Mean First-passage Times for a <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> Sytem with Repair

Consider a k-out-of-n system where the lifetimes of the components are independent and identically distributed exponential (λ) random variables. Each component has its own repair facility, the repair times being independent and identically distributed exponential (μ) random variables, independent of the failure times. The mean operating time and mean repair time during the cycle between two successive breakdowns are found using renewal theory and the expression for the system availability. Using these, the mean first-passage times from any of the operating states of the system to the down state, and the mean first-passage times from any of the down states to the operating state are found recursively.     

On reliability analysis of consecutive k-out-of-n systems with arbitrarily dependent components

In this paper, we consider the linear and circular consecutive k-out-of-n systems consisting of arbitrarily dependent components. Under the condition that at least n?r+1 components (r ≤ n) of the system are working at time t, we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive k-out-of-n systems. In the following, we investigate the inactivity time of the component with lifetime T_r:n at the system level for the consecutive k-out-of-n systems under the condition that the system is not working at time t > 0, and obtain some stochastic properties of this conditional random variable.     

A Generalized Block Replacement Policy with Minimal Repair and General Random Repair Costs for a Multi-unit System

A generalization of the block replacement policy (BRP) is proposed and analysed for a multi-unit system which has the specific multivariate distribution. Under such a policy an operating system is preventively replaced at times kT (k = 1, 2, 3,...), as in the ordinary BRP, and the replacement of the failed system at failure is not mandatory; instead, a minimal repair to the component of the system can be made. The choice of these two possible actions is based on some random mechanism which is age-dependent. The cost of the ith minimal repair of the component at age y depends on the random part C(y) and the deterministic part C_i(y). The aim of the paper is to find the optimal block interval T which minimizes the long-run expected cost per unit time of the policy.     

Costing a Finite Minimal Repair Replacement Policy

In this paper an integral equation technique is used to evaluate the expected cost for the period (0, t] of a policy involving minimal repair at failure with replacement after N failures. This cost function provides an appropriate criterion to determine the optimal replacement number N* for a system required for use over a finite time horizon. In an example, it is shown that significant cost savings can be achieved using N* from the new finite time horizon model rather than the value predicted by the usual asymptotic model.     

Reliability Analysis of a One-Unit System with Unrepairable Spare Units and its Optimization Applications

It is assumed that a unit is either in operation or is in repair. When the main unit is under repair, spare units which cannot be repaired are used. In this system the following quantities are of interest: (i) The time distribution and the mean time to first-system failure, given that the n spare units are provided at time 0. (ii) The probability that the number of the failed spare units are equal to exactly n during the interval (0, t], and its expected number during the interval (0, t]. These quantities are derived by solving the renewal-type equations.Two optimization problems are discussed using the results obtained, viz.: (i) The expected cost of two systems, one with both a main unit and spare units and the other with only spare units is considered. (ii) A preventive maintenance policy of the main unit is considered in order to minimize the expected cost rate. Some policies of the two problems are discussed under suitable conditions. Numerical examples are also presented.     

Estimation Problems for Periodically Correlated Isotropic Random Fields

Spectral theory of isotropic random fields in Euclidean space developed by M. I. Yadrenko is exploited to find a solution to the problem of optimal linear estimation of the functional

$$ A\zeta ={\sum\limits_{t=0}^{\infty}}\,\,\,{\int_{S_n}} \,\,a(t,x)\zeta (t,x)\,m_n(dx) $$

which depends on unknown values of a periodically correlated (cyclostationary with period T) with respect to time isotropic on the sphere S _n in Euclidean space E ⁿ random field ζ(t, x), t?∈?Z, x?∈?S _n . Estimates are based on observations of the field ζ(t, x)?+?θ(t, x) at points (t, x), t?=???1,???2, ..., x?∈?S _n , where θ(t, x) is an uncorrelated with ζ(t, x) periodically correlated with respect to time isotropic on the sphere S _n random field. Formulas for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimate of the functional Aζ are obtained. The least favourable spectral densities and the minimax (robust) spectral characteristics of the optimal estimates of the functional Aζ are determined for some special classes of spectral densities.

     

Probabilities of high extremes for a Gaussian stationary process in a random environment

Let ξ(t) be a zero-mean stationary Gaussian process with the covariance function r(t) of Pickands type, i.e., r(t) = 1 ? |t| ^α + o(|t| ^α ), t → 0, 0 < α ≤ 2, and η(t), ζ(t) be periodic random processes. The exact asymptotic behavior of the probabilities P(max _t∈[0,T] η(t)ξ(t) > u), P(max _t∈[0,T] (ξ(t) + η(t)) > u) and P(max _t∈[0,T] (η(t)ξ(t) + ζ(t)) > u) is obtained for u → ∞ for any T > 0 and independent ξ(t), η(t), ζ(t).     

