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Showing papers on "Renewal theory published in 1991"


Journal ArticleDOI
TL;DR: In this paper, an inventory problem where the supply is available only during an interval of (random) length X and the unavailability of supply lasts for a random duration Y is considered.
Abstract: This article deals with an inventory problem where the supply is available only during an interval of (random) length X. The unavailability of supply lasts for a random duration Y. Using concepts from renewal theory, we construct an objective function (average cost/time) in terms of the order-quantity decision variable Q. We develop the individual cost components as order, holding, and shortage costs after introducing two important random variables. Due to the complexity of the objective function when X and Y are general random variables, we discuss two special cases and provide numerical examples with sensitivity analysis on the cost and noncost parameters. The article concludes with a discussion of the comparison of the current model with random yield and random lead-time models. Suggestions for further research are also provided.

205 citations


Journal ArticleDOI
TL;DR: Two statistical models are considered: one is the standard variance component model adapted to censored data, and the other is a recent intensity based model with a random proportionality factor representing interindividual variation.
Abstract: For each of several individuals a sequence of repeated events, forming a renewal process, is observed up to some censoring time. The object is to estimate the average interevent time over the population of individuals as well as the variation of interevent times within and between individuals. Medical motivation comes from gastroenterology, and concerns the occurrence of certain cyclic movements in the small bowel during the fasting state. Two statistical models are considered: one is the standard variance component model adapted to censored data, and the other is a recent intensity based model with a random proportionality factor representing interindividual variation. These models are applied to the motility data, and their advantages are discussed. The intensity based model allows simple empirical Bayes estimation of the expected interevent times for an individual in the presence of censoring.

154 citations


Journal ArticleDOI
TL;DR: The main conclusion is that both the MMPP model and the fluid flow approximation can provide accurate loss predictions for parameter ranges of practical interest.
Abstract: Three different approximation techniques are examined. The performance models studied differ primarily in the manner in which the superposition of the voice sources (i.e., the arrival process) is modeled. The first approach models the superimposed voice sources as a renewal process, and performance calculations are based only on the first two moments of the renewal process. The second approach is based on modeling the superimposed voice sources as a Markov modulated Poisson process (MMPP). The choice of parameters for the MMPP attempts to capture aspects of the arrival process in a more intuitive manner than previously proposed approaches for determining the MMPP parameters and is shown to compute loss more accurately. Finally, a fluid flow approximation for computing packet loss is evaluated. For all three approaches, a unifying example, the case of multiplexing voice sources over a T1-rate link is considered. The main conclusion is that both the MMPP model and the fluid flow approximation can provide accurate loss predictions for parameter ranges of practical interest. >

132 citations


Journal ArticleDOI
TL;DR: In this article, the optimal block replacement policy can be described as a so-called one-opportunity-look-ahead policy, and an exact optimisation algorithm in case of K 2-distributed times between opportunities is presented.

78 citations


Journal ArticleDOI
TL;DR: In this article, the collision probability treatment for regions comprised of a uniform background medium with a random dispersion of small heterogeneities is analyzed using recently developed statistical techniques, by assuming that the chord length distributions in such regions follow a renewal process.

75 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation.
Abstract: This paper studies a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where the server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation, i.e. a repeated vacation strategy; (ii) a limited service discipline, where the server begins a vacation either if K customers have been served in the same busy period or if the system is empty and then a repeated vacation strategy is followed. The input process is a general Markovian arrival process introduced by Lucantoni, Meier-Hellstern and Neuts, which as special cases includes the Markov modulated Poisson process and the phase-type renewal process. The service times and vacation times each are generally distributed random variables. For both models, we obtain the queue length distribution at departures, at an arbitrary time instant and at arrival time. We also derive the loss probability of an arriving customer. We obtain formulae for the LST of the virtual waiting time distribution and for the LST of the waiting time distribution at arrival epochs.

44 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the stochastic ordering of random measures and point processes generated by a partial order for all bounded Borel subsets B of the state space and derived a condition that the former can be realized as a thinning of the latter.

