEEE Trans. Inf. Theory, vol. 42, no. 6, pp. 2065-2072, Nov. 1996. This result not only complements previous results on the adaptivity of the Context-Tree Weighting method on the relatively small class of all finite context-tree sources (which encompasses the class of all finite order Markov sources), it shows that almost minimax redundancy can be achieved on massive classes of sources (classes that cannot be smoothly parameterized by subsets of finite-dimensional spaces). Moreover, it shows that (almost) adaptive compression can be achieved in a computationally efficient way on those massive classes. While previous adaptivity results for CTW could rely on the fact that any Markov source is a finite-context-tree source, this is no longer the case for renewal sources. In order to prove almost adaptivity of CTW over renewal sources, it is necessary to establish that CTW carefully balances estimation error and approximation error
TL;DR: A substitutable two-product inventory system with joint-ordering policy with cost analysis is considered andumerical example illustrated the results obtained.
TL;DR: In this article, the authors considered a class of risk processes with delayed claims, and provided ruin probability estimates under heavy tail conditions on the claim size distribution under the assumption of a subexponential distribution.
Abstract: We consider a class of risk processes with delayed claims, and we provide ruin probability estimates under heavy tail conditions on the claim size distribution. Keywords : Extreme value theory; Poisson process; regular variation; renewal process; ruin probability; shot noise process; subexponential distribution.
TL;DR: A stochastic version of economic tool life models for machines with finite capacity tool magazines and a variable processing speed capability, where the tool life is a random variable is presented.
Abstract: We present a stochastic version of economic tool life models for machines with finite capacity tool magazines and a variable processing speed capability, where the tool life is a random variable. Using renewal theory to express the expected number of tool setups as a function of cutting speed and magazine capacity, we extend previously published deterministic mathematical programming models to the case of minimizing the expected total processing time. A numerical illustration with typical cutting tool data shows the deterministic model underestimates the optimal expected processing time by more than 8% when the coefficient of variation equals 0.3 (typical for carbide tools), and the difference exceeds 15% for single-injury tools having an exponentially distributed economic life (worst case).
TL;DR: In this article, the effect of transition of distributions of sums of independent identically distributed non-negative integer-valued random variables from one lattice to another in the framework of the generalised allocation scheme is considered.
Abstract: We consider the effect of transition of distributions of sums of independent identically distributed non-negative integer-valued random variables from one lattice to another in the framework of the generalised allocation scheme.
TL;DR: This work considers a more general situation where this time point is random and related to a specific event, for example, status change or onset of a disease, and provides formulae for the estimation of the distribution of interarrival times based on the observed event intervals.
Abstract: The analysis of length-biased data has been mostly limited to the interarrival interval of a renewal process covering a specific time point. Motivated by a surveillance problem, we consider a more general situation where this time point is random and related to a specific event, for example, status change or onset of a disease. We also consider the problem when additional information is available on whether the event intervals (interarrival intervals covering the random event) end within or after a random time period (which we call a window period) following the random event. Under the assumptions that the occurrence rate of the random event is low and the renewal process is independent of the random event, we provide formulae for the estimation of the distribution of interarrival times based on the observed event intervals. Procedures for testing the required assumptions are also furnished. We apply our results to human immunodeficiency virus (HIV) test data from public test sites in Seattle, Washington, where the random event is HIV infection and the window period is from the onset of HIV infection to the time at which a less sensitive HIV test becomes positive. Results show that the estimator of the intertest interval length distribution from event intervals ending within the window period is less biased than the estimator from all event intervals; the latter estimator is affected by right truncation. Finally, we discuss possible applications to estimating HIV incidence and analyzing length-biased samples with right or left truncated data.
TL;DR: In this paper, analytical bounds and approximations for a recursive algorithm for numerical computation are studied based on a simple iterative procedure which provides some analytical results and nice convergence properties when the number of iteration increases.
Abstract: Renewal-type equations are frequently encountered in the study of reliability, warranty analysis, replacement and maintenance policies, and inventory control. Renewal equations usually do not have analytical solutions, and hence, bounds or approximations are very useful. In this article, analytical bounds are studied based on a simple iterative procedure which provides some analytical results and nice convergence properties when the number of iteration increases. Bounds and approximations are also investigated for a recursive algorithm for numerical computation. In addition, some interesting monotonicity properties are introduced and discussed. The approximation error, which is important for determining the stopping rule of the iterative procedure and the numerical algorithm, is also studied.
TL;DR: In this article, the authors compare the waiting times of customers in multiple-server queues, where the idle times are removed, with different numbers of servers, and develop some results regarding the vector-valued marked point process whose points are arrival epochs of the superposition of renewal processes with different continuous inter-arrival distribution.
TL;DR: In this paper, the concept of the renewal property is extended to processes indexed by a multidimensional time parameter, including partial sum processes, Poisson processes and many other point processes whose jump points are not totally ordered.
Abstract: The concept of the renewal property is extended to processes indexed by a multidimensional time parameter. The definition given includes not only partial sum processes, but also Poisson processes and many other point processes whose jump points are not totally ordered. A new version of the waiting time paradox is proven for multidimensional Poisson processes, and is shown to imply the renewal property. Finally, martingale properties of renewal processes are studied.