Random process in a homogeneous Gaussian field

We consider a random process in a spatial-temporal homogeneous Gaussian field V (q , t) with the mean E V = 0 and the correlation function W(|q ? q′|, |t ? t′|) ≡ E[V (q, t)V (q′, t′)], where \( \bold{q} \in {\mathbb{R}^d} \), \( t \in {\mathbb{R}^{+} } \), and d is the dimension of the Euclidean space \( {\mathbb{R}^d} \). For a “density” G(r, t) of the familiar model of a physical system averaged over all realizations of the random field V, we establish an integral equation that has the form of the Dyson equation. The invariance of the equation under the continuous renormalization group allows using the renormalization group method to find an asymptotic expression for G(r, t) as r → ∞ and t → ∞.     

Limit theorems for supremum of Gaussian processes over a random interval

Let {X(t), t ≥ 0} be a centered stationary Gaussian process with correlation r(t)such that 1-r(t) is asymptotic to a regularly varying function. With T being a nonnegative random variable and independent of X(t), the exact asymptotics of P(sup_(t∈[0,T])X(t) x) is considered, as x →∞.     

Simulation of a random fuzzy queuing system with multiple servers

The paper considers a queuing system that has k servers and its interarrival times and service times are random fuzzy variables.We obtain a theorem concerning the average chance of the event “r servers (r ≤ k) are busy at time t”, provided that all the servers work independently. We simulate the average chance using fuzzy simulation method and obtain some results on the number of servers that are busy. Some examples to illustrate the simulation procedure are also presented.     

Almost Sure Approximation of the Superposition of the Random Processes

This article presents sufficient conditions, which provide almost sure (a.s.) approximation of the superposition of the random processes S(N(t)), when càd-làg random processes S(t) and N(t) themselves admit a.s. approximation by a Wiener or stable Lévy processes. Such results serve as a source of numerous strong limit theorems for the random sums under various assumptions on counting process N(t) and summands. As a consequence we obtain a number of results concerning the a.s. approximation of the Kesten–Spitzer random walk, accumulated workload input into queuing system, risk processes in the classical and renewal risk models with small and large claims and use such results for investigation the growth rate and fluctuations of the mentioned processes.     

Optimal Reliability Design of <Emphasis Type="Italic">K</Emphasis>-out-of-<Emphasis Type="Italic">N</Emphasis> Systems Subject to Two Kinds of Failure

K-out-of-N systems formed from N identical and independent components are considered in which the components can take two states: 0 (open) or 1 (closed) on command (e.g. Electromagnetic Relays and Solid State Switches). The components are subject to two kinds of failure on command: failure to open and failure to close. A K-out-of-N system is closed if and only if at least K of its components are closed. The system is considered open or closed depending on the states of its components. The optimum system is taken to be that system which maximizes the reliability. This paper finds the optimum K-out-of-N system given a fixed number of components.     

Weak convergence of the past and future of Brownian motion given the present

In this paper, we show that for t > 0, the joint distribution of the past {W _t?s : 0 ≤ s ≤ t} and the future {W _{t + s} :s ≥ 0} of a d-dimensional standard Brownian motion (W _s ), conditioned on {W _t ∈ U}, where U is a bounded open set in ? ^d , converges weakly in C[0,∞)×C[0,∞) as t→∞. The limiting distribution is that of a pair of coupled processes Y + B ¹,Y + B ² where Y,B ¹,B ² are independent, Y is uniformly distributed on U and B ¹,B ² are standard d-dimensional Brownian motions. Let σ _t ,d _t be respectively, the last entrance time before time t into the set U and the first exit time after t from U. When the boundary of U is regular, we use the continuous mapping theorem to show that the limiting distribution as t → ∞ of the four dimensional vector with components \((W_{\sigma _{t}},t-\sigma _{t},W_{d_{t}},d_{t}-t)\), conditioned on {W _t ∈U}, is the same as that of the four dimensional vector whose components are the place and time of first exit from U of the processes Y + B ¹ and Y + B ² respectively.     

Error estimates for a projection-difference method for a linear differential-operator equation

We study a projection-difference method for approximately solving the Cauchy problem u′(t) + A(t)u(t) + K(t)u(t) = h(t), u(0) = 0 for a linear differential-operator equation in a Hilbert space, where A(t) is a self-adjoint operator and K(t) is an operator subordinate to A(t). Time discretization is based on a three-level difference scheme, and space discretization is carried out by the Galerkin method. Under certain smoothness conditions on the function h(t), we obtain estimates for the convergence rate of the approximate solutions to the exact solution.     

