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Showing papers on "Renewal theory published in 1998"


Journal ArticleDOI
TL;DR: In this article, the authors show how methods that have been applied to derive results for the classical risk process can be adapted for a class of risk processes in which claims occur as a renewal process.
Abstract: In this paper I show how methods that have been applied to derive results for the classical risk process can be adapted to derive results for a class of risk processes in which claims occur as a renewal process. In particular, claims occur as an Erlang process. I consider the problem of finding the survival probability for such risk processes and then derive expressions for the probability and severity of ruin and for the probability of absorption by an upper barrier. Finally, I apply these results to consider the problem of finding the distribution of the maximum deficit during the period from ruin to recovery to surplus level 0.

82 citations


Journal ArticleDOI
TL;DR: In this paper, the authors show that when the information in the chronological order of interarrival times is ignored, they often appear spuriously exponential, leading to the impression that the system can be modeled using a homogeneous Poisson process.
Abstract: Failure data for a repairable system can be represented either by a set of chronologically ordered arrival times at which the system failed, or by a set of interarrival times defined as the times observed between successive failures (ignoring repair times in both cases). The two representations are mathematically equivalent if the chronological order of the interarrival times is maintained. Methods aimed at describing the distribution of the observed interarrival times are meaningful only if the interarrival times are identically distributed. In all other cases, such analyses are meaningless and often result in maximally misleading impressions about the system behavior, as demonstrated here by several examples. That is, when the information in the chronological order of interarrival times is ignored, they often appear spuriously exponential, leading to the impression that the system can be modeled using a homogeneous Poisson process. Misunderstandings of this nature can be avoided by applying an appropriate test for trend before attempting to fit a distribution to the interarrival times. If evidence of trend is determined, then a nonstationary model such as the nonhomogeneous Poisson process should be fitted using the chronologically ordered data.

54 citations


Journal ArticleDOI
TL;DR: In this paper, Arc-sine laws in the sense of renewal theory are proved for return time processes generated by transformations with infinite invariant measure on sets satisfying a type of Darling-Kac condition.
Abstract: Arc-sine laws in the sense of renewal theory are proved for return time processes generated by transformations with infinite invariant measure on sets satisfying a type of Darling-Kac condition, and an application to real transformations with indifferent fixed points is discussed.

50 citations


Journal ArticleDOI
TL;DR: This paper models a single-stage, single-product, stochastic assembly system, operating according to an Materials Requirements Planning controlled (MRP) ordering philosophy, enabling production lead times to be treated as independent and generally distributed random variables.

33 citations


Journal ArticleDOI
A. Garg1, J.R. Kalagnanam1
TL;DR: An accurate, computable approximation for evaluating the renewal function (RF) that yields a set of approximants to the RF that are re-usable, and can be used to compute the derivative and the integral of the RF.
Abstract: This paper describes an accurate, computable approximation for evaluating the renewal function (RF). The method uses Pade approximants to compute the RF near the origin and switches to the asymptotic values farther from the origin. There is a polynomial switch-over function in terms of the coefficient of variation of the distribution, enabling one to determine a priori if the asymptotic value can be used instead of computing the Pade approximant. The results are tested with the truncated Gaussian distribution. The method yields a set of approximants to the RF that are re-usable, and can be used to compute the derivative and the integral of the RF. Results for the RF are within 1% of the optimal solution for most coefficients of variation.

33 citations


Proceedings ArticleDOI
07 Jun 1998
TL;DR: An efficient and easy technique to generate fractional Gaussian noise traffic based on the spatial renewal process is developed and demonstrated, and the synthetically generated trace reproduces the desired marginal, autocorrelation and Hurst parameters well.
Abstract: An efficient and easy technique to generate fractional Gaussian noise traffic based on the spatial renewal process is developed and demonstrated. The synthetically generated trace reproduces the desired marginal, autocorrelation and Hurst parameters well. The model is particularly suitable for use in the discrete-event simulation of queueing systems involving VBR compressed video and aggregated LAN traffic.

31 citations


Journal ArticleDOI
TL;DR: It is shown that the performability of multicomponent systems that do not satisfy these rules can be bounded by tractable modifications.
Abstract: We consider a class of models for multicomponent systems in which components can break down and be repaired in a dependent manner and where breakdown and repair times can be arbitrarily distributed. The problem of calculating the equilibrium distribution and, from this, the expected performability for these models is intractable unless certain assumptions are made about breakdowns and repairs. In this paper we show that the performability of multicomponent systems that do not satisfy these rules can be bounded by tractable modifications. Our results are proved by stochastic comparability arguments and a Markov reward technique, which is of interest in itself as it enables one to prove that the equilibrium distribution of one process can be bounded by that of another even when the sample paths of the process are not. This is illustrated by an example.

