Showing papers on "Renewal theory published in 1986"

Optimal control of production rate in a failure prone manufacturing system

Abstract: We address the problem of controlling the production rate of a failure prone manufacturing system so as to minimize the discounted inventory, cost, where certain cost rates are specified for both positive and negative inventories, and there is a constant demand rate for the commodity produced. The underlying theoretical problem is the optimal control of a continuous-time system with jump Markov disturbances, with an infinite horizon discounted cost criterion. We use two complementary approaches. First, proceeding informally, and using a combination of stochastic coupling, linear system arguments, stable and unstable eigenspaces, renewal theory, parametric optimization, etc., we arrive at a conjecture for the optimal policy. Then we address the previously ignored mathematical difficulties associated with differential equations with discontinuous right-hand sides, singularity of the optimal control problem, smoothness, and validity of the dynamic programming equation, etc., to give a rigorous proof of optimality of the conjectured policy. It is hoped that both approaches will find uses in other such problems also. We obtain the complete solution and show that the optimal solution is simply characterized by a certain critical number, which we call the optimal inventory level. If the current inventory level exceeds the optimal, one should not produce at all; if less, one should produce at the maximum rate; while if exactly equal, one should produce exactly enough to meet demand. We also give a simple explicit formula for the optimal inventory level.

A useful generalization of renewal theory: counting processes governed by non-negative markovian increments

Abstract: Let N(t) be a counting process associated with a sequence of non-negative random variables (X,)' where the distribution of X+i, depends only on the value of the partial sum S = X.., X,. In this paper, we study the structure of the function H(t)= E[N(t)], extending the ordinary renewal theory. It is shown under certain conditions that h(t) = (d/dt)H(t) exists and is a unique solution of an extended renewal equation. Furthermore, sufficient conditions are given under which h(t) is constant, monotone decreasing and monotone increasing. Asymptotic behavior of h(t) and H(t) as t-oo is also discussed. Several examples are given to illustrate the theoretical results and to demonstrate potential use of the study in applications.

Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift

Abstract: Consider the sequence of partial sums of a sequence of i.i.d. random variables with positive expectation. We study various random quantities defined by the sequence of partial sums, e.g. the time at which the first or last crossing of a given level occurs, the value of the partial sum immediately before or after the crossing, the minimum of all partial sums. Necessary and sufficient conditions are given for the existence of moments of these quantities. RANDOM WALK; RENEWAL THEORY

Nonparametric Renewal Function Estimation

Abstract: : The renewal function is a basic tool used in many probabilistic models and sequential analysis. Based on a random sample of size n, a nonparametric estimator of the renewal function is introduced. Asymptotic properties of the estimator such as the almost sure consistency and local asymptotic normality are developed. A discussion of an application of the estimator is also provided. Keywords: U-statistics, reverse martingales, warranty analysis.

Uniformization and Hybrid Simulation/Analytic Models of Renewal Processes

Abstract: In this paper we demonstrate the use of uniformization in the simulation of renewal processes. Using uniformization, we represent the continuous random variable of interest as the first passage time of a continuous-time stochastic process associated with a Poisson process. We then use this result to develop a hybrid simulation/analytic method to model renewal processes. The estimators obtained from the hybrid simulation/analytic models have lower variance than the variance of the estimators of the traditional simulation models. We also discuss the possible impact of this method on the future of simulation methodology.

Renewal Tables: Tables of Functions Arising in Renewal Theory.

Abstract: : The generalized cubic splining algorithm enables us to evaluate recursively-defined convolutions for a wide variety of distribution functions. The algorithm has been applied to evaluate the renewal function, variance function and the integral of the renewal function for five distributions (gamma, inverse Gaussian, lognormal, truncated normal and Weibull) for a wide range of values of the shape parameter. The results of the computations are discussed and a comparison is made with previous tabulations. (Author)

Renewal theorem for a class of stationary sequences

Abstract: A renewal theorem is obtained for stationary sequences of the form ξn=ξ(...,Xn-1,Xn,Xn+1...), whereXn,\(n \in \mathbb{Z}\), are i.i.d. r.v.s. valued in a Polish space. This class of processes is sufficiently broad to encompass functionals of recurrent Markov chains, functionals of stationary Gaussian processes, and functionals of one-dimensional Gibbs states. The theorem is proved by a new coupling construction.

