Showing papers in "Journal of Applied Probability in 1986"

Discrimination between monotonic trends and long-range dependence

Abstract: We show that the periodogram behaves differently for a weakly dependent process with a small monotonic trend and a stationary strongly dependent process. In the former case it has a non-central -distribution with noncentrality parameter tending to 0 uniformly outside intervals . In the latter case it has λ α −1 times a central -distribution, 0 < α < 1.

Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules

Abstract: Processors handling multi-class traffic typically alternate between serving a particular class of traffic and performing other tasks, e.g., secondary service tasks or routine maintenance. The stochastic behavior of such systems is modeled by a newly introduced class of Bernoulli GI/G/1 vacation models. For this model, when a vacation is completed and customers are present, a customer is served. When a customer has just been served and other customers are present, the server accepts a customer with fixed probability p or commences a vacation of prespecified random duration with probability 1 – p. Whenever no customers are present, a vacation is taken. When p = 0 or p = 1 this schedule reduces to the previously introduced single service schedule and the exhaustive service schedule, respectively. An analysis of all three schedules on a state space incorporating server vacations is presented using simple methods in the complex plane. It is shown that the recent decomposition results for exhaustive service extend to the more general class of Bernoulli schedules.

Multivariate distributions for the life lengths of components of a system sharing a common environment

Abstract: In assessing the reliability of a system of components, it is usual to suppose the components to fail independently of each other. Often this is inappropriate because the common environment acting on all components induces correlation. For example, a harsh environment will encourage early failure of all components. A simple model that incorporates such dependencies is described, and several properties of this model investigated. Calculations are carried out for a parallel system of two components. Inequalities for multicomponent systems are suggested. The results generalize easily.

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.

Periodic and sequential preventive maintenance policies

Abstract: This paper considers periodic and sequential preventive maintenance (PM) policies for the system with minimal repair at failure: the PM is done (i) at periodic times kx and (ii) at constant intervals xk (k = 1, 2, ···, N). The system has a different failure distribution between PM'S and is replaced at the Nth PM. The optimal policies which minimize the expected cost rates are discussed. The optimal x and N of periodic PM and {xk } of sequential PM are easily computed in a Weibull distribution case.

Aspects of positive ageing

Abstract: The concept of positive ageing describes the adverse effects of age on the lifetime of units. Various aspects of this concept are described in terms of conditional probability distributions of residual lifetimes, failure rates, equilibrium distributions, etc. In this paper we further analyse this concept and relate it to stochastic dominance of first and higher orders. In the process we gain many insights and are able to define several new kinds of ageing criteria which supplement those existing in the literature.

Symmetric sampling procedures, general epidemic processes and their threshold limit theorems

Abstract: Iterative sampling procedures of a general type in a finite population are considered. They generalize the Reed-Frost process in that binomial sampling is replaced by an arbitrary symmetric sampling defined by a factorial series distribution. Threshold limit theorems are proved saying that the total number of sampled objects is either small with a certain limit distribution, or a finite fraction of the population with a Gaussian limit distribution as the size of the population gets large. These results extend earlier ones for the Reed-Frost

On aggregated Markov processes

Abstract: A finite-state Markov process is aggregated into several groups. What can be learned about the underlying process from the aggregated one? We provide some partial answers to this question.

Scheduling jobs with stochastically ordered processing times on parallel machines to minimize expected flowtime

Abstract: A number of jobs are to be processed using a number of identical machines which operate in parallel. The processing times of the jobs are stochastic, but have known distributions which are stochastically ordered. A reward r(t) is acquired when a job is completed at time t. The function r(t) is assumed to be convex and decreasing in t. It is shown that within the class of non-preemptive scheduling strategies the strategy SEPT maximizes the expected total reward. This strategy is one which whenever a machine becomes available starts processing the remaining job with the shortest expected processing time. In particular, for r(t) = – t, this strategy minimizes the expected flowtime.