24 citations


Journal ArticleDOI
TL;DR: In this paper, the authors derived the two-dimensional transforms of the emptiness function, the transient workload and queue-length distributions in the single-server queue with general service times and a batch Markovian arrival process (BMAP).
Abstract: Previously, we derived the two-dimensional transforms of the emptiness function, the transient workload and queue-length distributions in the single-server queue with general service times and a batch Markovian arrival process (BMAP). This arrival process includes the familiar phase-type renewal process and the Markov modulated Poisson process as special cases, as well as superpositions of these processes, and allows correlated interarrival times and batch sizes We continue the transient analysis of this model in this paper by deriving explicit expressions for the transforms of the queue length at the n-th departure (assuming a departure at time t = 0), and the delay of the n-th arrival (keeping track of the appropriate phase changes). Also, the departure process is characterized by the double transform of the probability that the n -th departure occurs at time less than or equal to time x

23 citations



Journal Article
TL;DR: In this paper, the authors show how methods that have been applied to derive results for the classical risk process can be adapted for a class of risk processes in which claims occur as a renewal process.
Abstract: In this paper I show how methods that have been applied to derive results for the classical risk process can be adapted to derive results for a class of risk processes in which claims occur as a renewal process. In particular, claims occur as an Erlang process. I consider the problem of finding the survival probability for such risk processes and then derive expressions for the probability and severity of ruin and for the probability of absorption by an upper barrier. Finally, I apply these results to consider the problem of finding the distribution of the maximum deficit during the period from ruin to recovery to surplus level 0.

17 citations


Journal ArticleDOI
01 Sep 1998
TL;DR: A system of N statistically identical machines is modeled using renewal theory through a unique architecture and state space, which is very useful in evaluating flexible manufacturing systems in particular.
Abstract: A system of N statistically identical machines is modeled using renewal theory through a unique architecture and state space. The model is very useful in evaluating flexible manufacturing systems (FMS) in particular. Each machine consists of multiple part types that are subject to individual failure. A multiple stage repair process composed of integrable sojourn time distributions requires access to spare parts to repair down machines. A performability measure is used to gauge system effectiveness for this partially degradable system, and an example is given.

Journal ArticleDOI
Tae-Jin Lim1
TL;DR: In this paper, a Markov switching model is proposed for analyzing the failure process of a repairable system, where a finite number of states governs the distinct lifetime distributions, and the state makes transitions according to a discrete-time Markov chain.

Journal ArticleDOI
TL;DR: In this article, approximation formulae for solutions of renewal-type equations are derived by finding the first and higher Frechet derivatives of the functional that has the underlying lifetime density as input and a normalised version of the solution of the renewal type equation as output.

Proceedings ArticleDOI
01 Dec 1998
TL;DR: The capacity of a bottleneck node is related to performance measures of interest for best-effort traffic, such as the mean file transfer time or probability of congestion, through simple models of congestion control in high-speed networks.
Abstract: We consider simple models of congestion control in high-speed networks and develop diffusion approximations which could be useful for resource allocation. We first show that, if the arrival process is Poisson and the service times are exponential, then, under a certain scaling, the steady-state distribution of the number of sources in the system consists of appropriately normalized and truncated Gaussian and exponential distributions. We then consider the case where the arrival process is a general renewal process with finite coefficient of variation and service-time distributions that are phase-type, and show the impact of these distributions on the steady-state distribution. We use these results to relate the capacity of a bottleneck node to performance measures of interest for best-effort traffic, such as the mean file transfer time or probability of congestion.

Journal ArticleDOI
TL;DR: An explicit formula is derived being a function of the Laplace transform of the renewal distribution evaluated at the discount rate, the probability for a low bid, and the ratio between the two bid-values for n, to maximize the total expected discounted revenue from the sale of the n assets.

