Showing papers on "Renewal theory published in 2015"
TL;DR: An analytically tractable model is proposed, in which interactions among the agents are ruled by a renewal process, that is able to reproduce aging behavior in temporal social networks, and an analytic solution for the topological properties of the integrated network produced by the model is developed.
Abstract: The presence of burstiness in temporal social networks, revealed by a power-law form of the waiting time distribution of consecutive interactions, is expected to produce aging effects in the corresponding time-integrated network. Here, we propose an analytically tractable model, in which interactions among the agents are ruled by a renewal process, that is able to reproduce this aging behavior. We develop an analytic solution for the topological properties of the integrated network produced by the model, finding that the time translation invariance of the degree distribution is broken. We validate our predictions against numerical simulations, and we check for the presence of aging effects in a empirical temporal network, ruled by bursty social interactions.
94 citations
TL;DR: In this article, an uncertain random variable has been proposed as a generalization of both the stochastic process and the uncertain process, and some special types of uncertain random processes such as stationary increment process and renewal process are discussed.
Abstract: To deal with a system with both randomness and uncertainty, chance theory has been built and an uncertain random variable has been proposed as a generalization of random variable and uncertain variable Correspondingly, as a generalization of both the stochastic process and the uncertain process, this paper will propose an uncertain random process In addition, some special types of uncertain random processes such as stationary increment process and renewal process will also be discussed
72 citations
TL;DR: An alternating renewal theorem is proved, which gives the limit chance distribution of the interval availability of the uncertain random alternating renewal system.
Abstract: As a mixture of a random variable and an uncertain variable, an uncertain random variable is a tool to deal with the indeterminacy quantities involving randomness and human uncertainty. This paper aims at proposing a concept of an uncertain random alternating renewal process to model a repairable system with random on-times and uncertain off-times. An alternating renewal theorem is proved, which gives the limit chance distribution of the interval availability of the uncertain random alternating renewal system.
39 citations
TL;DR: In this article, a system subject to simultaneous shocks is considered and the evolution of the system is studied using matrix-analytic methods, the model for the arrivals of shocks being a batch Markovian arrival process.
33 citations
04 Aug 2015
TL;DR: In this paper, the renewal counting number process N = N(t) is considered as a forward march over the non-negative integers with independent identically distributed waiting times, and the Laplace transform with respect to both variables x and t is applied.
Abstract: We consider the renewal counting number process N = N(t) as a forward march over the non-negative integers with independent identically distributed waiting times. We embed the values of the counting numbers N in a “pseudo-spatial” non-negative half-line x ≥ 0 and observe that for physical time likewise we have t ≥ 0. Thus we apply the Laplace transform with respect to both variables x and t. Applying then a modification of the Montroll-Weiss-Cox formalism of continuous time random walk we obtain the essential characteristics of a renewal process in the transform domain and, if we are lucky, also in the physical domain. The process t = t(N) of accumulation of waiting times is inverse to the counting number process, in honour of the Danish mathematician and telecommunication engineer A.K. Erlang we call it the Erlang process. It yields the probability of exactly n renewal events in the interval (0; t]. We apply our Laplace-Laplace formalism to the fractional Poisson process whose waiting times are of Mittag-Leffler type and to a renewal process whose waiting times are of Wright type. The process of Mittag-Leffler type includes as a limiting case the classical Poisson process, the process of Wright type represents the discretized stable subordinator and a re-scaled version of it was used in our method of parametric subordination of time-space fractional diffusion processes. Properly rescaling the counting number process N(t) and the Erlang process t(N) yields as diffusion limits the inverse stable and the stable subordinator, respectively.
33 citations
TL;DR: A mathematical model of the renewal access protocol (RAP) is developed and it is shown that the throughput performance of the RAP depends only on the expectation of the selection distribution where the backoff counter is selected, provided that the number of terminals is fixed.