37 citations


Journal ArticleDOI
TL;DR: In this article, the authors established strong consistency (i.e., almost sure convergence) of infinitesimal perturbation analysis (IPA) estimators of derivatives of steady-state means for a broad class of systems.
Abstract: We establish strong consistency (i.e., almost sure convergence) of infinitesimal perturbation analysis (IPA) estimators of derivatives of steady-state means for a broad class of systems. Our results substantially extend previously available results on steady-state derivative estimation via IPA.Our basic assumption is that the process under study is regenerative, but our analysis uses regenerative structure in an indirect way: IPA estimators are typically biased over regenerative cycles, so straightforward differentiation of the regenerative ratio formula does not necessarily yield a valid estimator of the derivative of a steady-state mean. Instead, we use regeneration to pass from unbiasedness over fixed, finite time horizons to convergence as the time horizon grows. This provides a systematic way of extending results on unbiasedness to strong consistency.Given that the underlying process regenerates, we provide conditions under which a certain augmented process is also regenerative. The augmented process includes additional information needed to evaluate derivatives; derivatives of time averages of the original process are time averages of the augmented process. Thus, through this augmentation we are able to apply standard renewal theory results to the convergence of derivatives.

35 citations


Journal ArticleDOI
TL;DR: In this article, the authors present the partial differential equations (PDEs) describing the transients of the probability density functions (PDFs) characterizing the statistical evolution of a manufacturing system producing a single product under hedging-point control policies.
Abstract: The authors present the partial differential equations (PDEs) describing the transients of the probability density functions (PDFs) characterizing the statistical evolution of a manufacturing system producing a single product under hedging-point control policies. The authors demonstrate the Markov renewal nature of the dynamics of the controlled process and use the system of PDEs to compute the transition kernel of that renewal process. This Markov renewal viewpoint is particularly useful in discussing ergodicity in view of the abundant literature on the asymptotic behavior of Markov renewal processes. Moreover, besides allowing direct determination of system steady state, when it exists, it permits the computation of various statistics, as well as, in some cases, the derivation of bounds on the speed of convergence to steady state. >

34 citations


Journal ArticleDOI
TL;DR: In this article, the authors extended the asymptotic results for ordinary renewal processes to the superposition of independent renewal processes and applied the key superposition renewal theorem to the study of renewal superpositions.
Abstract: This paper extends the asymptotic results for ordinary renewal processes to the superposition of independent renewal processes. In particular, the ordinary renewal functions, renewal equations, and the key renewal theorem are extended to the superposition of independent renewal processes. We fix the number of renewal processes, p, and study the asymptotic behavior of the superposition process when time, t, is large. The key superposition renewal theorem is applied to the study of

30 citations


Journal ArticleDOI
TL;DR: Two approaches to thin the arrival process of a stationary waiting time random variable W â‰i WS, T in a GI/G/1 queueing system with generic service and inter-arrival time random variables S and T respectively, with ES 0} and EW are studied in light traffic conditions.
Abstract: For a stationary waiting time random variable W â‰i WS, T in a GI/G/1 queueing system with generic service and inter-arrival time random variables S and T respectively, with ES 0} and EW are studied in light traffic conditions. One way of attaining these conditions, as considered in a previous paper, is to replace T by γT for large γ; another way is to thin the arrival process with small but positive retention probability π. These two approaches are compared, the thinning approach being applied to queues with either a renewal or a periodic Poisson arrival process. Results are also given for GI/M/k and GI/D/k queues. The variety of queueing systems studied is reflected in the different behaviour both of the quantities calculated directly and of the derived quantity EW | W > 0. The dominant feature of light traffic characteristics is their dependence on the clustering tendency and related properties of the arrival process.

Journal ArticleDOI
Attila Csenki1
TL;DR: In this paper, the authors derived the probability mass function and cumulative distribution function of the joint distribution of the first m sojourn times for absorbing Markov chains for a fault-tolerant multiprocessor system.

Journal ArticleDOI
TL;DR: In this article, an irreducible semi-Markov process is considered whose finite state space is partitioned into two non-empty sets A and B. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB(t) are easily expressible.
Abstract: In this note, an irreducible semi-Markov process Y = { Y,: t 0} is considered whose finite state space is partitioned into two non-empty sets A and B. Let MB(t) stand for the number of visits of Y to B during the time interval [0, t], t > 0. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB(t) are easily expressible. The theory is applied to a small reliability model in conjunction with a Tauberian argument to evaluate the behaviour of the first two moments of MB(t) as t -oc. RENEWAL THEORY; LAPLACE TRANSFORM; RELIABILITY; TAUBERIAN THEOREM

Journal ArticleDOI
TL;DR: Analytical results are obtained for the generating functions and binomial moments of both the continuous time system size and pre-arrival system size by investigating a general infinite server system with batch arrivals following a Markov renewal input process.
Abstract: An important property of most infinite server systems is that customers are independent of each other once they enter the system. Though this non-interacting property (NIP) has been instrumental in facilitating excellent results for infinite server systems in the past, the utility of this property has not been fully exploited or even fully recognized. This paper exploits theNIP by investigating a general infinite server system with batch arrivals following a Markov renewal input process. The batch sizes and service times depend on the customer types which are regulated by the Markov renewal process. By conditional approaches, analytical results are obtained for the generating functions and binomial moments of both the continuous time system size and pre-arrival system size. These results extend the previous results on infinite server queues significantly.

