Showing papers on "Renewal theory published in 1982"
TL;DR: The purpose here is to provide a better understanding of these procedures and a general framework for making new approximations, in particular, the two basic procedures can be used as building blocks to construct refined composite procedures.
Abstract: This paper initiates an investigation of simple approximations for stochastic point processes. The goal is to develop methods for approximately describing complex models such as networks of queues and multiechelon inventory systems. The proposed approach is to decouple or decompose the model by replacing all the component flows point processes by independent renewal processes. Here attention is focused on ways to approximate a single point process by a renewal process. This is done in two steps: First, properties of the point process are used to specify a few moments of the interval between renewals; then a convenient distribution is fit to these moments. Two different methods are suggested for specifying the moments of the renewal interval. The stationary-interval method equates the moments of the renewal interval with the moments of the stationary interval in the point process to be approximated. The asymptotic method, in an attempt to account for the dependence among successive intervals, determines the moments of the renewal interval by matching the asymptotic behavior of the moments of the sums of successive intervals. These two procedures are applied to approximate the superposition merging of point processes. The purpose here is to provide a better understanding of these procedures and a general framework for making new approximations. In particular, the two basic procedures can be used as building blocks to construct refined composite procedures. Composite procedures for the âGi/G/1 queue with a superposition arrival process are discussed by Albin in Part II. Albin has developed a hybrid procedure for approximating the mean sequence length and other characteristics in the âGi/G/1 queue for which the average error when compared with simulated values was 3% over a large number of test systems.
548 citations
TL;DR: In this paper, mathematical renewal theory is used to make a general model of the combined processes of search, encounter, capture, and handling, and a model based on minimization of the probability of death due to an energetic shortfall is presented.
Abstract: Some simple stochastic models of optimal foraging are considered. Firstly, mathematical renewal theory is used to make a general model of the combined processes of search, encounter, capture and handling. In the case where patches or prey items are encountered according to a Poisson process the limiting probability distribution of energy gain is found. This distribution is found to be normal and its mean and variance are specified. This result supports the use of Holling's disc equation to specify the rate of energy intake in foraging models. Secondly, a model based on minimization of the probability of death due to an energetic shortfall is presented. The model gives a graphical solution to the problem of optimal choices when mean and variance are related. Thirdly, a worked example using these results is presented. This example suggests that there may be natural relationships between mean and variance which make solutions to the problems of ‘energy maximization’ and ‘minimization of the probability of starvation’ similar. Finally, current trends in stochastic modeling of foraging behavior are critically discussed.
343 citations
TL;DR: In this article, the convergence of a positive recurrent Harris chain on a general state space with invariant probability measure π is studied and necessary and sufficient conditions for the geometric convergence of λP n f towards its limit π( f ), and when such convergence happens it is uniform over f and in L 1 (π)-norm.
130 citations
TL;DR: In this paper, the authors derive an algorithm that produces the nonparametric maximum likelihood estimator (i.e., the analog of the single-sample empirical distribution function) of the common lifetime distribution, based on such data.
Abstract: Data collected from many independent identically distributed renewal processes, each of which is observed for an arbitrary period of time, is usually affected by censoring coupled with length biased sampling. In this paper we derive an algorithm that produces the nonparametric maximum likelihood estimator (i.e., the analog of the single-sample empirical distribution function) of the common lifetime distribution, based on such data.
79 citations
TL;DR: In this article, it was shown that Krakowski's relevation transform generates the nonhomogeneous Poisson process in an analogous fashion to the way in which Stieltjes convolution generates the renewal process.
Abstract: It is shown that Krakowski's relevation transform generates the nonhomogeneous Poisson process in an analogous fashion to the way in which Stieltjes convolution generates the renewal process. Properties of failure rates and of inter-failure times are discussed and an application to warranty analysis is described.
70 citations
TL;DR: It is proved in this paper that E's1Δ, Δ = 0, Δ â¥ 0, is both necessary and sufficient for a global minimum if the underlying renewal function is concave, and the optimal stationary policy can be computed efficiently by a one-dimensional search routine.