Order estimation by accumulated prediction errors

Abstract: This paper presents a new criterion based on prediction error which allows the estimation of the number of parameters as well as structures in statistical models. The criterion is valid for short and long samples alike. Unlike Akaike's earlier criterion, also based on prediction error, the criterion proposed here appears to produce consistent error estimates in ARMA processes. ARMA PROCESSES; MINIMUM DESCRIPIION LENGTH; MODEL COST; PARAMETERS; CONSISTENCY

Mixing properties of harris chains and autoregressive processes

Abstract: Soit {Yn: n≥1} une chaine de Markov recurrente au sens de Harris sur un espace d'etat general. On montre que {Yn} est fortement melangeante sachant qu'il existe une distribution de probabilite stationnaire π(•) pour {Yn}

Optimal replacement times - a general set-up

Abstract: For a large class of replacement models for stochastically deteriorating systems the optimality criteria of total expected discounted cost and long-run (expected) average cost per unit time have a common structure. In the present paper a formal description of this structure is given and the optimal rule is determined. A so-called 'A-minimization technique' is applied. This method is discussed in general terms. AVERAGE COST; EXPECTED DISCOUNTED COST; STOPPING TIME; OPTIMIZATION; A -PROBLEMS

The M/M /∞ queue in a random environment

Abstract: The M/M/oo queue in a random environment is an infinite-server queue where arrival and service rates are stochastic processes. Here we study the steady-state behavior of such a system. Explicit results are obtained for the factorial moments, the impossibility of a 'matrix-Poisson' steady-state distribution is demonstrated and two numerical examples are presented.

Optimal allocation of components in parallel−series and series−parallel systems

Abstract: This paper shows how majorization and Schur-convex functions can be used to solve the problem of optimal allocation of components to parallel-series and series-parallel systems to maximize the reliability of the system. For parallel-series systems the optimal allocation is completely described and depends only on the ordering of component reliabilities. For series-parallel systems, we describe a partial ordering among allocations that can lead to the optimal allocation. Finally, we describe how these problems can be cast as integer linear programming problems and thus the results obtained in this paper show that when some linear integer programming problems are recast in a different way and the techniques of Schur functions are used, complete solutions can be obtained in some instances and better insight in others.

Probabilistic analysis of optimum partitioning

Abstract: Given a set of n items with real-valued sizes, the optimum partition problem asks how it can be partitioned into two subsets so that the absolute value of the difference of the sums of the sizes over the two subsets is minimized. We present bounds on the probability distribution of this minimum under the assumption that the sizes are independent random variables drawn from a common distribution. For a large class of distributions, we determine the asymptotic behavior of the median of this minimum, and show that it is exponentially small.

The extinction time of a general birth and death process with catastrophes

Abstract: The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfy λ 0 = 0, λ j > 0 for each j > 0, and . Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λ j = jλ, µj = jμ) with catastrophes of several different types.

The effect of increasing service rates in a closed queueing network

Abstract: In this paper we study the equilibrium behavior of the queue lengths in a product-form closed queueing network when the service rates at a subset of stations (nodes) are increased. Univariate and multivariate likelihood ratio orderings as well as multivariate stochastic ordering of the queue lengths are established to indicate the effect of increasing the service rates. Relations among these orderings are also developed.

Bisexual Galton−Watson branching processes with superadditive mating functions

Abstract: For a bisexual Galton-Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

Some remarks on the Rayleigh process

Abstract: The transition p.d.f. for a one-dimensional Rayleigh process in the presence of an absorption condition or a zero-flux condition in the origin is obtained in closed form. The first-passage-time problem through an arbitrary constant boundary is then considered and the moment-generating function is determined. In some particular cases the first-passage-time p.d.f. is explicitly derived. Use of some of these results is finally made to obtain the transition p.d.f. of the affine drift-linear infinitesimal-variance diffusion process when the origin is an entrance or a regular boundary in the presence of a reflection condition.

Extreme values of non-stationary random sequences

Abstract: We extend some results of the extreme-value theory of stationary random sequences to non-stationary random sequences. The extremal index, defined in the stationary case, plays a similar role in the extended case. The details show that this index describes not only the behaviour of exceedances above a high level but also above a non-constant high boundary.