Journal ArticleDOI
TL;DR: In this article, it was shown that every irreducible shift of finite type is conjugate to a renewal system, and that every shift can be approximated from above by renewal systems by placing finite-type constraints on possible concatenations.
Abstract: Renewal systems are symbolic dynamical systems originally introduced by Adler IfW is a finite set of words over a finite alphabetA, then the renewal system generated byW is the subshiftX W ⊂A Z formed by bi-infinite concatenations of words fromW Motivated by Adler’s question of whether every irreducible shift of finite type is conjugate to a renewal system, we prove that for every shift of finite type there is a renewal system having the same entropy We also show that every shift of finite type can be approximated from above by renewal systems, and that by placing finite-type constraints on possible concatenations, we obtain all sofic systems

Book ChapterDOI
01 Jan 1991
TL;DR: A temporal point process must be both orderly and without aftereffects to be a Poisson process and this chapter discusses self-exciting point processes.
Abstract: A temporal point process must be both orderly and without aftereffects to be a Poisson process. The orderliness restriction that points be isolated from one another is relaxed in Ch. 4 where a generalized Poisson process is defined and studied. The restriction that the process be without aftereffects is removed in this chapter about self-exciting point processes.

01 Jan 1991
TL;DR: In this article, the authors studied a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation.
Abstract: This paper studies a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where the server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation, i.e. a repeated vacation strategy; (ii) a limited service discipline, where the server begins a vacation either if K customers have been served in the same busy period or if the system is empty and then a repeated vacation strategy is followed. The input process is a general Markovian arrival process introduced by Lucantoni, Meier-Hellstern and Neuts, which as special cases includes the Markov modulated Poisson process and the phase-type renewal process. The service times and vacation times each are generally distributed random variables. For both models, we obtain the queue length distribution at departures, at an arbitrary time instant and at arrival time. We also derive the loss probability of an arriving customer. We obtain formulae for the LST of the virtual waiting time distribution and for the LST of the waiting time distribution at arrival

Journal ArticleDOI
J.M. Dickey1
TL;DR: In this article, a series expansion for the n th order distribution functions of the Weibull distribution and the coefficients are found by solving recurrence equations are developed for the availability and the renewal function of an alternating renewal process.

Journal ArticleDOI
TL;DR: In this article, it was shown that in a superposition of finitely many independent renewal processes, an observation from the limiting (when t → ∞) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1 − U)W), where U and Ware independent random variables with U ~ uniform(0, 1) and W distributed according to the limiting total life distribution of the superposition process.
Abstract: It is shown that, in a superposition of finitely many independent renewal processes, an observation from the limiting (when t →∞) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1 – U)W), where U and Ware independent random variables with U ~ uniform(0, 1) and W distributed according to the limiting total life distribution of the superposition process.

Journal ArticleDOI
TL;DR: In this paper, boundary crossing behavior of conditional random walks is studied and asymptotic distributions of the exit time and the excess over the boundary are derived for modified repeated significance tests and change-point problems.
Abstract: Herein boundary crossing behavior of conditional random walks is studied. Asymptotic distributions of the exit time and the excess over the boundary are derived. In the course of derivation, two results of independent interest are also obtained: Lemma 4.1 shows that a conditional random walk behaves like an unconditional one locally in a very strong sense. Theorem B.1 describes a class of distributions over which the renewal theorem holds uniformly. Applications are given for modified repeated significance tests and change-point problems.