Abstract: The most common measure of effectiveness used in determining the optimal s, S inventory policies is the total cost function per unit time, Es, Δ, Δ = S-s. In stationary analysis, this function is constructed through the limiting distribution of on-hand inventory, and it involves some renewal-theoretic elements. For Δ â¥ 0 given, Es, Δ turns out to be convex in s, so that the corresponding optimal reorder point, s1Δ, can be characterized easily. However, Es1Δ, Δ is not in general unimodal on Δ â¥ 0. This requires the use of complicated search routines in computations, as there is no guarantee that a local minimum is global.
Both for periodic and continuous review systems with constant lead times, full backlogging and linear holding and shortage costs, we prove in this paper that E's1Δ, Δ = 0, Δ â¥ 0, is both necessary and sufficient for a global minimum Es1Δ, Δ is pseudoconvex on Δ â¥ 0 if the underlying renewal function is concave. The optimal stationary policy can then be computed efficiently by a one-dimensional search routine. The renewal function in question is that of the renewal process of periodic demands in the periodic review model and of demand.sizes in the continuous review model.
66 citations
51 citations
TL;DR: In this article, a modified block replacement policy was proposed in which a unit is replaced at failure during (0, T0) and at scheduled replacement time T. The model with two variables was transformed into one variable and the optimum policy was discussed.
Abstract: This paper considers a modified block replacement policy in which a unit is replaced at failure during (0, T0) and at scheduled replacement time T. If a failure occurs in an interval (T0, T), then the unit remains as it is until T. The mean cost rate is obtained, using the results of renewal theory. The model with two variables is transformed into one variable and the optimum policy is discussed. An example shows how to compute the optimum T0* and T* when the failure time of the unit has a gamma distribution.
43 citations
01 Jan 1982
TL;DR: The asymptotic joint distribution of the residual life and total life of a B-interval is that of a renewal process generated by {Bn, n ⥠1}.
Abstract: Consider a renewal process {X,, n 2 1} for which there is defined an associated sequence of independent and identically distributed random variables {Bn, n 2 1 } such that Bn is the length of a subinterval of Xn. We show that when attention is restricted only to B-intervals, the asymptotic joint distribution of the residual life and total life of a B-interval is that of a renewal process generated by (Bn, n 2 1}. TN THE STUDY of stochastic systems, successful analysis is often dependent upon the process being regenerative. As part of the analysis, it may be essential to focus attention on a particular event that occurs during the regeneration cycle. For example, in Oliver's [1964] derivation of the expected waiting time in the M/G/1 queue, it is necessary to calculate the expected remaining service time of the customer in service, if any, at an arrival epoch. By considering only those times when the server is working, Oliver implied that the service times generate a renewal process and so the remaining service time is the equilibrium excess random variable. (Terms are defined in Section 1.) Though this argument is not rigorous, it provides the correct expression for the expected remaining service times as confirmed by Wolff [1970] and Brumelle [1971]. Questions remain, however, as to what other characteristics of a renewal process are inherited by these service times and whether such characteristics are also inherited by other types of events within regeneration cycles. Consider the general setting of a renewal process in which each renewal interval X, contains a subinterval Bn such that {Bn, n ? 1} is a sequence of nonnegative independent and identically distributed (i.i.d.) random variables. We prove that the limiting distributions of excess (residual) life and total life (spread) of such subintervals are the equilibrium distributions for the corresponding quantities in a renewal process generated by {Bn, n - 1}. This is true even if Bn is dependent on another part of the regeneration cycle. Such a case arises in Kleinrock's ([19751,
27 citations
TL;DR: In this paper, a summing up of the author's papers concerning the probability of ruin in a risk business is presented, and results as well as proofs are reviewed in a more systematic manner.
Abstract: In this article a summing up is made of the author's papers concerning the probability of ruin in a risk business. Results as well as proofs are reviewed. In certain cases not covered in the earlier papers a more systematic treatment is given. Primarily the probability of ruin for a finite time period is dealt with. The corresponding problem for an infinite period is treated as a limit case in a more cursory way. It is throughout assumed that the epochs of claims follow a renewal process, i.e. we adhere to E. Sparre Andersen's model. The case of an ordinary renewal process as well as the case of a stationary such one are treated for both positive and negative capital risks in the spirit of H. Cramer's treatment of the Poisson case. Since the author La. has aimed at facilitating the numerical calculation of ruin probabilities, an essential complement to the article in this respect is the numerical implementation of the author's formulas performed by Nils Wikstad. The reader is, therefore, referred...