Extremes of Gaussian processes with smooth random expectation and smooth random variance

Let ξ(t), t ∈ [0, T],T > 0, be a Gaussian stationary process with expectation 0 and variance 1, and let η(t) and μ(t) be other sufficiently smooth random processes independent of ξ(t). In this paper, we obtain an asymptotic exact result for P(sup _t∈[0,T](η(t)ξ(t) + μ(t)) > u) as u→∞.     

Connecting orbits of Lagrangian systems in a nonstationary force field

We study connecting orbits of a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential U(q, t) = f(t)V(q). It is assumed that the factor f(t) tends to ∞ as t→±∞ and vanishes at a unique point t ₀ ∈ ?. Let X ₊, X _? denote the sets of isolated critical points of V (x) at which U(x, t) as a function of x distinguishes its maximum for any fixed t > t ₀ and t < t ₀, respectively. Under nondegeneracy conditions on points of X _± we prove the existence of infinitely many doubly asymptotic trajectories connecting X _? and X ₊.     

Small-time sampling behavior of a Fleming-Viot process

The Fleming-Viot process with parent-independent mutation process is one particular neutral population genetic model.As time goes by,some initial species are replaced by mutated ones gradually.Once the population mutation rate is high,mutated species will elbow out all the initial species very quickly.Small-time behavior in this case seems to be the key to understand this fast transition.The small-time asymptotic results related to time scale t/θ and a(θ)t,where lim_θ→∞~(θa(θ))=0,are obtained by Dawson and Shui(1998,2001),Shui and Xiong(2002),and Xiang and Zhang(2005),respectively.Only the behavior under the scale t(θ),where lim_θ→∞~(t(θ))=0 and lim_θ→∞~(θt(θ))=∞,was left untouched.In this paper,the weak limits under various small-time scales are obtained.Of particular interest is the large deviations for the small-time transient sampling distributions,which reveal interesting phase transition.Interestingly,such a phase transition is uniquely determined by some species diversity indices.     

Torsional Rigidity for Regions with a Brownian Boundary

Let ?? ^m be the m-dimensional unit torus, m ∈ ?. The torsional rigidity of an open set Ω ? ?? ^m is the integral with respect to Lebesgue measure over all starting points x ∈ Ω of the expected lifetime in Ω of a Brownian motion starting at x. In this paper we consider Ω = ?? ^m \β[0, t], the complement of the path ß[0, t] of an independent Brownian motion up to time t. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as t → ∞. For m = 2 the main contribution comes from the components in ??²\β0, t] whose inradius is comparable to the largest inradius, while for m = 3 most of ??³\β[0, t] contributes. A similar result holds for m ≥ 4 after the Brownian path is replaced by a shrinking Wiener sausage W _r(t)[0, t] of radius r(t) = o(t ^-1/(m-2)), provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of ß[0, t] in ?³ and W ₁[0, t] in ? ^m , m ≥ 4, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on ?? ^m , which has received a lot of attention in the literature in past years.     

The 〈G<Emphasis Type="Italic">/M/r/</Emphasis>FIFO〉 Machine-Interference Model with State-Dependent Speeds

This article treats a version of the multiple machine-interference problem with r operatives under FIFO repair discipline. The running times of machine i are supposed to be identically and arbitrarily distributed random variables with density function f _i (x), i = 1,…, n. The repair times of all machines are assumed to be identically and exponentially distributed random variables with mean 1/μ. The paper provides the main steady-state operational characteristics of the system when the running and repair speeds are dependent on the number of machines in working order.     

Certain bivariate distributions and random processes connected with maxima and minima

The minimum and the maximum of t independent, identically distributed random variables have \(\bar F^{t}\) and F ^t for their survival (minimum) and the distribution (maximum) functions, where \(\bar F = 1-F\) and F are their common survival and distribution functions, respectively. We provide stochastic interpretation for these survival and distribution functions for the case when t >?0 is no longer an integer. A new bivariate model with these margins involve maxima and minima with a random number of terms. Our construction leads to a bivariate max-min process with t as its time argument. The second coordinate of the process resembles the well-known extremal process and shares with it the one-dimensional distribution given by F ^t . However, it is shown that the two processes are different. Some fundamental properties of the max-min process are presented, including a distributional Markovian characterization of its jumps and their locations.