Periodicity and absolute regularity

Abstract: For a stationary ergodic process it is proved that the dependence coefficient associated with absolute regularity has a limit connected with a periodicity concept. Similar results can then be obtained for stronger dependence coefficients. The periodicity concept is studied separately and it is seen that the double tailσ-field can be trivial while the period is 2. The paper imbeds renewal theory in ergodic theory. The total variation metric is used.

Return probabilities for correlated random walks

Abstract: A correlated random walk on a d-dimensional integer lattice is studied in which, at any stage, the probabilities of the next step being in the various possible directions depend on the direction of the previous step. Using a renewal argument, asymptotic estimates are obtained for the probability of

Queueing analysis of fault-tolerant computer systems (extended abstract)

Abstract: Queueing models provide a useful tool for predicting the performance of many service systems including computer systems, telecommunication systems, computer/communication networks and flexible manufacturing systems. Traditional queueing models predict system performance under the assumption that all service facilities provide failure-free service. It must, however, be acknowledged that service facilities do experience failures and that they get repaired. In recent years, it has been increasingly recognized that this separation of performance and reliability/availability models is no longer adequate.An exact steady-state queueing analysis of such systems is considered by several authors and is carried out by means of generating functions, supplementary variables, imbedded Markov process and renewal theory, or probabilistic techniques [1,2,7,8]. Another approach is approximate, in which it is assumed that the time to reach the steady-state is much smaller than the times to failures/repairs. Therefore, it is reasonable to associate a performance measure (reward) with each state of the underlying Markov (or semi-Markov) model describing the failure/repair behavior of the system. Each of these performance measures is obtained from the steady-state queueing analysis of the system in the corresponding state [3,5].Earlier we have developed models to derive the distribution of job completion time in a failure-prone environment [3,4]. In these models, we need to consider a possible loss of work due to the occurrence of a failure, i.e., the interrupted job may be resumed or restarted upon service resumption. Note that the job completion time analysis includes the delays due to failures and repairs. The purpose of this paper [9] is to extend our earlier analysis so as to account for the queueing delays. In effect, we consider an exact queueing analysis of fault-tolerant systems in order to obtain the steady-state distribution and the mean of the number of jobs in the system. In particular, we study a system in which jobs arrive in a Poisson fashion and are serviced according to FCFS discipline. The service requirements of the incoming jobs form a sequence of independent and identically distributed random variables. The failure/repair behaviour of the system is modelled by an irreducible continuous-time Markov chain, which is independent of the number of jobs in the system. Let the state-space be {1,2, …,n}. When the computer system is in state i it delivers service at rate ri ≥ 0. Furthermore, depending on the type of the state, the work done on the job is preserved or lost upon entering that state. The actual time required to complete a job depends in a complex way upon the service requirement of the job and the evolution of the state of the system. Note that even though the service requirements of jobs are independent and identically distributed, the actual times required to complete these jobs are neither independent nor identically distributed, and hence the model cannot be reduced to a standard M/G/1 queue [8]. As loss of work due to failures and interruptions is quite a common phenomenon in fault-tolerant computer systems, the model proposed here is of obvious interest.Using our earlier results on the distribution of job completion time we set up a queueing model and show that it has the block M/G/1 structure. Queueing models with such a structure have been studied by Neuts, Lucantoni and others [6]. We demonstrate the usefulness of our approach by performing the numerical analysis for a system with two processors subject to failures and repairs.

Counting by Weighing: An Approach Using Renewal Theory

Abstract: Renewal theory and Bayesian decision theory are used to solve a problem related to counting a large number of items by weighing them. Specifically, a batch is to be obtained containing a given number of items by adding items until their total weight reaches a critical value that can depend on the results of a preliminary sample. Furthermore, the optimal sample size for this preliminary sample is to be determined. The distribution of individual weights is assumed to be normal.

