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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-5217
    Keywords: Reliability ; replacement ; optimal stopping
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Ersetzungsmodelle beschreiben im Rahmen der Zuverlässigkeitstheorie das Problem, einen optimalen Zeitpunkt für die vorbeugende Ersetzung eines technischen Systems festzulegen. Eine große Vielfalt solcher Modelle ist in letzter Zeit in der Literatur zu finden. Ziel dieser Arbeit ist es, einen einheitlichen Zugang zu dem Ersetzungsproblem anzugeben, indem dieses Problem als Problem des optimalen Stoppens aufgefaßt wird. Die Verallgemeinerung vorhandener Modelle erlaubt es, eine notwendige Monotoniebedingung abzuschwächen und unterschiedliche Informationsniveaus zu betrachten. An einem Beispiel wird gezeigt, wie sich explizite Lösungen des Ersetzungsproblems gewinnen lassen und wie sich das Informationsniveau auf die optimale Ersetzungsstrategie auswirkt.
    Notes: Summary Replacement models deal with the problem of finding the optimal random time for a preventive replacement of a technical system. A large class of such models has been considered in the literature recently. In this paper a unifying approach to the replacement problem is given regarding it as an optimal stopping problem. The generalization allows to weaken a necessary monotonicity condition and to consider different information levels. Measuring information in this context leads to discussing the problem in general terms of stochastic process theory. An example shows how to get explicit solutions and how the information level influences the optimal replacement policy.
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-5217
    Keywords: Generalized convexity ; duality ; root term ; optimal solution
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We extend the duality theorems for a class of nondifferentiable problems with Mond-Weir type duals.
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-5217
    Keywords: Stochastic Processes ; Amplitude Process ; Reliability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract This paper introduces a definition of reliability based on a process range. Thus, process failure is defined when the range of a process first reaches a given and unacceptable level. The Mean Time To Failure (MTTF) which is denned as the mean of the first time for a range to attain a given amplitude is then calculated for an asymmetric random walk process. The probability distribution of the range is then given and the process reliability over long periods of system operations are then calculated. Applications such as the control of wings movements, stock price and exchange rates volatility (defined in terms of reliability) are also used to motivate the usefulness of range processes in reliability studies. Finally, we point out that there is necessarily a relationship between the range reliability and the propensity of a series to become chaotic.
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  • 4
    ISSN: 1432-5217
    Keywords: Queu(e)ing System(s) ; Repairable System(s) ; Reliability ; Availability ; Insensitivity ; Light Traffic ; Fast Repair ; Special Random Process(es) ; Ergodic Theorem(s) ; Ergodicity ; Perturbation Analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract A repairable system is composed of components ofI types. A component can be loaded, put on standby, queued or repaired. The repair facility is here assumed to be a queueing system of a rather general structure though interruption of repairs is not allowed. Typei components possess a lifetime distributionA i (t) and repair time distributionB t (t). The lifetime of componentj is exhausted with a state-dependent rate α j (t). A Markov process Z(t) with supplementary variables is built to investigate the system behaviour. An ergodic result, Theorem 1, is established under a set of conditions convenient for light traffic analysis. In Theorems 2 to 6, a light traffic limit is derived for the joint steady state distribution of supplementary variables. Applying these results, Theorems 7 to 10 derive light traffic properties of a busy period-measured random variable. Essentially, the concepts of light traffic equivalence due to Daley and Rolski (1992) and Asmussen (1992) are used. The asymptotic (light traffic) insensitivity of busy period and steady state parameters to the form ofA i(t) [given their means and (in some cases) values of density functions for smallt], is observed under some analytic conditions.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-5217
    Keywords: Quasi-inverse ; epi-inverse ; hypo-inverse ; nondecreasing functions ; quasiconvex functions ; convex functions ; duality ; rearrangement ; modulus of continuity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung In dieser Arbeit wird systematisch die Umkehrung monoton nichtfallender Funktionenf: ℝ → ℝ ∪ {−∞, +∞} studiert. Die Ergebnisse bilden die Grundlage für eine neue Dualitätstheorie quasikonvexer Probleme [6]. Da jedoch die Fragestellung bei einer ganzen Anzahl weiterer Situationen auftritt, verdient sie eine gesonderte Behandlung. Anwendungen in der Topologie, Wahrscheinlichkeitstheorie, monotonen Umordnungen und in der konvexen Analysis werden aufgezeigt und skizziert.
    Notes: Abstract This work is devoted to a systematic study of the inversion of nondecreasing one variable extended real-valued functions. Its results are preparatory for a new duality theory for quasiconvex problem [6]. However the question arises in a variety of situations and as such deserves a separate treatment. Applications to topology, probability theory, monotone rearrangements, convex analysis are either pointed out or sketched.
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-5217
    Keywords: vectorial control-approximation problem ; vectorial location problem ; duality ; efficiency
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung In der Arbeit werden für eine Klasse von vektoriellen Steuer-Approximationsproblemen in reellen reflexiven Banachräumen vektorielle Dualprobleme konstruiert und Dualitätseigenschaften hergeleitet. Als Spezialfall ergeben sich entsprechende Aussagen für vektorielle Standortprobleme.
    Notes: Abstract The author formulates vectorial dual problems for a certain class of vectorial control-approximation problems in real reflexive Banach spaces. A number of propositions concerning duality are derived. Corresponding propositions are mentioned for the special case of the vectorial location problems.
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  • 7
    ISSN: 1432-5217
    Keywords: Markov decision processes ; countable state space ; Linear programming ; duality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We present an Linear Programming formulation of MDPs with countable state and action spaces and no unichain assumption. This is an extension of the Hordijk and Kallenberg (1979) formulation in finite state and action spaces. We provide sufficient conditions for both existence of optimal solutions to the primal LP program and absence of duality gap. Then, existence of a (possibly randomized) average optimal policy is also guaranteed. Existence of a stationary average optimal deterministic policy is also investigated.
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  • 8
    ISSN: 1432-5217
    Keywords: Reliability ; optimal stopping ; martingale
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract The assumption that components which are part of a technical system work independently seems not appropriate in a number of applications. A lot of multivariate extensions of the univariate exponential distribution have been suggested as lifetime distributions. But only the models of J. E. Freund and of A. W. Marshall and I. Olkin seem to be physically motivated. A combination of these approaches yields a bivariate lifetime distribution which is investigated in some detail. Applications of this bivariate lifetime model are considered in preventive maintenance. In a two-component parallel system the optimal replacement time shall be determined with respect to the total expected discounted cost criterion. Results of the theory of stochastic processes are used to obtain the optimal strategy for different information levels. Some numerical examples based on a two-component parallel system with dependent component lifetimes show how the optimal replacement policy depends on the different information levels and on the degree of dependence of the components.
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-5217
    Keywords: Optimization theory ; mathematical programming ; duality ; half spaces ; minimum norm duality ; separating hyperplanes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract This note shows that half spaces play a very special role in the development of duality. In addition to the minimum norm duality, the duality in linear programming, and Wolfe's and Johri's formulations in nonlinear programming can all be derived via half spaces by following an identical five step procedure.
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