Abstract: We consider a simple MAC protocol, called the renewal access protocol (RAP), that adopts all of the legacy 802.11 standard but the backoff stage feature. To meet two objectives in the design of the RAP---optimal throughput and high short-term fairness---we develop a mathematical model of the RAP and rigorously analyze the performance of the RAP. First, we show that the throughput performance of the RAP depends only on the expectation of the selection distribution where the backoff counter is selected, provided that the number of terminals is fixed, which is in accordance with a well-known result. Second, with the help of renewal and reliability theories, we analyze the short-term fairness of the RAP. We also show that if the RAP has a selection distribution of the New Better than Used in Expectation (NBUE) type, the RAP can guarantee high short-term fairness. Third, we construct a special binomial distribution that is obviously of the NBUE type that can achieve high short-term fairness as well as optimal throughput when used as the selection distribution of the RAP. Furthermore, by the Poisson approximation for binomial distributions, we propose to use in practice a Poisson distribution corresponding to the special binomial distribution. Numerical and simulation results are provided to validate our analysis.
28 citations
TL;DR: In this article, a new technique for operator renewal sequences associated with dynamical systems preserving an infinite measure is introduced, which improves the results on mixing rates obtained by Melbourne and Terhesiu.
Abstract: In this work, we introduce a new technique for operator renewal sequences associated with dynamical systems preserving an infinite measure that improves the results on mixing rates obtained by Melbourne and Terhesiu [Operator renewal theory and mixing rates for dynamical systems with infinite measure. Invent. Math. 1 (2012), 61–110]. Also, this technique allows us to offer a very simple proof of the key result of Melbourne and Terhesiu that provides first-order asymptotics of operator renewal sequences associated with dynamical systems with infinite measure. Moreover, combining techniques used in this work with techniques used by Melbourne and Terhesiu, we obtain first-order asymptotics of operator renewal sequences under some relaxed assumption on the first return map.
21 citations
TL;DR: In this paper, the authors present recursive formulae for the ruin probability at or before a certain claim arrival instant for some particular continuous time risk model, where the claim number process underlying this risk model is a renewal process with either Erlang or a mixture of exponentials inter-claim times.
Abstract: In this paper, we present recursive formulae for the ruin probability at or before a certain claim arrival instant for some particular continuous time risk model. The claim number process underlying this risk model is a renewal process with either Erlang or a mixture of exponentials inter-claim times (ICTs). The claim sizes (CSs) are independent and distributed in Erlang's family, i.e., they can have different parameters, which yields a non-homogeneous risk process. We present the corresponding recursive algorithm used to evaluate the above mentioned ruin probability and we illustrate it on several numerical examples in which we vary the model's parameters to assess the impact of the non-homogeneity on the resulting ruin probability.
20 citations
TL;DR: A stochastic approach to analyze instantaneous unavailability of standby safety equipment caused by latent failures is presented and an integral equation for point unavailability is derived and numerically solved for a given maintenance policy.
19 citations
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TL;DR: In this paper, a new functional relation for the probability density function of the exponential functional of a Levy process is established, which allows to significantly simplify the techniques commonly used in the study of these random variables and hence provide quick proofs of known results, derive new results, as well as sharpening known estimates for the distribution.
Abstract: We establish a new functional relation for the probability density function of the exponential functional of a Levy process, which allows to significantly simplify the techniques commonly used in the study of these random variables and hence provide quick proofs of known results, derive new results, as well as sharpening known estimates for the distribution. We apply this formula to provide another look to the Wiener-Hopf type factorisation for exponential functionals obtained in a series of papers by Pardo, Patie and Savov, derive new identities in law, and to describe the behaviour of the tail distribution at infinity and of the distribution at zero in a rather large set of situations.
18 citations
TL;DR: In this article, the relationship among intermittency with fractal waiting time distribution, Continuous Time Random Walk (CTRW) and the emergence of Fractional Calculus (FC) is reviewed.
Abstract: The relationships among intermittency with fractal Waiting Time distribution, Continuous Time Random Walk (CTRW) and the emergence of Fractional Calculus (FC) are reviewed. The derivation, in the long-time limit, of Time Fractional Diffusion Equation (TFDE) is shown and compared with the case of normal diffusion equation. Emphasis is given to the underlying connections of CTRW with concepts and results from probability theory and stochastic processes: conditional probabilities, the law of total probability, Central and (Levy) Generalized limit theorems, renewal theory. It is shown how the emergence of a well-defined scaling rigorously emerges by imposing the invariance of the probability distribution under a group of self-similarity transformations involving space and time. The physical interpretation of some crucial mathematical passages is explained. In particular, the physical meaning of self-similarity coupled with the long-time limit is explained, having in mind a experimental point of view. Finally, the emergence of FC in complexity is discussed and associated with the ubiquitous generation of short-time transition events in the dynamics of complex systems. These renewal events are associ- ated with the dynamical emergence (birth) and decay (death) of cooperative long-lived structures, thus giving rise to a intermittent birth-death process of cooperation.