On inference and transient response for M/G/1 models

Abstract: : This paper addresses two problems of interest in service system analysis: (a) that of making statistical, data-driven estimates of the long-run probability of a long delay, and (b) the assessment of rate of approach to a long-run system performance measure such as expected delay, the rate being characterized by a simple exponential, at least initially Both are illustrated by reference to M/G/1 and related systems Keywords: Estimation of virtual waiting time distribution; Terminating renewal process; Bootstrap; Jacknife; Transient behavior of queueing systems (Author)

Stochastic Damage Models and Dependence Effects in the Survivability Analysis of Communication Networks

Abstract: Stochastic analyses of the Survivability of communication networks often include a simplifying assumption that failures of, or damages to, various components of the network are statistically independent. This assumption can be quite unrealistic and can lead one to conclusions that are grossly in error. Survivability analyses and syntheses of robust networks should incorporate dependencies introduced by single events that affect large geographical areas. In this paper, we construct a stochastic damage model, analyze it, and apply the results to the survivability analysis of some simple network topologies. We demonstrate how the results can differ significantly from those obtained when independence of damage is assumed. The damage model consists of a Poisson ensemble of events (damage centers) on the plane, of given intensity (level of attack), and a network resource is damaged, and hence dysfunctional, if it lies within a radius ρ (damage radius) of some damage-causing event. Statistical properties of the damage process are obtained (e.g., the covariance function, mean and variance of the damage extent on a line resulting from the Poisson ensemble) and used to evaluate dependence effects. The damage process on a line is shown to be an alternating renewal process corresponding to the busy/idle process of an appropriately defined M/G/\infty queue, and standard M/G//infty and Type-II counter results can thus be exploited to obtain some desired quantities.

Analysis of (S, s) Inventory System with Decaying Items

Abstract: This article obtains the stationary distribution of the inventory level of an (S, s) inventory model with decaying items. The demand to this inventory system is governed by a general renewal process. Items decay at a constant rate independently and identically. When the inventory reduces to a level

The asymptotic joint distribution of the state occupancy times in an alternating renewal process

Abstract: An alternating renewal process starts at time zero and visits states 1,2,…,r, 1,2, …,r 1,2, …,r, … in sucession. The time spent in state i during any cycle has cumulative distribution function Fi, and the sojourn times in each state are mutually independent, positive and nondegenerate random variables. In the fixed time interval [0,T], let Ui(T) denote the total amount of time spent in state i. In this note, a central limit theorem is proved for the random vector (Ui(T), 1 ≤ i ≤ r) (properly normed and centered) as T ∞.

The non-renewal nature of the quasi-input process in the M/G/1/∞ queue

Abstract: On montre que, excepte dans le cas trivial de service instantane, le processus de quasi-entree n'est jamais de renouvellement

On the process of stabilization in the renewal model: approximations for the time to convergence.

Abstract: The transient behaviour of the renewal model leading to the stable age distribution is studied for weakly skewed net maternity functions (found in human as well as in some animal populations). The study, which is partly based on heuristic arguments, first provides approximate expressions for the damping constant and the circular frequency (in terms of the moments of the net maternity function) belonging to the principal oscillatory component of the birth trajectory. The time to stability (defined as the time interval after which the principal oscillatory component has become less than a certain fraction of the stable solution) is then determined in two cases: For the genesis model and for a stable population in which the net reproduction rate is reduced to one. The results are applied to a problem which arises in the mass rearing of pest insects.

Stationary regenerative systems

Abstract: The theory of regenerative systems on the real line unifies the notions of strong Markov processes indexed by ℝ, recurrent events, regenerative processes, semi-Markov processes, etc.

The non-renewal nature of the quasi-input process

Abstract: Falin (1984) examined the quasi-input process (the flow of service starting times) in the M/G/1/oo queue and raised the question as to whether this process is a renewal process. We show that, except in the trivial case of instantaneous service, the quasi-input process is never renewal.