Showing papers on "Renewal theory published in 1983"

General shock models associated with correlated renewal sequences

Abstract: In this paper we define and analyze a general shock model associated with a correlated pair (Xn, Yn ) of renewal sequences, where the system fails when the magnitude of a shock exceeds (or falls below) a prespecified threshold level. Two models, depending on whether the nth shock Xn is correlated to the length Yn of the interval since the last shock, or to the length Yn of the subsequent interval until the next shock, are considered. The transform results, an exponential limit theorem, and properties of the associated renewal process of the failure times are obtained. An application in a stochastic clearing system with numerical results is also given.

The coupling of regenerative processes

Abstract: A distributional coupling concept is defined for continuous-time stochastic processes on a general state space and applied to processes having a certain non-time-homogeneous regeneration property: regeneration occurs at random times So, S1, forming an increasing Markov chain, the post-S, process is conditionally independent of So,... , S,_ given S,, and the conditional distribution is independent of n. The coupling problem is reduced to an investigation of the regeneration times So, S1, , and a successful coupling is constructed under the condition that the recurrence times X,+1 =S,+ -S, given that S, = s, s e[0, oo), are stochastically dominated by an integrable random variable, and that the distributions P,(A) = P(X, eA I S,= s), s E [0, oo), have a common component which is absolutely continuous with respect to Lebesgue measure (or aperiodic when the S,'s are lattice-valued). This yields results on the tendency to forget initial conditions as time tends to 00. In particular, tendency towards equilibrium is obtained, provided the post-S, process is independent of S,. The ergodic results cover convergence and uniform convergence of distributions and mean measures in total variation norm. Rate results are also obtained under moment conditions on the P,'s and the times of the first regeneration. RENEWAL PROCESS: MARKOV CHAIN; REGENERATIVE RANDOM MEASURE: TIMEDEPENDENCE

Renewal Theory for $M$-Dependent Variables

Abstract: Let $S_n, n = 1,2,\cdots$, denote the partial sums of a stationary $m$-dependent sequence of random variables with positive expectation. The first passage times $\min\{n: S_n > t\}$ are investigated. Several results are extended from the case of independent variables.

Transport properties of the one-dimensional stochastic Lorentz model: I. Velocity autocorrelation function

Abstract: Point scatterers are placed on the real line such that the distances between scatterers are independent identically distributed random variables (stationary renewal process). For a fixed configuration of scatterers a particle performs the following random walk: The particle starts at the pointx with velocityυ, ¦υ¦=1. In between scatterers the particle moves freely. At a scatterer the particle is either transmitted or reflected, both with probability 1/2. For given initial conditions of the particle the velocity autocorrelation function is a random variable on the scatterer configurations. If this variable is averaged over the distribution of scatterers, it decays not faster thant −3/2.

Functions of discrete probability measures: Rates of convergence in the renewal theorem

Abstract: Let ν and τ be finite measures on the set of integers such that the Fourier transform of ν is an analytic function of the τ transform. The central result shows how ν may be approximated by linear combinations of convolution powers of τ. Applications are given to renewal theory, infinitely divisible measures and age-dependent branching processes.

On queueing systems with renewal departure processes

Abstract: It is proved that, for a large class of stable stationary queueing systems with renewal arrival processes and without losses, a necessary condition for the departure process also to be a renewal process is that its interval distribution be the same as that of the arrival process. This result is then applied to the classical GI/G/s queueing systems. In particular, alternative proofs of known characterizations of the M/G/1 and GI/M/1 systems are given, as well as a characterization of the GI/G/∞ system. In the course of the proofs, sufficient conditions for the existence of all the moments of the stationary queue-size distributions of both the GI/G/1 and GI/G/∞ systems are derived.

Numerical Evaluation of Instantaneous Availability

Abstract: Instantaneous availability (the probability that the system is operational at any time t) is calculated via renewal theory by representing the system as a 2-state stochastic process; the defined states are on-time and off-time which are combined to form the total cycle time. The analytic solution to the above problem is quite general and can be applied to any system adjusted cycle time and on-time distribution. Using this model, a numerical solution is obtained. If the on-time and cycle-time are gamma distributed, then the analytic solution, in both complex and real forms, can be derived for the instantaneous availability of a system. if they are not gamma distributed then the analytic approach is difficult (if not impossible) and an alternative method is required. Thus a numerical approach is used, since it is distribution-free. A numerical example of small sample size illustrates the numerical approach.

Inequalities and bounds for variances of point processes and fibre processes

Abstract: For the variances of numbers of points and total fibre lengths in compact convex sets, corresponding to point processes and fibre processes, bounds are proved. They base on monotonicity and convexity properties of a function of geometrical nature occuring in the formula for the variance when using the reduced second moment measure. As particular cases, stationary renewal processes. Cox processes and Neyman-Scott processes are considered. For certain fibre process models Poisson point processes and segment processes are used as approximative models.

A renewal theorem and its applications to some sequential procedures

Abstract: A renewal theorem of the elementary type for some stopping times which arise from some statistical estimation problems has been established. It is applied to prove the asymptotic risk efficiency for the problem of estimating the mean when the loss function is a weighted sum of the squared error and the sample size, and the variance is unknown. It is also applied to verify a conjecture of Robbins and Siegmund (1974) on the evaluation of the variance of the estimator of the logarithm of the odds ratio for a sequential procedure.

Point Processes and Renewal Theory: A Brief Survey

Abstract: A brief survey is given of recent and not-so-recent work on point processes and renewal theory. The emphasis is on the ‘applied mathematics’ of the subject and to a lesser extent on the statistical analysis of empirical data.

Estimation of frequency by random sampling

Abstract: We consider the estimation of frequency ω of a sinusoidal oscillation contaminated by a stationary noise under a random sampling scheme according to a stationary point processN. We prove the strong consistency and the asymptotic normality for a certain estimator of ω. Then we apply these results to the case whereN is a stationary delayed renewal process.

Complex System Modelling with Fault Trees and Stochastic Processes

Abstract: A general introduction to fault tree analysis is given. Basic concepts of fault tree representation are introduced and relations to switching theory emphasized. The probabilistic evaluation of coherent systems is discussed. This is an application of alternating renewal processes. It is possible to use for evaluation minimal cuts, expansion, or modular decomposition. For decomposition, interesting relations to switching theory exist.

A Simplified Method to Calculate Failure Times in Fault-Tolerant Systems

Abstract: A simplified method is presented to calculate moments of failure time and residual lifetime of a fault-tolerant system. The method is based on recent results in queueing theory. Its effectiveness is illustrated by considering a dual repairable system from the literature.

Some results on first passage times in one dimensional random walks

Abstract: Analytic expressions are presented for the characteristic function of the first passage time distribution for biased random walk on a finite chain (and diffusion with drift on a finite line); of the first passage time distribution for a random walk on a chain, in which the events (jumps) are governed by an arbitrary renewal process; and of the distribution of the time of escape from a bounded set of points in the latter case. A fundamental relation between the first passage time distribution and the conditional probability for random walk (or diffusion) in one dimension is analyzed and generalized.

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.

An operational view on renewal theory

Abstract: In this paper we derive a formula for the moments of the residual life in operational context, and show that the Paradox of Residual Life holds also in a finite queueing model. In addition, we prove the renewal theorem, show that forward and backward times are independent, and state the memoryless property. As applications we point out how to derive Takacs recurrence formula for the moments of the waiting time and how to base the Markovian state theory on this.