Showing papers on "Renewal theory published in 2010"
TL;DR: This paper applies Markov renewal processes for both mobility modeling and predicting the likelihoods of the next-cell transition, along with anticipating the duration between the transitions, for an arbitrary user in a wireless network and shows how the proposed prediction method can lead to a significant improvement over some of the conventional methods.
Abstract: An understanding of the network traffic behavior is essential in the evolution of today's wireless networks and thus leads to a more efficient planning and management of the network's scarce bandwidth resources. Prior reservation of radio resources at future locations of a user's mobile trajectory can assist in optimizing the allocation of the network's limited resources and sustaining a desirable quality-of-service (QoS) level. This can also help to ensure that the network service can be available anywhere and anytime, which is only possible if, at any time, we can predict from where a user is going to make its demands. In this paper, we apply Markov renewal processes for both mobility modeling and predicting the likelihoods of the next-cell transition, along with anticipating the duration between the transitions, for an arbitrary user in a wireless network. Our proposed prediction technique will also be extended to compute the likelihoods of a user being in a particular state after N transitions. The proposed technique can also be used to estimate the expected spatial-temporal traffic load and activity at each location in a network's coverage area. Using some real traffic data, we illustrate how our proposed prediction method can lead to a significant improvement over some of the conventional methods.
84 citations
TL;DR: This work directly lower bound the information capacity for channels with independent identically distributed (i.i.d.) deletions and duplications using ideas from renewal theory, rather than focusing on the transmission capacity by analyzing the error probability of some randomly generated code using a combinatorial argument.
Abstract: In this paper, we directly lower bound the information capacity for channels with independent identically distributed (i.i.d.) deletions and duplications. Our approach differs from previous work in that we focus on the information capacity using ideas from renewal theory, rather than focusing on the transmission capacity by analyzing the error probability of some randomly generated code using a combinatorial argument. Of course, the transmission and information capacities are equal, but our change of perspective allows for a much simpler analysis that gives more general theoretical results. We then apply these results to the binary deletion channel to improve existing lower bounds on its capacity.
66 citations
TL;DR: In this article, a theory of operator renewal sequences in the context of infinite ergodic theory was developed, and the asymptotic behaviour of iterates of the transfer operator was determined for large classes of dynamical systems.
Abstract: We develop 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 $L^n$ of the transfer operator. This was previously an intractable problem.
Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points.
In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of $\sum_{j=1}^nL^j$) for the class of systems under consideration.
In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for $L^n$ and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.
55 citations
01 Nov 2010
TL;DR: Warranty cost models are presented based on the quasi-renewal processes and exponential distribution, including repairable products with a given warranty period considering conditional probabilities and renewal theory.
Abstract: In this paper, warranty cost models are presented based on the quasi-renewal processes and exponential distribution. Cost analyses are conducted for various systems under the basic assumption that a repair service is imperfect. We develop warranty cost models, reliability, and other measures for several systems, including multicomponent systems. This paper focuses on warranty cost analysis, including repairable products with a given warranty period considering conditional probabilities and renewal theory. The exponential distribution is used to analyze and obtain the warranty cost. Numerical examples are discussed to demonstrate the applicability of the methodology derived in this paper.
51 citations
TL;DR: In this article, the authors improved the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process, assuming certain regularity conditions.
Abstract: Leveille & Garrido (2001a, 2001b) have obtained recursive formulas for the moments of compound renewal sums with discounted claims, which incorporate both, Andersen's (1957) generalization of the classical risk model, where the claim number process is an ordinary renewal process, and Taylor's (1979), where the joint effect of the claims cost inflation and investment income on a compound Poisson risk process is considered. In this paper, assuming certain regularity conditions, we improve the preceding results by examining more deeply the asymptotic and finite time moment generating functions of the discounted aggregate claims process. Examples are given for claim inter-arrival times and claim severity following phase-type distributions, such as the Erlang case.