25 citations
TL;DR: In this article, the coupling method is applied to renewal theory in continuous time and old and new rate results for tendency towards equilibrium and forgetfulness of initial delay are obtained. And the coupling approach also provides a new way of analysing the asymptotics of functions of renewal processes which avoids the route via a renewal equation.
Abstract: This paper shows how the coupling method applies to renewal theory in continuous time. Old and new rate results for tendency towards equilibrium and forgetfulness of initial delay are obtained. The coupling approach also provides a new way of analysing the asymptotics of functions of renewal processes which avoids the route via a renewal equation.
TL;DR: In this article, the Lai strong law of Lai for random walks with positive mean n and an absolutely continuous component was shown to hold in an r-quick version, assuming the existence of a suitable higher moment of the step distribution.
Abstract: Let {S„} be a random walk whose step distribution has positive mean n and an absolutely continuous component. For any bounded measurable function/, a Marcinkiewicz-Zygmund strong law in an r-quick version (a 'Lai strong law') is proved for /(£„), assuming existence of a suitable higher moment of the step distribution. This is extended to show n~\"ÇZÏ f(Sk) J\"8/(/\")<#} -»0 (rquickly). These results remain true when the step distribution is lattice, provided / is constant between lattice points. Certain intermediate results on renewal theory, mixing, local limit theory, ladder height, and a strong law of Lai for mixing random variables are of independent interest.
TL;DR: In this article, it was shown that when attention is restricted only to B-intervals, the asymptotic joint distribution of the residual life and total life of a Binterval is that of a renewal process generated by {Bn, n ⥠1}.
Abstract: Consider a renewal process {Xn, n ⥠1} for which there is defined an associated sequence of independent and identically distributed random variables {Bn, n ⥠1} such that Bn is the length of a subinterval of Xn. We show that when attention is restricted only to B-intervals, the asymptotic joint distribution of the residual life and total life of a B-interval is that of a renewal process generated by {Bn, n ⥠1}.
TL;DR: In this article, the moments of the forward recurrence time of an ordinary renewal process are derived in terms of the renewal function and the moment of the common lifetime distribution, and asymptotic formulae as the process is allowed to run on for a fixed long time are given.
TL;DR: Whenever insensitivity prevails towards an i.i.d. sequence or---equivalently---a renewal process, then replacing this process by an arbitrary point process with the same intensity when stationary still does not change the stationary distribution of the system state.
Abstract: Where sequences of iid random variables are used for modelling successive repair limes, life times, service times, etc, it can happen that stationary probabilities are found for the---suitably defined---system state, which depend only on the means of those variables, not on any further characteristics of their distributions The well-known Erlang loss model provides an example For a large class of models, Jansen, Koenig, and Nawrotzki Jansen, U, D Koenig, K Nawrotzki 1979 A criterion of insensitivity for a class of queuing systems with random marked point processes Math Operationsforsch Statist10 379--403 have shown that, whenever such insensitivity prevails towards an iid sequence or---equivalently---a renewal process, then replacing this process by an arbitrary point process with the same intensity when stationary still does not change the stationary distribution of the system state We extend their model and prove the analogous result by way of a different technique of general interest
TL;DR: In this article, the authors generalize renewal theory to the case of a Markov chain on the real line with stationary transition probabilities satisfying a drift condition, and show that these expectations are unique solutions of the generalized renewal equations they satisfy.
Abstract: Standard renewal theory is concerned with expectations related to sums of positive i.i.d. variables, $S_n = \sum^n_{i=1} Z_i$. We generalize this theory to the case where $\{S_i\}$ is a Markov chain on the real line with stationary transition probabilities satisfying a drift condition. The expectations we are concerned with satisfy generalized renewal equations, and in our main theorems, we show that these expectations are the unique solutions of the equations they satisfy.