Journal ArticleDOI
TL;DR: The renewal-theory paradox is ubiquitous in queueing theory, and it sometimes produces anomalous results as discussed by the authors, e.g., the length of the renewal interval that covers a randomly selected time epoch tends to be longer than an ordinary renewal interval.
Abstract: that the length of the renewal interval that covers a randomly-selected time epoch tends to be longer than an ordinary renewal interval. This paradox manifests itself in numerous interesting ways in queueing theory, a prime example being the celebrated Pollaczek-Khintchine formula for the mean waiting time in the M/G/1 queue. In this expository paper, we give intuitive arguments that "explain" why the renewal-theory paradox is ubiquitous in queueing theory, and why it sometimes produces anomalous results. In particular, we use these intuitive arguments to explain decomposition in vacation models, and to derive formulas that describe some recent

Journal ArticleDOI
TL;DR: A method based on the asymptotic results of a renewal process to obtain the equivalent MMPP(2) parameters is proposed, which is compared with those of Heffes and Lucantoni and with a Poisson approximation.
Abstract: This paper considers packetized voice and video traffic incoming to an ATM multiplexer. Each voice and video source is approximated by a two-state Markov–modulated Poisson process (MMPP(2)). We propose a method based on the asymptotic results of a renewal process to obtain the equivalent MMPP(2) parameters. The integrated voice and video sources are modelled by the superposition of two MMPP(2) processes, denoted MMPP(4). The analysis of the superposed traffic in terms of buffer occupancy, cell loss probability and delay is performed with an MMPP(4)/Er(k)/1/B queuing model considering different buffer sizes and numbers of sources. Results of the proposed method are also compared with those of Heffes and Lucantoni (IEEE J. Select. Areas Commun., SAC-4, 856–868 (1986)) and with a Poisson approximation. © 1998 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this article, the maximum of the difference of two renewal processes with discrete time is shown to be semicontinuous in discrete topology. But the distribution of the maximum is not known.
Abstract: We find the distribution of the maximum of the difference of two renewal processes with discrete time that is semicontinuous in discrete topology.

Journal ArticleDOI
TL;DR: In this paper, a multitype Markov branching process under the influence of disasters (catastrophes) arriving as a renewal process leading to possible mutation of particles to other types is considered.

Proceedings ArticleDOI
13 Oct 1998
TL;DR: This paper introduces models based on a 2D Poisson process for both intermittent and continuous motion feeding and shows how to optimize the throughput when there is a constraint on the expected number of times a part should go through the system.
Abstract: We study a programmable robotic part feeder that relies on a sequence of three conveyor belts to singulate and re-circulate parts. In industrial practice, belt speeds are set in an ad-hoc fashion. Experience with real feeders reveals that throughput can suffer due to (1) starvation where no parts are visible to the camera and (2) saturation, where too many parts are visible, which prevents identifying part pose or grasping due to obstruction by nearby parts. This motivates our search for a systematic approach to setting belt speeds. This paper introduces models based on a 2D Poisson process for both intermittent and continuous motion feeding. For intermittent motion feeding we apply renewal theory to approximate and optimize the theoretical throughput. For continuous motion feeding we use a M/G/1 queue with customer impatience to approximate and optimize the theoretical throughput. We show that the analytic theory compares very well with simulation studies. For both models we show how to optimize the throughput when there is a constraint on the expected number of times a part should go through the system.

Journal ArticleDOI
TL;DR: In this article, a neural model with N interacting neurons is considered, and the stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes.
Abstract: A neural model with N interacting neurons is considered. A firing of neuron i delays the firing times of all other neurons by the same random variable theta((i)), and in isolation the firings of the neuron occur according to a renewal process with generic interarrival time Y-(i). The stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes. Further, the distribution of the period of activity of a neuron is studied for the symmetric case where theta((i)) and Y-(i) do not depend upon i. The tools are probabilistic and involve path decompositions, Palm theory and random walks.

Journal ArticleDOI
01 Apr 1998
TL;DR: In this paper, the problem of sequential estimation for stochastic processes in the presence of a nuisance parameter is dealt with, using the approach to estimation through estimating equations, optimum estimating functions based on a random observation time.
Abstract: The paper deals with the problem of sequential estimation for stochastic processes in the presence of a nuisance parameter. Using the approach to estimation through estimating equations, optimum estimating functions based on a random observation time are investigated in some models for processes appearing in reliability systems theory.

Proceedings ArticleDOI
Serap A. Savari1
16 Aug 1998
TL;DR: It is shown that for this class of sources the redundancy of a Tunstall code with a small to moderate dictionary size is considerably smaller than suggested by an exact asymptotic analysis of the code.
Abstract: Renewal theory is a powerful tool in the asymptotic analysis of source codes. We investigate its limitations in answering fundamental questions about predictable, binary memoryless sources. We show that for this class of sources the redundancy of a Tunstall code with a small to moderate dictionary size is considerably smaller than suggested by an exact asymptotic analysis of the code.