31 Aug 2015
TL;DR: A compact-form representation for the mixed double transform of probability distribution of the number of packets transmitted down the node till fixed time epoch is derived.
Abstract: Transient departure process of outgoing packets in a node of a wireless sensor network with energy saving mechanism based on threshold activation of the radio transmitter/receiver is investigated. A finite-buffer queueing system of the GI/M/1-type with N-policy, in which the transmission after each idle period is being initialized only after reaching a threshold of N packets waiting in the buffer, is considered as a mathematical model of the node operation. Applying the analytical approach based on the paradigm of embedded Markov chain, total probability law, linear algebra and renewal theory, a compact-form representation for the mixed double transform of probability distribution of the number of packets transmitted down the node till fixed time epoch is derived. A numerical illustrating example is attached as well.
TL;DR: In this article, the authors discuss the ballistic phase of quenched and annealed stretched polymers in random environment on ρ, with an emphasis on the natural renormalized renewal structures which appear in such models.
Abstract: In these lecture notes we discuss ballistic phase of quenched and annealed stretched polymers in random environment on \(\mathbb{Z}^{d}\) with an emphasis on the natural renormalized renewal structures which appear in such models. In the ballistic regime an irreducible decomposition of typical polymers leads to an effective random walk reinterpretation of the latter. In the annealed case the Ornstein-Zernike theory based on this approach paves the way to an essentially complete control on the level of local limit results and invariance principles. In the quenched case, the renewal structure maps the model of stretched polymers into an effective model of directed polymers. As a result one is able to use techniques and ideas developed in the context of directed polymers in order to address issues like strong disorder in low dimensions and weak disorder in higher dimensions. Among the topics addressed: Thermodynamics of quenched and annealed models, multi-dimensional renewal theory (under Cramer’s condition), renormalization and effective random walk structure of annealed polymers, very weak disorder in dimensions d ≥ 4 and strong disorder in dimensions d = 2, 3.
TL;DR: In this paper, the authors consider items that are incepted into operation having already a random initial age and define the corresponding remaining lifetime and show that these lifetimes are identically distributed when the age distribution is equal to the equilibrium distribution of the renewal theory.
Abstract: We consider items that are incepted into operation having already a random initial age and define the corresponding remaining lifetime. We show that these lifetimes are identically distributed when the age distribution is equal to the equilibrium distribution of the renewal theory. Then we develop the population studies approach to the problem and generalize the setting in terms of stationary and stable populations of items. We obtain new stochastic comparisons for the corresponding population ages and remaining lifetimes that can be useful in applications. Copyright © 2014 John Wiley & Sons, Ltd.
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TL;DR: In this article, the authors considered the GI/GI/N queueing network, where a stream of jobs with independent and identically distributed service times arrive according to a renewal process to a common queue served by identical servers in a First-Come-First-Serve manner.
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 It\^{o} 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 [1] 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 1981. The methods introduced here are more generally applicable for the analysis of a broader class of networks.
TL;DR: The main novelty of this work is the process followed to estimate the warranty period, which can be defined according to the risk that a company is willing to assume in the market.
Abstract: This paper considers selecting the warranty period after the completion of a series of successive repairs on a product. Two stochastic failure models are used: the general renewal process (GRP) model and the non-homogeneous Poisson process (NHPP) model. Both use a Weibull distribution for the life time of the product, allowing the possibility to renewal (GRP) or not (NHPP) when successive repairs are performed. The NHPP is applied to estimate the warranty period after simple repairs (minimal), while GRP is used when repairs are complex (overhauls). The main novelty of this work is the process followed to estimate the warranty period, which can be defined according to the risk that a company is willing to assume in the market. Our procedure provides a useful tool for a maintenance company or for a manufacturer’s after-sales division.