TL;DR: In this paper, renewal theorems are obtained for a class of perrturbed arithmetic random walks, and an application to the study of fixed-width confidence intervals is discussed, where the authors apply the renewal theorem to the case of fixed width confidence intervals.
Abstract: Renewal theorems are obtained for a class of perrturbed arithmetic random walks. An application to the study of fixed-width confidence intervals is discussed
Posted Content•
TL;DR: In this paper, a variant of Shanbhag's result is introduced, and several characterizations of truncated and untruncated distributions are obtained using this result and renewal theory.
Abstract: In a recent paper, Shanbhag (1977) uses an elementary approach from renewal theory to give an extension of a characterization of the Poisson law by Rao-Rubin (1964). In the present paper, a variant of Shanbhag's result is introduced. Using Shanbhag's and this result, several characterizations of truncated and untruncated distributions are obtained
TL;DR: In this article, the concept of availability measures is extended to formulae for the joint prediction of availability and number of breakdowns (or repairs) of the system during a fixed interval.
Abstract: An availability measure is the probability that a two-state system modeled by an alternating renewal process is available at one or more points or intervals. The concept of availability measures is extended to formulae for the joint prediction of availability and numbers of breakdowns (or repairs) of the system during a fixed interval.
TL;DR: For the semi-Markov model in risk and queueing theories, this article introduced the natural extension of the concept of stationarity in classical models (i.e., the passage of a renewal process to a general stationary renewal process for the arrivals) and showed that some interesting relations can be obtained between positive capital risk models using a new concept of duality called *-duality.
Abstract: For the semi-Markov model in risk and queueing theories, we introduce the natural extension of the concept of stationarity in classical models (ie the passage of a renewal process to a general stationary renewal process for the arrivals) It is shown that some interesting relations can be obtained between positive capital risk models using a new concept of duality called *-duality
TL;DR: In this article, the authors compare several competing estimates of the availability of a system which alternates between two states, "up" and "down", in accordance with an alternating renewal process.
Abstract: We compare several competing estimates of the availability of a system which alternates between two states, “up” and “down,” in accordance with an alternating renewal process. Both interval and point estimators are compared under several special but representative situations. The comparison reaffirms the validity and robustness of the log-logistic jackknifed estimates. However, when the point estimates are compared from the intrinsic criterion of probability of concentration, the uniformly minimum variance estimate obtained for the Markov model performs very well.
TL;DR: In this article, the flexibility of the filtered renewal process compared with the filtered Poisson process as a model for traffic noise is demonstrated by proving that for the family of gamma distributed headways with coefficient of variation less than one, the variance of the traffic noise signal never exceeds that of the corresponding filtered poisson process.
TL;DR: In this article, random variables are constructed for a Bellman-Harris branching process which are the extensions of backward and forward recurrence times from an ordinary renewal process, and asymptotic limits for all three time-dependent distributions are also obtained.
TL;DR: The GBRP provides for re-installation of used, non-failed items under stochastically optimal conditions determined from the cost model, and the renewal theory based cost model described is more realistic than other cost models.
TL;DR: In this article, a renewal of a nonhomogeneous Markov chain with incomplete information given by a point process is treated and a long run average cost is studied, where the authors apply the "crossing level" method given by Davis.
01 Dec 1982
TL;DR: In this article, the authors consider the case of Renewal Theory, which in the context of Reliability has led to the characterization of many classes of repair/replacement policies, and which appears to depend crucially on the assumption of independence for times between successive failures.
Abstract: : The greatest success, as well as the most severe limitations, of standard Reliability Theory have been due to its restriction to the study of independent failure-time random variables. Consider the case of Renewal Theory, which in the context of Reliability has led to the characterization of many classes of repair/replacement policies, and which appears to depend crucially on the assumption of independence for times between successive failures. In practical life, it is clear that successive replacements of failed components in a complicated assembly (say, an aircraft) may have some cumulative effect tending to shorten future times between replacements. Additionally, one can imagine that shocks to the system from failures of single components can affect the lifetimes of the remaining components, or even that the age of important components can be reflected in the operating characteristics and therefore in the hazard of failure of the system. (Author)