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: k -m T(k,r) =X C'_kp^k 'q'T(r - i) +1, (1) i=0 which is used for finding the function K₀(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 T(m+2, r) - T(m+1, r)< 0 (2) is used. Since T(r) can be found explicitly from the conditions of the problem, inequality (2) is " ln C' where K0(r) = m+1, is m +1, m + 2 + easy resolved. Therefore, the reserve interval ln A obtained. The algorithm for optimal strategy computing consists of the following steps: 1) for r = m, we have K0(m) = m and T(m) = p^m/(1 - p^m). 2) then, if we find K₀(m+1), K₀(m+2), ... , and K₀(r-1) to define K₀(r), it is sufficient to 1 {k-m Л compare J(K₀(r-1), r) > fK„(r-1)+1, r), where f (k, r) =-^ ( g C_kp^k-iqⁱT(r - i) + 1j . Results of the numerical simulation are represented in the final section of the paper.

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