Preventive maintenance in a 1 out of n system: The uptime,downtime and costs

In repairable systems with redundancy, failed units can be replaced by spare units in order to reduce the system downtime. The failed units are sent to a repair shop or manufacturer for corrective maintenance and subsequently are returned for re-use. In this paper we consider a 1 out of n system with cold standby and we assume that repaired units are “as good as new”.When a unit has an increasing failure rate it can be advantageous to perform preventive maintenance in order to return it to its “as good as new” state, because preventive maintenance will take less time and tends to be cheaper. In the model we present we use age-replacement; a machine is taken out for preventive maintenance and replaced by a standby one if its age has reached a certain value, T_pm. In this paper we derive an approximation scheme to compute the expected uptime, the expected downtime and the expected costs per time unit of the system, given the total number of units and the age-replacement value, T_pm. Consequently the number of units and the value T_pm can be determined for maximum long-term economy.     

Factorizing τ-spectra of mappings

In this paper by a spectrum of mappings we mean a morphism of spectra of spaces. However, using the notion of a mapping of mappings, we give the definition of a spectrum of mappings similar to that of a spectrum of spaces. In this case, the formulations of the given results are also similar to the formulations of the corresponding results concerning the spectra of spaces.For the spectra of mappings we define the notion of a τ-spectrum of mappings factorizing in a special sense and prove a version of the Spectral Theorem for such spectra. Furthermore, to a given indexed collection F of mapping we associate a τ-spectrum factorizing in the above special sense whose mappings are Containing Mappings for F constructed in Iliadis (2005) [4]. These associated τ-spectra and the corresponding version of the Spectral Theorem imply that for a given indexed collection F of mappings any so-called “natural” τ-spectrum for F factorizing in the special sense contains a cofinal and τ-closed subspectrum whose mappings are Containing Mapping for F. Thus, Containing Mappigs for F appear here without any concrete construction. The associated τ-spectra are used also in order to define and characterize the so-called second-type saturated classes of mappings (which are “saturated” by universal elements).     

Some inequalities for the Hadamard product and the Fan product of matrices

If A and B are nonsingular M-matrices, a sharp lower bound on the smallest eigenvalue τ(A★B) for the Fan product of A and B is given, and a sharp lower bound on τ(A°B^-1) for the Hadamard product of A and B^-1 is derived. In addition, we also give a sharp upper bound on the spectral radius ρ(A°B) for nonnegative matrices A and B.     

Maintenance models based on the <Emphasis Type="Italic">np</Emphasis> control charts with respect to the sampling interval

This paper develops models for the maintenance of a system based on np control charts with respect to the sampling interval. At any given time, the system is assumed to be in one of the three possible states; in-control, out-of-control and failure. If the control chart signals, suggesting the possibility of an out-of-control state, an investigation will be carried out. We assume that this investigation is perfect in that it reveals the true state of the system. If an assignable cause is confirmed by the investigation, a minor repair will be carried out to remove the cause. If the assignable cause is not attended to, it will gradually develop into a failure. When a failure occurs, the system cannot operate and a major repair is needed. We discuss three models depending on the assumptions related to the renewal mechanism, the occurrence of failures, and the time between minor repairs. The paper seeks to optimise the performance of such a system in terms of the sampling interval. Geometric processes are utilised for modelling the lifetimes between minor repairs if the minor repair cannot bring the system back to an as good as new condition. The expected cost per unit time for maintaining the systems with respect to the sampling interval of the control chart is obtained. Numerical examples are conducted to demonstrate the applicability of the methodology derived.     

A general model for consecutive-k-out-of-n: F repairable system with exponential distribution and (k−1)-step Markov dependence

In this paper, a general model for consecutive-k-out-of-n: F repairable system with exponential distribution and (k−1)-step Markov dependence is introduced. The lifetime of a component is an exponential random variable, its parameter depends on the number of consecutive failed components that precede the component. The repair time is also an exponential random variable. A priority repair rule on the basis of the system failure risk is adopted. Then the transition density matrix of the system is determined. Some reliability indices, including the system availability, rate of occurrence of failures and reliability are evaluated accordingly. For the demonstration of the model and methodology, a linear system example and a circular system example are investigated.     

Asymptotics of repeated interaction quantum systems

A quantum system S interacts in a successive way with elements E of a chain of identical independent quantum subsystems. Each interaction lasts for a duration τ and is governed by a fixed coupling between S and E. We show that the system, initially in any state close to a reference state, approaches a repeated interaction asymptotic state in the limit of large times. This state is τ-periodic in time and does not depend on the initial state. If the reference state is chosen so that S and E are individually in equilibrium at positive temperatures, then the repeated interaction asymptotic state satisfies an average second law of thermodynamics.     

