MATHEMATICS
On an algorithm for calculating optimal strategies on an infinite time interval
V. N. Gubin Tomsk Polytechnic University, Tomsk State University, Tomsk, Russian Federation
Abstract:
In this paper, a system where the interval between check times is discrete and constant is considered. The probability of failure for one element between check times is equal to
$p$. The redundancy criterion satisfies the following equation:
\begin{equation}
T(k,r)=\sum_{i=0}^{k-m}C_k^i p^{k-i}q^i T(r-i)+1,\tag{1}
\end{equation}
which is used for finding the function
$K_0(r)$.
Then, previous results related to properties of optimal strategies are stated. The main result of
the paper is the solution of the problem about saving the reserve consumption. In the case
$m=1$,
this problem was solved by the author earlier. To solve this problem in the general case, the
inequality
\begin{equation}
T(m+2,r)-T(m+1,r)\leqslant 0\tag{2}
\end{equation}
is used. Since
$T(r)$ can be found explicitly from the conditions of the problem, inequality (2) is
easy resolved. Therefore, the reserve interval $\left[m+1,m+2+\left[\frac{\ln C}{\ln A}\right]\right]$, where
$K_0(r)=m+1$, is
obtained. The algorithm for optimal strategy computing consists of the following steps:
- for $r=m$, we have $K_0(m)=m$ and $T(m)=p^m/(1-p^m)$.
- then, if we find $K_0(m+1)$, $K_0(m+2)$, …, and $K_0(r-1)$ to define $K_0(r)$, it is sufficient to
compare $f(K_0(r-1),r)\geqslant f(K_0(r-1)+1,r)$, where $f(k,r)=\frac{1}{1-p^k}\left(\sum\limits_{i=1}^{k-m}C_k^i p^{k-i}q^i T(r-i)+1\right)$.
Results of the numerical simulation are represented in the final section of the paper.
Keywords:
mean time between failures, element failure, system, reliability, redundancy strategy,
optimal strategy, redundancy criterion.
UDC:
519.873 Received: 22.02.2017
DOI:
10.17223/19988621/47/1
Fulltext:
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