On a class of reserved devices
In this paper, we consider three models of redundancy: (1) By use of the mean time between system failures on a finite interval; (2) By use of the mean time between system failures on a infinite interval; (3) By use of the system reliability on a finite interval. For all three models, the redundancy criterion has the following form: k -m T(k,r) =£ C[p <sup>k</sup> ''q''T(r - i). (1) i=0 Using the sigma-operator turns out to be an effective way for proving many properties of optimal strategies. Let T(r) > 0 and T(r) increase. The following properties are proved: T(r + 2) < T(r +1) <sup>( )</sup> T(r +1) T(r) '' Under more restrictive conditions, this inequality was obtained in the thesis of L.V. Ushakova. (4) The function ln T(r) is convex; (5) T(k +1,r) - T(k,r) increases with an increase in r. To find the optimal strategy, a simplified algorithm is obtained using the properties. This algorithm is based on a modification of the Bellman dynamic programming method mentioned in V.V. Travkina''s work. The essence of the algorithm is as follows. (1) If there are m elements, then we have k <sub>0</sub> (m) = m. For each model, T(m) is calculated. (2) All further calculations for the three models are similar. Then it is necessary to calculate the values of the function T(r) by means of its previous values using the formula 1 k -m T(k, r) =--- X Cip <sup>k-i</sup>qr (r - i), (2) 1 - p <sup>k</sup> 1=1 (3) Suppose that k <sub>0</sub>(m), k <sub>0</sub> (m + 1),...k <sub>0</sub> (r-1), T(m), T(m+1),..., T(r-1) have already calculated. To find k <sub>0</sub>(r), we need to calculate k <sub>0</sub>(r- 1) and k <sub>0</sub>(r-1)+1. Then, using (2), we calculate T(k <sub>0</sub>(r- 1), r) and T(k <sub>0</sub>(r- 1), r) and compare them. If T(k <sub>0</sub> (r-1), r) > T(k <sub>0</sub> (r-1)+1, r) then k <sub>0</sub>(r) = k <sub>0</sub>(r-1) and T(r) = T(k <sub>0</sub> (r-1), r). If T(k <sub>0</sub>(r-1)+1, r) > T(k <sub>0</sub> (r-1), r) then k <sub>0</sub>(r) = k <sub>0</sub>(r-1)+1, and T(r) = T(k <sub>0</sub>(r-1)+1, r).