Transversal numbers of translates of a convex body

Let F be a family of translates of a fixed convex set M in Rⁿ. Let τ(F) and ν(F) denote the transversal number and the independence number of F, respectively. We show that ν(F)?τ(F)?8ν(F)-5 for n=2 and τ(F)?2^n-1nⁿν(F) for n?3. Furthermore, if M is centrally symmetric convex body in the plane, then ν(F)?τ(F)?6ν(F)-3.     

A deteriorating cold standby repairable system with priority in use

In this paper, a cold standby repairable system consisting of two dissimilar components and one repairman is studied. In this system, it is assumed that the working time distributions and the repair time distributions of the two components are both exponential and component 1 is given priority in use. After repair, component 2 is “as good as new” while component 1 follows a geometric process repair. Under these assumptions, using the geometric process and a supplementary variable technique, some important reliability indices such as the system availability, reliability, mean time to first failure (MTTFF), rate of occurrence of failure (ROCOF) and the idle probability of the repairman are derived. A numerical example for the system reliability R(t) is given. And it is considered that a repair-replacement policy based on the working age T of component 1 under which the system is replaced when the working age of component 1 reaches T. Our problem is to determine an optimal policy T^∗ such that the long-run average cost per unit time of the system is minimized. The explicit expression for the long-run average cost per unit time of the system is evaluated, and the corresponding optimal replacement policy T^∗ can be found analytically or numerically. Another numerical example for replacement model is also given.     

Pathwidth, trees, and random embeddings

We prove that, for every integer k≥1, every shortest-path metric on a graph of pathwidth k embeds into a distribution over random trees with distortion at most c=c(k), independent of the graph size. A well-known conjecture of Gupta, Newman, Rabinovich, and Sinclair [12] states that for every minor-closed family of graphs F, there is a constant c(F) such that the multi-commodity max-flow/min-cut gap for every flow instance on a graph from F is at most c(F). The preceding embedding theorem is used to prove this conjecture whenever the family F does not contain all trees.     

Schrödinger equations with unique positive isolated singularities

We investigate the class of nonnegative potentialsV(x) for which the Schrödinger equation ?Δu+V u=0 admits a unique type of singular solution such thatu(x)→∞ asx→0. This class includes the potentials with inverse-square growth at 0, i.e. 0≤V(x)≤C|x|^?2. If for instance we fix boundary datau=g at |x|=1 then the singular solution is unique up to a multiplicative factor.     

Sp(2N)-covers for self-contragredient supercuspidal representations of GL(N)

Let F be a non-archimedean local field of odd residual characteristic. Let (J,τ) be a maximal simple type in GLN(F) for the inertial class [GLN(F),π]_GLN(F) of a self-contragredient supercuspidal irreducible representation π of GLN(F). Identify GLN(F) to the standard Siegel Levi subgroup in Sp2N(F). We construct, in Sp2N(F), a type for the inertial class [GLN(F),π]_Sp2N(F), as a Sp2N(F)-cover of (J,τ), strongly related to the GL2N(F)-cover of (J×J,τ⊗τ) in GL2N(F) constructed by Bushnell and Kutzko and which induces to a simple type in GL2N(F). In the process, we show that if τ has positive level, then the maximal simple type (J,τ) may be attached to a simple stratum [A,n,0,β] where the field F[β] is a quadratic extension of F[β²], and to a simple character θ in C(A,0,β) Galois conjugate of its inverse.     

Conjugate transforms for τT semigroups of probability distribution functions

This paper introduces the use of conjugate transforms in the study of τ_T semigroups of probability distribution functions. If Δ⁺ denotes the space of one-dimensional distribution functions concentrated on [0, ∞) and T is a t-norm, i.e., a suitable binary operation on [0, 1], then the operation τ_T is defined for F, G in Δ⁺by τ_T(F, G)(x) = sup_{u+v = x}T(F(u), G(v)) for all x. The pair (Δ⁺, τ_T) is then a semigroup. For any Archimedean t-norm T, a conjugate transform C_T is defined on (Δ⁺, τ_T). These transforms are shown to play a role similar to that played by the Laplace transform on the convolution semigroup. Thus a theory of “characteristic functions” for τ_T semigroups is developed. In addition to establishing their basic algebraic properties, we also use conjugate transforms to study the algebraic questions of the cancellation law, infinitely divisible elements, and solutions of equations in τ_T semigroups.     

Interval uneffectiveness distribution for a k-out-of-n multistate reliability system with repair

This paper deals with a parallel load-sharing reliability system with cold standby redundancy and ample repair facilities. That is, we have n identical parallel units, of which at most k units are operating simultaneously. If less than k units are available, the system operates at a proportionally reduced level. For this system, an approximate method is given for the calculation of the probability distribution of that proportion of the system capacity that cannot be used in a given time period. The method is based on an approximation of the k-out-of-n multistate system by a two-state single component. Validation of the approximation using Monte-Carlo simulation shows satisfactory performance. Also, sensitivity results are given, showing in particular a decreasing sensitivity of the measures of performance to the distributional form of the unit lifetimes and repair times as the size of the system increases. Furthermore, it is found that the effect of the distributional form of the unit lifetimes dominates that of the unit repair times.     