46 citations
TL;DR: This work characterize mean time to failure and system availability under the maximal repair policy where the whole system is replaced by a brand new after successfully completing a phase before the next phase starts.
46 citations
15 May 2010
TL;DR: In this paper, a point process describing the occurrence of the claims, a model for the cost of claims, and a premium are discussed, and examples of Cox processes which have been used in risk models are given.
Abstract: An insurance risk model consists of three parts: a point process describing the occurrence of the claims, a model for the cost of the claims, and a premium. Here Poisson processes, renewal processes, and Cox processes are discussed. Examples of Cox processes which have been used in risk models are given. Classes of distributions for the cost of the claims mentioned are exponentially bounded distributions or light-tailed distributions, subexponential distributions or heavy-tailed distributions, and an intermediate case.
Keywords:
poisson process;
renewal process;
cox process;
exponentially bounded distributions;
light tails;
subexponential distributions;
heavy tails
44 citations
TL;DR: In this article, the authors considered a many-server queueing system where customers arrive according to a renewal process and have service and patience times that are drawn from two independent sequences of independent, identically distributed random variables.
Abstract: A many-server queueing system is considered in which customers arrive according to a renewal process and have service and patience times that are drawn from two independent sequences of independent, identically distributed random variables. Customers enter service in the order of arrival and are assumed to abandon the queue if the waiting time in queue exceeds the patience time. The state of the system with $N$ servers is represented by a four-component process that consists of the forward recurrence time of the arrival process, a pair of measure-valued processes, one that keeps track of the waiting times of customers in queue and the other that keeps track of the amounts of time customers present in the system have been in service and a real-valued process that represents the total number of customers in the system. Under general assumptions, it is shown that the state process is a Feller process, admits a stationary distribution and is ergodic. It is also shown that the associated sequence of scaled stationary distributions is tight, and that any subsequence converges to an invariant state for the fluid limit. In particular, this implies that when the associated fluid limit has a unique invariant state, then the sequence of stationary distributions converges, as $N\rightarrow \infty$, to the invariant state. In addition, a simple example is given to illustrate that, both in the presence and absence of abandonments, the $N\rightarrow \infty$ and $t\rightarrow \infty$ limits cannot always be interchanged.
42 citations
TL;DR: A rigorous probabilistic framework to analytically characterize the service reliability of a DCS in the presence of communication uncertainties and stochastic topological changes due to node deletions is presented.
Abstract: In distributed computing systems (DCSs) where server nodes can fail permanently with nonzero probability, the system performance can be assessed by means of the service reliability, defined as the probability of serving all the tasks queued in the DCS before all the nodes fail. This paper presents a rigorous probabilistic framework to analytically characterize the service reliability of a DCS in the presence of communication uncertainties and stochastic topological changes due to node deletions. The framework considers a system composed of heterogeneous nodes with stochastic service and failure times and a communication network imposing random tangible delays. The framework also permits arbitrarily specified, distributed load-balancing actions to be taken by the individual nodes in order to improve the service reliability. The presented analysis is based upon a novel use of the concept of stochastic regeneration, which is exploited to derive a system of difference-differential equations characterizing the service reliability. The theory is further utilized to optimize certain load-balancing policies for maximal service reliability; the optimization is carried out by means of an algorithm that scales linearly with the number of nodes in the system. The analytical model is validated using both Monte Carlo simulations and experimental data collected from a DCS testbed.
36 citations
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TL;DR: In this article, the authors focus on the number of boxes occupied by at least one of the balls in the Bernoulli sieve and derive a variety of limiting distributions from the properties of associated perturbed random walks.
Abstract: The Bernoulli sieve is a version of the classical `balls-in-boxes' occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative renewal process, also known as the residual allocation model or stick-breaking. We focus on the number $K_n$ of boxes occupied by at least one of $n$ balls, as $n\to\infty$. A variety of limiting distributions for $K_n$ is derived from the properties of associated perturbed random walks. Refining the approach based on the standard renewal theory we remove a moment constraint to cover the cases left open in previous studies.