An optimal inspection and replacement policy for a deteriorating system

Abstract: This paper investigates a system whose deterioration is expressed as a continuous-time Markov process. It is assumed that the state of the system cannot be identified without inspection. This paper derives an optimal policy minimizing the expected total long-run average cost per unit time. It gives the optimal time interval between successive inspections and determines the states at which the system is to be replaced. Furthermore, under some reasonable assumptions reflecting the practical meaning of the deterioration, it is shown that the optimal policy has monotonic properties. A control limit rule holds for replacement, and the time interval between successive inspections decreases as the degree of deterioration increases.

Approximation of the queue-length distribution of an M/GI/s queue by the basic equations

Abstract: We give a unified way of obtaining approximation formulas for the steady-state distribution of the queue length in the M/GI/s queue. The approximations of Hokstad (1978) and Case A of Tijms et al. (1981) are derived again. The main interest of this paper is in considering the theoretical meaning of the assumptions given in the literature. Having done this, we derive new approximation formulas. Our discussion is based on one version of the steady-state equations, called the basic equations in this paper. The basic equations are derived for M/GI/s/k with finite and infinite k. Similar approximations are possible for M/GI/s/k (k < +∞).

Expected waiting times in a multiclass batch arrival retrial queue

Abstract: Expressions are derived for the expected waiting times for the customers of two types who arrive in batches (in a compound Poisson fashion) at a single-server queueing station with no waiting room. Those who cannot get served immediately keep returning to the system after random exponential amounts of time until they get served. The result is shown to agree with similar results for three special cases studied in the literature.

On recurrence and transience of growth models

Abstract: Let Xn be non-negative random variables, possessing the Markov property. We given criteria for deciding whether Pr(X. -- oo) is positive or 0. It turns out that essentially this depends on the magnitude of E(X+., I X. = x) compared to that of E(X+, |I Xn = x) for large x. The assumptions are chosen such that

Consistency of model selection and parameter estimation

Abstract: The relationship between consistency of model selection and that of parameter estimation is investigated. It is shown that the consistency of model selection is achieved at the cost of a lower order of consistency of the resulting estimate of parameters in some domain. The situation is different when selecting autoregressive moving average models, since the information matrix becomes singular when overfitted. Some detailed analyses of the consistency are given in this case. AUTOREGRESSIVE PROCESSES; AKAIKE INFORMATION CRITERION; FINITE PARAMETER ESTIMATION

Optimum replacement of a system subject to shocks

Abstract: A system is subject to shocks. Each shock weakens the system and makes it more expensive to run. It is desirable to determine a replacement time for the system. Boland and Proschan [4] consider periodic replacement of the system and give sufficient conditions for the existence of an optimal finite period, assuming that the shock process is a non-homogeneous Poisson process and the cost structure does not depend on time. Block et al. [3] establish similar results assuming that cost structure is time dependent, still requiring that the shock process is a non-homogeneous Poisson process. We show via a sample path argument that the results of [3] and [4] hold for any counting process whose jump size is of one unit magnitude.

Point-Process Models with Linearly Parametrized Intensity for Application to Earthquake Data

Abstract: It is demonstrated that linear parametrization of the conditional intensity provides systematic classes of flexible models which are reasonably useful for calculating maximum likelihoods. To exemplify the modelling, seismic activity around Canberra is decomposed into components of evolutionary trend, clustering and periodicity. The causal relationship between earthquake sequences from two seismic regions is also analysed for a certain Japanese earthquake data set. Some technical aspects of the modelling and calculations are descrnbed.

A planar random motion governed by the two-dimensional telegraph equation

Abstract: On presente un mouvement aleatoire planaire gouverne par l'equation des telegraphistes a 2 dimensions. On demontre que la particule effectuant le mouvement est a chaque temps t dans un cercle centre au point de depart et de rayon ct/(2) 1/2 . On obtient la densite explicite de la position de particule

Convergence of quasi-stationary to stationary distributions for stochastically monotone Markov processes

Abstract: : It is shown that if a stochastically monotone Markov process of (0, infinity) with stationary distribution H has its state space truncated by making all states in (B, infinity) absorbing, then the quasi-stationary distribution of the new process coverages to H as B approaches limit of infinity. (Author)