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Sirous Fathi Manesh

Sirous Fathi Manesh

Academic rank: Assistant Professor
ORCID:
Education: PhD.
ScopusId: 6473
HIndex:
Faculty: Faculty of Science
Address: Department of Statistics, Faculty of Sciences, University of Kurdistan
Phone:

Research

Title
Stochastic Comparisons of Replacement Policies
Type
Presentation
Keywords
Arrangement increasing density function, Log concave density function, Majorization, Preventative maintenance, Schur convcave function, Standby system, Stochastic orderings.
Year
2015
Researchers Sirous Fathi Manesh ، Baha-Eldin Khaledi

Abstract

‎Consider a system that has one component in operation and $n-1$ components in stand by condition‎. ‎To improve the reliability of this system‎, ‎we use the preventive maintenance ($PM$) with block replacement policy ${\bf a}=(a_1‎ , ‎\ldots a_n)$ such that $\sum_{i=1}^{n}a_{i}=a$‎. ‎That is‎, ‎we replace the $i^,$th component by $i+1^,$th component either it fails or at time $a_i$‎. ‎Now‎, ‎let $X_i‎, ~ ‎i=1,\ldots‎ , ‎n$ be the life time of the $i^,$th component and every component is successfully switched with probability one‎. ‎Then‎, ‎the life time of this system is $\sum_{i=1}^{n} (X_i\wedge a_i)$ where $(X_i\wedge a_i)=\min (X_i‎, ‎a_i)$‎. ‎In this paper‎, ‎we compare different allocation of $(a_1‎ , ‎\ldots a_n)$ to find the best or worst allocation by the sense of expectation or usual stochastic ordering and prove the following results.\\‎ % ‎\begin{itemize}‎ % ‎\item{1.}‎ ‎1‎. ‎If $X_1,\ldots,X_n$ are exchangeable‎, ‎then‎ ‎$${\bf a}\geq_m {\bf a}^* \Longrightarrow \sum_{i=1}^{n} (X_i\wedge a_i) \leq_{icv} \sum_{i=1}^{n} (X_i\wedge a_i^*).$$‎ ‎2‎. ‎If $X_1,\ldots,X_n$ are exchangeable and have log-concave joint density function‎, ‎then‎ ‎$${\bf a}\geq_m {\bf a}^* \Longrightarrow \sum_{i=1}^{n} (X_i\wedge a_i) \leq_{st} \sum_{i=1}^{n} (X_i\wedge a_i^*).$$‎ ‎As intuitively expected‎, ‎under the conditions $(1)$ and $(2)$ the best allocation is $(\bar{a}‎, ‎\ldots‎, ‎\bar{a})$ by respect to expectation and survival function of life time of the system‎, ‎respectively.\\‎ ‎3‎. ‎If $X_1,\ldots,X_n$ have arrangement increasing joint density function and $(a_1^*‎, ‎\ldots‎, ‎a_n^*)$ is the best allocation by the sense of usual stochastic ordering‎, ‎then $a_1^*\leq a_2^*‎, ‎\ldots \leq a_n^*$.\\‎ ‎4‎. ‎If $X_1,\ldots,X_n$ are have arrangement increasing and log-concave joint density function‎, ‎then‎ ‎$${\bf a}\geq_m {\bf a}^* \Longrightarrow \sum_{i=1}^{n} (X_i\wedge a_{(n-i+1)}) \leq_{st} \sum_{i=1}^{n} (X_i\wedge a_i^*).$$‎