32 citations
TL;DR: This paper provides a compound variation of the Newsboy problem, instead of demand simply being known as to its distribution, here demand is generated by customers arriving at different points in time requiring amounts of varying size.
TL;DR: In this article, a new formulation of Stein's method is developed to obtain computable upper bounds on the total variation distance between the geometric distribution and a distribution of interest, which is based on the construction of a coupling between the original distribution and the discrete equilibrium distribution from renewal theory.
Abstract: We develop a new formulation of Stein's method to obtain computable upper bounds on the total variation distance between the geometric distribution and a distribution of interest. Our framework reduces the problem to the construction of a coupling between the original distribution and the "discrete equilibrium" distribution from renewal theory. We illustrate the approach in four non-trivial examples: the geometric sum of independent, non-negative, integer-valued random variables having common mean, the generation size of the critical Galton-Watson process conditioned on non-extinction, the in-degree of a randomly chosen node in the uniform attachment random graph model and the total degree of both a fixed and randomly chosen node in the preferential attachment random graph model.
TL;DR: The theory enables the effect of refractoriness on the time-dependent rate of an ensemble of encoding point processes in response to modulation of the input to be uncovered and represents a widely applicable framework to define and analyze nonstationary renewal processes.
Abstract: Stochastic point processes with refractoriness appear frequently in the quantitative analysis of physical and biological systems, such as the generation of action potentials by nerve cells, the release and reuptake of vesicles at a synapse, and the counting of particles by detector devices. Here we present an extension of renewal theory to describe ensembles of point processes with time varying input. This is made possible by a representation in terms of occupation numbers of two states: active and refractory. The dynamics of these occupation numbers follows a distributed delay differential equation. In particular, our theory enables us to uncover the effect of refractoriness on the time-dependent rate of an ensemble of encoding point processes in response to modulation of the input. We present exact solutions that demonstrate generic features, such as stochastic transients and oscillations in the step response as well as resonances, phase jumps and frequency doubling in the transfer of periodic signals. We show that a large class of renewal processes can indeed be regarded as special cases of the model we analyze. Hence our approach represents a widely applicable framework to define and analyze nonstationary renewal processes.
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TL;DR: In this article, Steutel et al. discuss the relation of size bias to the waiting time paradox, renewal theory, sampling, tightness and uniform integrability, compound Poisson distributions, infinite divisibility, and the lognormal distributions.
Abstract: With $X^*$ denoting a random variable with the $X$-size bias distribution, what are all distributions for $X$ such that it is possible to have $X^*=X+Y$, $Y\geq 0$, with $X$ and $Y$ {\em independent}? We give the answer, due to Steutel \cite{steutel}, and also discuss the relations of size biasing to the waiting time paradox, renewal theory, sampling, tightness and uniform integrability, compound Poisson distributions, infinite divisibility, and the lognormal distributions.
01 Dec 2010
TL;DR: Novel results regarding the impact of each mobility model on the traffic nature are found: random walk, random waypoint, and discrete Brownian motion can only induce short range dependent traffic, whose autocorrelation function decays exponentially fast.
Abstract: Recently, there has been a great deal of research on investigating the effects of mobility on network attributes such as capacity, connectivity, and coverage. In this paper, the node mobility is studied from a new perspective with an objective to reveal the inherent impact of different mobility models on the the traffic patterns in wireless sensor networks. Specifically, the transmission pattern of a mobile sensor node is first characterized by an alternating renewal process that changes states between the active and the inactive. Then, the active state distribution is investigated under four commonly used mobility models: random walk, random waypoint, discrete Brownian motion, and extended Levy walk. For each mobility model, the spectrum of the traffic oriented from a single node is analyzed based on renewal theory. According to this analysis, novel results regarding the impact of each mobility model on the traffic nature are found: random walk, random waypoint, and discrete Brownian motion can only induce short range dependent traffic, whose autocorrelation function decays exponentially fast. In contrast, the traffic under extended Levy walk exhibits pseudo long range dependence, in which the autocorrelation function decays slower than exponential and follows a power law form at large time lags. Finally, the revealed findings are verified by the statistical analysis on the collected traffic traces from the simulated transmissions.
