scholarly journals A Markov time related to a priority system

2006 ◽  
Vol 2006 ◽  
pp. 1-9 ◽  
Author(s):  
Edmond J. Vanderperre
Keyword(s):  
Stochastic Process ◽  
Differential Equations ◽  
Stochastic Analysis ◽  
Recovery Time ◽  
Holomorphic Functions ◽  
Priority Rule ◽  
Stieltjes Transform ◽  
Duplex System ◽  
Cold Standby ◽  
A Priority

We consider a basic renewable duplex system characterized by cold standby and subjected to a priority rule. Apart from a general stochastic analysis presented in the previous literature, we introduce a Markov time called the recovery time of the system. In order to obtain the corresponding Laplace-Stieltjes transform, we employ a stochastic process endowed with transition measures satisfying generalized coupled differential equations. The solution is provided by the theory of sectionally holomorphic functions.

scholarly journals Long-run availability of a priority system: a numerical approach

2005 ◽  
Vol 2005 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Edmond J. Vanderperre ◽  
Stanislav S. Makhanov
Keyword(s):  
Stochastic Process ◽  
Numerical Approach ◽  
Priority Rule ◽  
Twin System ◽  
System A ◽  
Long Run ◽  
Random Behavior ◽  
Cold Standby ◽  
Standby System ◽  
A Priority

We consider a two-unit cold standby system attended by two repairmen and subjected to a priority rule. In order to describe the random behavior of the twin system, we employ a stochastic process endowed with state probability functions satisfying coupled Hokstad-type differential equations. An explicit evaluation of the exact solution is in general quite intricate. Therefore, we propose a numerical solution of the equations. Finally, particular but important repair time distributions are involved to analyze the long-run availability of theT-system. Numerical results are illustrated by adequate computer-plotted graphs.

scholarly journals Reliability of an engineering system characterized by hot and cold standby: A compound boundary value problem

2021 ◽  
pp. 16-16
Author(s):  
Edmond Vanderperre ◽  
Stanislav Makhanov
Keyword(s):  
Boundary Value Problem ◽  
Survival Time ◽  
Boundary Value ◽  
Survival Function ◽  
Holomorphic Functions ◽  
Engineering System ◽  
Entire System ◽  
Duplex System ◽  
Cold Standby ◽  
Auxiliary Unit

We analyse the reliability (survival function) of a duplex system characterized by hot standby and sustained by an auxiliary unit in cold standby. The entire system is attended by two heterogeneous repairmen. Our methodology is based on the theory of sectionally holomorphic functions combined with the notion of dual transforms. Finally, we also study the total occupational time of the repairman responsible for the repair of the failed priority unit during the survival time of the system.

scholarly journals Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction

2019 ◽  
Vol 52 (1) ◽  
pp. 482-489 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv ◽  
Liana Smolovyk
Keyword(s):  
Partial Differential Equations ◽  
Continuous Function ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Positive Continuous Function ◽  
Partial Differential ◽  
Partial Derivatives ◽  
Fixed Direction

AbstractIn the paper we investigate slice holomorphic functions F : ℂn → ℂ having bounded L-index in a direction, i.e. these functions are entire on every slice {z0 + tb : t ∈ℂ} for an arbitrary z0 ∈ℂn and for the fixed direction b ∈ℂn \ {0}, and (∃m0 ∈ ℤ+) (∀m ∈ ℤ+) (∀z ∈ ℂn) the following inequality holds{{\left| {\partial _{\bf{b}}^mF(z)} \right|} \over {m!{L^m}(z)}} \le \mathop {\max }\limits_{0 \le k \le {m_0}} {{\left| {\partial _{\bf{b}}^kF(z)} \right|} \over {k!{L^k}(z)}},where L : ℂn → ℝ+ is a positive continuous function, {\partial _{\bf{b}}}F(z) = {d \over {dt}}F\left( {z + t{\bf{b}}} \right){|_{t = 0}},\partial _{\bf{b}}^pF = {\partial _{\bf{b}}}\left( {\partial _{\bf{b}}^{p - 1}F} \right)for p ≥ 2. Also, we consider index boundedness in the direction of slice holomorphic solutions of some partial differential equations with partial derivatives in the same direction. There are established sufficient conditions providing the boundedness of L-index in the same direction for every slie holomorphic solutions of these equations.

scholarly journals Solutions de similitude d'un jeu différentiel stochastique

2006 ◽  
Vol Volume 5, Special Issue TAM... ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Similarity Solutions ◽  
Explicit Solutions ◽  
Two Dimensional ◽  
Controlled Processes ◽  
International Audience ◽  
First Time

International audience A two-dimensional controlled stochastic process defined by a set of stochastic differential equations is considered. Contrary to the most frequent formulation, the control variables appear only in the infinitesimal variances of the process, rather than in the infinitesimal means. The differential game ends the first time the two controlled processes are equal or their difference is equal to a given constant. Explicit solutions to particular problems are obtained by making use of the method of similarity solutions to solve the appropriate partial differential equation. On considère un processus stochastique commandé bidimensionnel défini par un ensemble d'équations différentielles stochastiques. Contrairement à la formulation la plus fréquente, les variables de commande apparaissent dans les variances infinitésimales du processus, plutôt que dans les moyennes infinitésimales. Le jeu différentiel prend fin lorsque les deux processus sont égaux ou que leur différence est égale à une constante donnée. Des solutions explicites à des problèmes particuliers sont obtenues en utilisant la méthode des similitudes pour résoudre l'équation aux dérivées partielles appropriée.

Pathwise convergent higher order numerical schemes for random ordinary differential equations

2007 ◽  
Vol 463 (2087) ◽  
pp. 2929-2944 ◽  
Author(s):  
Peter E Kloeden ◽  
Arnulf Jentzen
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Vector Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Time Variable ◽  
Numerical Schemes ◽  
Hölder Continuous ◽  
Usual Order

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) with a stochastic process in their vector field. They can be analysed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable, so traditional numerical schemes for ODEs do not achieve their usual order of convergence when applied to RODEs. Nevertheless deterministic calculus can still be used to derive higher order numerical schemes for RODEs via integral versions of implicit Taylor-like expansions. The theory is developed systematically here and applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes.

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