Journal ArticleDOI
TL;DR: A novel algorithm for discrete tracking of slow frequency hopping spread-spectrum signals is described and analyzed, which allows a large degree of flexibility with respect to trading off system performance, complexity, and the redundancy introduced for tracking purposes.
Abstract: For pt.I see ibid., vol.39, no.2, p.304 (1991). A novel algorithm for discrete tracking of slow frequency hopping spread-spectrum signals is described and analyzed. In this system, each Mth hopping interval is completely used for transmission of synchronization data. The motivation for introducing this algorithm is that it allows a large degree of flexibility with respect to trading off system performance, complexity, and the redundancy introduced for tracking purposes. The analysis of the system performance is based on the utilization of results from discrete renewal process theory. However, because of the difficulty in obtaining the exact values for the transition probabilities, approximate results are presented. >

Journal ArticleDOI
TL;DR: In this article, two methods of estimating the time between transmission overhauls and the number of replacement components needed were presented, assuming the transmission components follow a two-parameter Weibull failure distribution.
Abstract: Two methods of estimating the time between transmission overhauls and the number of replacement components needed were presented. The first method assumes replacement of all components during an overhaul of a failed transmission (full replacement method). The second method assumes replacement of failed components only (partial replacement method). Both methods assume the transmission components follow a two-parameter Weibull failure distribution. Renewal theory was presented to estimate the number of component replacements in a transmission for both methods. For the partial replacement method, renewal theory was used with the individual component life predictions to estimate the number of component replacements needed and the transmission time between overhauls. For the full replacement method, renewal theory was used with a transmission system life model to estimate the number of replacement transmissions needed and the transmission time between overhauls. Confidence statistics were applied to both methods to improve the statistical estimate of sample behavior. A transmission example was presented to illustrate use of both methods.

Journal ArticleDOI
TL;DR: In this article, a critical age-dependent branching process with state-dependent immigration at 0 is denoted Z(t), at IID times of an independent renewal process, IID {Zi(t)} processes are introduced.
Abstract: A critical age-dependent (Bellman-Harris) branching process with state-dep endent immigration at 0 is denoted Z(t). At IID times of an independent renewal process, IID {Zi(t)} processes are introduced. Asymptotic moments, and a limit th eorem for the overall process are given.

Book ChapterDOI
01 Jan 1991
TL;DR: This contribution introduces briefly regenerative stochastic processes and uses them to solve some reliability problems encountered in pratical applications.
Abstract: Stochastic processes are powerful tools for the investigation of the reliability and availability of repairable equipment and systems. Because of the involved models and in order to be mathematically tractable, these processes are generally confined to the class of regenerative stochastic processes with a finite state space, to which belong renewal processes, Markov processes, semi-Markov processes, and more general regenerative processes with only few (in the limit case only one) regeneration states. This contribution introduce briefly these processes and uses them to solve some reliability problems encountered in pratical applications. Investigations deal with different kinds of reliabilities and availabilities for one item, series, parallel, and series/ parallel structures. For series/parallel structures useful approximate expressions are developed.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the iterates of a Harris recurrent Markov operator can be represented as a (delayed) renewal sequence, provided only that certain filling schemes are successful.

Proceedings ArticleDOI
26 Jun 1991
TL;DR: A nonlinear estimation scheme for estimating a noisy alternating renewal process signal transmitted through an analog channel and results of computer simulation illustrating some potential applications are presented.
Abstract: A nonlinear estimation scheme for estimating a noisy alternating renewal process signal transmitted through an analog channel is considered. The derivation for a nonlinear smoother and results of computer simulation illustrating some potential applications are presented.

Journal ArticleDOI
TL;DR: In this article, the authors consider random walks Sℕ, adapted to a filtration, and derive their basic properties and give equivalent characterizations in terms of certain drift constants which are introduced before and of great importance for renewal theoretic analysis.
Abstract: We consider random walks Sℕ, adapted to a filtration % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $\cal F_N$ , whose conditional increment distribution functions are bounded from above and/or below by an integrable distribution function. A further stability condition on the conditional increment means is also introduced. Such random walks share a number of properties with those having i.i.d. increments, in particular a uniform law of large numbers. In this paper, which is accompanied by a second one on renewal theory, we derive their basic properties and give equivalent characterizations in terms of certain drift constants which are introduced before and of great importance for a renewal theoretic analysis.

Journal ArticleDOI
TL;DR: Several easy-to-compute optimal policy approximations based on the asymptotic renewal theory are evaluated for a wide range of parameter settings for gamma, log-normal, truncated normal, Weibull and inverse Gaussian batch size distributions.

Journal ArticleDOI
TL;DR: Finiteness of the moments of first-crossing times related to nonlinear renewal theory and widely used in sequential analysis is discussed in this article, where the renewal theory is applied to sequential analysis.
Abstract: Finiteness of the moments of first–crossing times related to nonlinear renewal theory and widely used in sequential analysis, is discussed