01 Jan 2010
TL;DR: In this paper, the authors focus on the number of boxes occupied by at least one of n balls, and derive a variety of limiting distributions from the properties of associated perturbed random walks.
Abstract: The Bernoulli sieve is a version of the classical `balls-in-boxes' occupancy scheme, in which random frequencies of in¯nitely many boxes are produced by a multiplicative renewal process, also known as the residual allocation model or stick-breaking. We focus on the number Kn of boxes occupied by at least one of n balls, as n ! 1. A variety of limiting distributions for Kn is derived from the properties of associated perturbed random walks. Re¯ning the approach based on the standard renewal theory we remove a moment constraint to cover the cases left open in previous studies.
TL;DR: In this article, the renewal sequence version of the model that seemingly generates a new class of stationary binomial series is shown to obey an AR(1) recursion in cases where the renewal lifetime has a constant hazard rate past lag one.
TL;DR: In this article, the quenched LDP is applied to the collision local time of transient random walks, and the existence of an intermediate phase for a class of interacting stochastic systems is established.
Abstract: When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. In a companion paper the annealed and the quenched LDP are applied to the collision local time of transient random walks, and the existence of an intermediate phase for a class of interacting stochastic systems is established.
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TL;DR: In this article, it was shown that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional poisson process with Mittag-Leffler waiting times, which unifies the two main approaches in the stochastic theory of time-fractional diffusion equations.
Abstract: The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also establishes an interesting connection between the fractional Poisson process and Brownian time.
TL;DR: In this article, the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit in the Galton-Watson process, which is a generalization of a result given in the classical treatise by Asmussen and Hering (1983).
Abstract: We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. We include in particular the case of Galton-Watson processes so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering (1983). As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be of interest in themselves and which correct, generalize, and simplify some earlier works.
TL;DR: The stochastic characteristics of a recently proposed greedy routing scheme is addressed, and asymptotic expressions for the hop length moments are obtained through fast Fourier transform convolutions.
Abstract: We address the stochastic characteristics of a recently proposed greedy routing scheme. The behavior of individual hop advancements is examined, and asymptotic expressions for the hop length moments are obtained. The change of the hop distribution as the sink distance is varied is quantified with a Kullback–Leibler analysis. We discuss the effects of the assumptions made, the inherent dependencies of the model, and the influence of a sleep scheme. We propose a renewal process model for multiple hop advancements and justify its suitability under our assumptions. We obtain the renewal process distributions via fast Fourier transform convolutions. We conclude by giving future research tasks and directions.
01 Jan 2010
TL;DR: In this paper, a statistical model is presented for decision making in repairing water pipes network which is considered as a "repairable" system which is under repeating failure modes from this, a practical model for anticipating the failure of the water pipes in repairable systems has been presented using the trend renewal process concept in this process, the statistical Power law has been used for projecting the failure rate to account for the effects of repairs and for different failure modes in estimation of failure intensity.
Abstract: In this paper, a statistical model is presented for decision making in repairing water pipes network The water distribution system has been considered as a "repairable" system which is under repeating failure modes From this, a practical model for anticipating the failure of the water pipes in repairable systems has been presented using the trend renewal process concept In this process, the statistical Power law has been used for projecting the failure rate to account for the effects of repairs and for different failure modes in estimation of failure intensity After finding the failures as a function of time, the reliability of the system efficiency is then estimated using survival analysis At the end, a sample pipes network has been modeled using presented statistical model and the values of failure intensities with respect to time and the curve for reliability function has been found
01 Jan 2010
TL;DR: In this article, the Fano factor of the spike count distribution depends on the coefficient of variation of the inter-spike interval (ISI) distribution, and the survival probability is derived.