TL;DR: In this paper, the authors present correlated fractional counting processes on a finite-time interval, where the correlation parameter is equal to 0 and the univariate distributions coincide with those of the space-time fractional Poisson process in Orsingher and Polito (2012).
Abstract: We present some correlated fractional counting processes on a finite-time interval. This will be done by considering a slight generalization of the processes in Borges et al. (2012). The main case concerns a class of space-time fractional Poisson processes and, when the correlation parameter is equal to 0, the univariate distributions coincide with those of the space-time fractional Poisson process in Orsingher and Polito (2012). On the one hand, when we consider the time fractional Poisson process, the multivariate finite-dimensional distributions are different from those presented for the renewal process in Politi et al. (2011). We also consider a case concerning a class of fractional negative binomial processes.
01 Jan 2015
TL;DR: The paper starts with a description of the basis of organizational renewal, which later allows to present an original proposal ofizational renewal process model and its components.
Abstract: The aim of this paper is to present the proposal of organizational renewal process model. The paper starts with a description of the basis of organizational renewal, which later allows to present an original proposal of organizational renewal process model and its components.
TL;DR: An easily implementable stochastic model for planning the inventory of spare components needed for corrective replacements of system components is presented and results obtained show that even small changes of the level of spares in the existing inventory can result in considerable changes in the spare shortage probability.
Abstract: An easily implementable stochastic model for planning the inventory of spare components needed for corrective replacements of system components is presented. The probability of spare shortage in a given time interval is chosen as the decision criterion. The model is based on the assumption that the system contains a great number of identical independent components subject to wear-out. It can be used to determine the minimal number of spare components needed at the beginning of the planning interval to fulfil the requirement for an acceptable shortage probability during this interval. Besides, the model enables calculation of the probability of spare shortage during a given time interval considering the existing inventory level at the beginning of this interval. The model contains a few constant parameters that can be estimated from the component field data. It also includes two time dependent parameters which are calculated using the renewal function from the renewal theory. A discrete approximati...
TL;DR: In this article, a probabilistic model is presented to quantify parameters that define the exceedance rates of earthquake magnitudes, which can be transformed into the parameter estimation of single Poisson or renewal process.
Abstract: A probabilistic model is presented to quantify parameters that define the exceedance rates of earthquake magnitudes. Incompleteness of seismic catalogues and superposition of Poisson-renewal earthquake generation processes are both taken into account within a Bayesian framework. The formulation can be transformed into the parameter estimation of single Poisson or renewal process. The incomplete exceedance rate parameters are estimated from incomplete data, so that the estimated values are equal to those of the complete rate. Two cases are studied: the first one corresponds to a seismic source in the Gulf of Mexico and the other to a seismic source in the southern Pacific coast of Mexico.
TL;DR: The comparison with stochastic counterparts shows an interesting and reasonable homology in convergence mode and limit value between the results obtained in fuzzy renewal processes and the corresponding results in Stochastic renewal processes, though they build on two essentially different mathematical cornerstones, possibility theory and probability theory, respectively.
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TL;DR: A necessary and sufficient condition for an asymptotically stable renewal process to satisfy the strong renewal theorem was established in this article, which is valid for all alpha in (0, 1), thus completing a result for alpha in 1/2, 1) which was proved in the 1963 paper of Garsia and Lamperti.
Abstract: A necessary and sufficient condition is established for an asymptotically stable renewal process to satisfy the strong renewal theorem. This result is valid for all alpha in (0, 1), thus completing a result for alpha in (1/2, 1) which was proved in the 1963 paper of Garsia and Lamperti [6]. This paper is superseded by arXiv:1612.07635.
TL;DR: In this paper, a renewal-reward process with multivariate rewards is considered, where rewards in each time period may depend on each other and on the period length, but not on the other time periods.
TL;DR: In this paper, a theory of operator renewal sequences in the context of infinite ergodic theory was developed for dynamical systems preserving an infinite measure, and the asymptotic behavior of iterates Ln of the transfer operator, yielding results on mixing and rates of mixing.
