On an algorithm for calculating optimal strategies on an infinite time interval

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.