Abstract: This chapter reviews the theory of stochastic point processes. For simple renewal processes, the relation between the stochastic intensity, the inter-spike interval (ISI) distribution, and the survival probability are derived. The moment and cumulant generating functions and the relation between the ISI distribution and the autocorrelation is investigated. We show how the Fano factor of the spike count distribution depends on the coefficient of variation of the ISI distribution. Next we investigate models of renewal processes with variable rates and CV2, which is often used to assess the variability of the spike train in this case and compare the latter to the CV. The second half of the chapter deals with stochastic point processes with correlations between the intervals. Several examples of such processes are shown, and the basic analytical techniques to deal with these processes are expounded. The effect of correlations in the ISIs on the Fano factor of the spike count and the CV2 are also explored.
TL;DR: It is found that interevent times are not independent and identically distributed random variables but that the conditional probability distribution functions in the tail obey the Weibull distribution.
Abstract: We construct a one-dimensional piecewise linear intermittent map from the interevent time distribution for a given renewal process. Then, we characterize intermittency by the asymptotic behavior near the indifferent fixed point in the piecewise linear intermittent map. Thus, we provide a framework to understand a unified characterization of intermittency and also present the Lyapunov exponent for renewal processes. This method is applied to the occurrence of earthquakes using the Japan Meteorological Agency and the National Earthquake Information Center catalog. By analyzing the return map of interevent times, we find that interevent times are not independent and identically distributed random variables but that the conditional probability distribution functions in the tail obey the Weibull distribution.
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02 Sep 2010
TL;DR: In this paper, the exponential distribution and Poisson process are used for lifetime distribution classes and multivariate Lifetime Distributions are used to model the relationship between families of lifetime distributions, as well as association and dependency.
Abstract: Preliminaries.- Exponential Distribution.- Poisson Process.- Parametric Families of Lifetime Distributions.- Lifetime Distribution Classes.- Multivariate Lifetime Distributions.- Association and Dependence.- Renewal Theory.- Risk Theory.- Asset Pricing Theory.- Credit Risk Modeling.
TL;DR: In this article, a generalized discounted penalty function is studied by using random walk techniques and the renewal theory, where the downward jumps represent the claims as usual and the upward jumps are also allowed to explain random gains.
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TL;DR: This note describes several situations where simple product limit estimators, though inefficient, may still be useful in survival analysis under particular sampling frames corresponding to how the renewal process is observed.
Abstract: Nonparametric estimation of the gap time distribution in a simple renewal process may be considered a problem in survival analysis under particular sampling frames corresponding to how the renewal process is observed. This note describes several such situations where simple product limit estimators, though inefficient, may still be useful.
TL;DR: In this article, the authors consider the problem of nonparametric estimation of the gap time distribution in a simple renewal process under particular sampling frames corresponding to how the renewal process is observed and describe several such situations where simple product limit estimators, though inefficient, may still be useful.
Abstract: Nonparametric estimation of the gap time distribution in a simple renewal process may be considered a problem in survival analysis under particular sampling frames corresponding to how the renewal process is observed. This note describes several such situations where simple product limit estimators, though inefficient, may still be useful.
TL;DR: In this paper, the process of deterioration of repairable systems with each repair is modeled using converging geometric-type processes and it is proved that the expectation of the number of repairs in each interval of time is infinite.
Abstract: The process of deterioration of repairable systems with each repair is modeled using converging geometric-type processes. It is proved that the expectation of the number of repairs in each interval of time is infinite. A new regularization procedure is suggested and the corresponding optimization problem is discussed.
TL;DR: In this paper, a generalized mixture of gamma distributions in terms of the confluent hypergeometric function, as the waiting time distribution, was obtained for renewal theory and various applications of the model to reliability.
Abstract: In the present paper, we study the properties of finite mixture of exponential model in the context of renewal theory. We obtain a generalized mixture of gamma distributions in terms of the confluent hypergeometric function, as the waiting time distribution. We present various applications of the model to reliability.