Journal ArticleDOI
TL;DR: In this article, the authors considered a renewal process where the underlying distribution displays non-monotonic ageing property of a specific kind and derived an important bound for the expected number of renewals by time t.
Abstract: In this note, we consider a renewal process where the underlying distribution displays non-monotonic ageing property of a specific kind. An important bound for the expected number of renewals by time t has been obtained. A characterization of the exponential distribution has also been derived as a consequence.

Journal ArticleDOI
TL;DR: In this article, the authors present a numerical scheme employing the film-renewal theory for gas-liquid mass transfer, which is capable of calculating mass transfer rates, conversion and remaining gas holdup from system and operating parameters.

Posted Content
TL;DR: In this article, the problem of choosing the cutting speed of a flexible machine-tool to minimize the expected total processing time is addressed and a numerical procedure is provided and a comparison is made with the deterministic and exponential cases.
Abstract: We address the problem of choosing the cutting speed of a flexible machine-tool to minimize the expected total processing time. The tool life is a random variable with a known distribution. Tool failures cause a set-up time and thus increase processing time. We assume the coefficient of variation is given and the mean tool life is a function of the cutting speed governed by Taylor's formula. We analyse the problem using renewal theory and obtain an exact solution for the gamma distribution. A numerical procedure is provided and a comparison is made with the deterministic and exponential cases.

Journal ArticleDOI
TL;DR: On the basis of renewal theory, generalized master equations for transport processes are derived covering normal processes such as the neutron transport process with delaying neutrons as well as anomalous processes of different kinds, including Levy-flights and trappings as mentioned in this paper.
Abstract: On the basis of the renewal theory, generalized master equations for transport processes are derived covering normal processes such as the neutron transport process with delaying neutrons as well as anomalous processes of different kinds, including Levy-flights and trappings. The finite-speed effect on the time-asymptotic behavior of the random displacement of a particle is investigated.

Proceedings ArticleDOI
07 Sep 1998
TL;DR: This work presents a new method to compute bounds and an approximation on the average state sojourn times for a special class of deterministic and stochastic Petri nets (DSPNs) and makes use of concepts from renewal theory.
Abstract: Transient analysis of non-Markovian Stochastic Petri nets is a theoretically interesting and practically important problem. We present a new method to compute bounds and an approximation on the average state sojourn times for a special class of deterministic and stochastic Petri nets (DSPNs). In addition to the idea of the subordinated Markov chain traditionally used for the stationary solution of DSPNs, our algorithm makes use of concepts from renewal theory. An application to a finite-capacity queue with a server subject to breakdowns is included.

Book ChapterDOI
01 Jan 1998
TL;DR: In this paper, the authors examined generalizations in another direction, stemming from the observation in Chapter 2 that, for a Poisson process, conditional on the total number of points in a bounded region of time or space, the individual points can be treated as independently and identically distributed over the region.
Abstract: The Poisson process can be generalized in many directions. We have already discussed some consequences of relaxing the independency assumptions while retaining those of stationarity and orderliness of a point process on the line. In this chapter we examine generalizations in another direction, stemming from the observation in Chapter 2 that, for a Poisson process, conditional on the total number of points in a bounded region of time or space, the individual points can be treated as independently and identically distributed over the region. This prompts an alternative approach to specifying the structure of point processes in a bounded domain or, more generally, of any point process in which the total number of points is finite with probability 1. Such a process is called a finite point process.

Book ChapterDOI
01 Jan 1998
TL;DR: This chapter shows how renewal theory can be used to construct a theoretically valid discretization method for general Markov chains, and considers some convergence control methods based on these discretized chains, even though the chains can be use in many alternative ways.
Abstract: As discussed in Chapter 2, an important drawback of Raftery and Lewis’ (1992a, 1996) convergence control method is that the discretized version of the Markov chain is not a Markov chain itself, unless a stringent lumpability condition holds (see Kemeny and Snell, 1960). This somehow invalidates the binary control method, although it provides useful preliminary information on the required number of iterations. However, the discrete aspect of the criterion remains attractive for its intuitive flavour and, while the Duality Principle of Chapter 1 cannot be invoked in every setting, this chapter shows how renewal theory can be used to construct a theoretically valid discretization method for general Markov chains. We then consider some convergence control methods based on these discretized chains, even though the chains can be used in many alternative ways (see also Chapter 5).