Abstract: We correct the statement and proof of Theorem 10.4 in our paper “Operator renewal theory and mixing rates for dynamical systems with infinite measure” [Inventiones Mathematicae 189 (2012) 61–110]. The main results in the paper are unaffected. In the original article, we developed a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates Ln of the transfer operator, yielding results on mixing and rates of mixing. The main results, presented in Section 2 of the original article, focus on the restricted transfer operator 1Y L1Y where Y is a suitable finite measure subset of the full infinite measure system, and accordingly yield mixing results for observables supported on Y . Extensions to more general observables were described in specific cases in Section 1 based on results described in Sections 10 and 11.
TL;DR: In this paper, the authors use point processes theory to describe the asymptotic distribution of all upper order statistics for observations collected at renewal times and obtain limiting theorems for corresponding extremal processes.
Journal Article•
TL;DR: In this article, a parametric estimator for the value of weibull renewal function is proposed based on the maximum likelihood estimators of the unknown parameters of Weibull distribution.
Abstract: In this study, the weibull renewal process in which interrenewal times are weibull distributed is considered. In case of random right censored sample, a parametric estimator for the value of weibull renewal function is proposed based on the maximum likelihood estimators of the unknown parameters of weibull distribution. Key Words: Weibull distribution, Renewal process, renewal function, censoring
TL;DR: In this paper, the authors considered the asymptotic behavior of the quantity E ( b Θ( t ) 1 Θ ( t ) > a t ), where a and b are suitable positive constants, and Θ t is an inhomogeneous renewal process generated by the sequence θ 1, θ 2, ….
01 Jan 2015
TL;DR: In this paper, a stochastic model is constructed and the variance of the time to recruitment is obtained when the inter-policy decision times form a geometric process and inter-exit times form an ordinary renewal process.
Abstract: In this paper, the problem of time to recruitment is studied using a univariate policy of recruitment involving two thresholds for a single grade manpower system with attrition generated by its policy decisions. Assuming that the policy decisions and exits occur at different epochs, a stochastic model is constructed and the variance of the time to recruitment is obtained when the inter-policy decision times form a geometric process and inter- exit times form an ordinary renewal process. The analytical results are numerically illustrated with relevant findings by assuming specific distributions.
TL;DR: This paper introduces a new approach, whereby an actual discrete time system (ADTS) is approximated using a virtual continuous timesystem (VCTS) and obtains an asymptotically delay optimal policy in the ADTS, which has both a priority feature and a safety stock feature.
Abstract: Delay optimal control of multi-hop networks remains a challenging problem even in the simplest scenarios. In this paper, we consider delay optimal control of a two-hop half-duplex network with independent identically distributed ON-OFF fading. Both the source node and the relay node are equipped with infinite buffers and have exogenous bit arrivals. We focus on delay optimal link selection to minimize the average sum queue length over a finite horizon subject to a half-duplex constraint. To solve the problem, we introduce a new approach, whereby an actual discrete time system (ADTS) is approximated using a virtual continuous time system (VCTS). We obtain an asymptotically delay optimal policy in the VCTS. Using the relationship between the VCTS and the ADTS, we obtain an asymptotically delay optimal policy in the ADTS. The obtained policy has both a priority feature and a safety stock feature. It offers good design insights for wireless relay networks. In addition, the obtained policy has a closed-form expression, does not require knowledge of arrival statistics, and can be implemented online. Finally, using renewal theory and the theory of random walks, we analyze the average delay resulting from the asymptotically delay optimal policy.
TL;DR: In this paper, the authors define the non-homogeneous time convolutions and try to give order to the nonhomogeneous renewal processes, and a real data application to an aspect of motor car insurance is proposed.
Abstract: Non-homogeneous renewal processes are not yet well established. One of the tools necessary for studying these processes is the non-homogeneous time convolution. Renewal theory has great relevance in general in economics and in particular in actuarial science, however, most actuarial problems are connected with the age of the insured person. The introduction of non-homogeneity in the renewal processes brings actuarial applications closer to the real world. This paper will define the non-homogeneous time convolutions and try to give order to the non-homogeneous renewal processes. The numerical aspects of these processes are then dealt with and a real data application to an aspect of motorcar insurance is proposed.