Formes antihermitiennes devenant hyperboliques sur un corps de déploiement

Let H=(a,b)_F be a division quaternion algebra over a field F of characteristic not 2. Denote by τ the canonical involution on H and by K a splitting field of H. If h is a skew-hermitian form over (H,τ) then, by extension of scalars to K and by Morita equivalence, we obtain a quadratic form h_K over K. This gives a map of Witt groups ρ:W⁻¹(H,τ)→W(K) induced by ρ(h)=h_K. When K is a generic splitting field of H we prove in this note that the map ρ is injective.     

Computability-theoretic and proof-theoretic aspects of partial and linear orderings

Szpilrajn’s Theorem states that any partial orderP=〈S,<p〉 has a linear extensionP=〈S,<L〉. This is a central result in the theory of partial orderings, allowing one to define, for instance, the dimension of a partial ordering. It is now natural to ask questions like “Does a well-partial ordering always have a well-ordered linear extension?” Variations of Szpilrajn’s Theorem state, for various (but not for all) linear order typesτ, that ifP does not contain a subchain of order typeτ, then we can chooseL so thatL also does not contain a subchain of order typeτ. In particular, a well-partial ordering always has a well-ordered extension.We show that several effective versions of variations of Szpilrajn’s Theorem fail, and use this to narrow down their proof-theoretic strength in the spirit of reverse mathematics.     

Hitting Time Distributions for Denumerable Birth and Death Processes

For an ergodic continuous-time birth and death process on the nonnegative integers, a well-known theorem states that the hitting time T _0,n starting from state 0 to state n has the same distribution as the sum of n independent exponential random variables. Firstly, we generalize this theorem to an absorbing birth and death process (say, with state ?1 absorbing) to derive the distribution of T _0,n . We then give explicit formulas for Laplace transforms of hitting times between any two states for an ergodic or absorbing birth and death process. Secondly, these results are all extended to birth and death processes on the nonnegative integers with ?? an exit, entrance, or regular boundary. Finally, we apply these formulas to fastest strong stationary times for strongly ergodic birth and death processes.     

Asymptotics for the Tail Probability of Random Sums with a Heavy-Tailed Random Number and Extended Negatively Dependent Summands

Let {X,X_k:k≥1} be a sequence of extended negatively dependent random variables with a common distribution F satisfying EX 0.Let r be a nonnegative integer-valued random variable,independent of {X,X_k:k≥1}.In this paper,the authors obtain the necessary and sufficient conditions for the random sums S_r =(?)X_n to have a consistently varying tail when the random number t has a heavier tail than the summands,i.e.,(P(Xx))/(P(r x))→0as x→∞     

Oscillation theorems for second-order functional differential equations

This paper presents some comparison theorems on the oscillatory behavior of solutions of second-order functional differential equations. Here we state one of the main results in a simplified form: Let q, τ₁, τ₂ be nonnegative continuous functions on (0, ∞) such that τ₁ ? τ₂ is a bounded function on [1, ∞) and t ? τ₁(t) → ∞ if t → ∞. Then $\text{y}\text{?}\text{(t) + q(t) y(t ? τ}{}_1\text{(t)) = 0}$ is oscillatory if and only if $\text{y}\text{?}\text{(t) + q(t) y(t ? τ}{}_2\text{(t)) = 0}$ is oscillatory.     

Robustness of F-tests in singular linear models

F-test is the most popular test in the general linear model. However, there is few discussions on the robustness of F-test under the singular linear model. In this paper, the necessary and sufficient conditions of robust F-test statistic are given under the general linear models or their partition models, which allows that the design matrix has deficient rank and the covariance matrix of error is a nonnegative definite matrix with parameters. The main results obtained in this paper include the existing findings of the general linear model under the definite covariance matrix. The usage of the theorems is illustrated by an example.     

Optimal replacement policy for a multistate repairable system

In this paper, a deteriorating simple repairable system with k + 1 states, including k failure states and one working state, is studied. The system after repair is not ‘as good as new’ and the deterioration of the system is stochastic. Under these assumptions, we study a replacement policy, called policy N, based on the failure number of the system. The objective is to maximize the long-run expected profit per unit time. The explicit expression of the long-run expected profit per unit time is derived and the corresponding optimal solution may be determined analytically or numerically. Furthermore, we prove that the model for the multistate system in this paper forms a general monotone process model which includes the geometric process repair model as a special case. A numerical example is given to illustrate the theoretical results.