On a problem of calculating a set of spare parts that have two types of failures

Abstract

Aim. The paper, that continues [24], aims to develop an algorithm that would allow finding the required number of spare items (SPTA) for a complex system, whose elements may or may not be maintainable. Unlike in [24], as a generalisation, the paper introduces additional inoperable states. Those states are characterised by system downtime associated with the replacement of a failed element with an element from the SPTA. If the time of replacement of a failed element is not a negligibly small value as compared to other time indices of the serviced system, it becomes necessary, as suggested, to account for additional inoperable states. Methods. Markov models are used for describing the technical system under consideration. The final probabilities were obtained using a developed system of Kolmogorov equations. A stationary solution was obtained for the system of Kolmogorov equations. Classical methods of the probability theory and mathematical theory of dependability, some special functions were used. Conclusions. The paper formalizes the problem of determining the required number of SPTAs for a system with items that may fail at a random moment in time. The failures may be of two types. The first type of failures leaves an item in an inoperable repairable state. In this case, the item can be repaired in the maintenance unit of the company that operates such item. The second type of failures, a more catastrophic one, leaves the item in an inoperable non-repairable state, and it can be repaired only by the manufacturer or a specialized maintenance company. A Markov graph was built for the respective birth and death process. Equations were formalised for typical states of the Markov graph. A stationary solution was obtained for the system of Kolmogorov equations using induction. The theorem of general solution was proven for all the states of the Markov graph. In case of unlimited repair, the solution is significantly simplified. It was shown that, on an assumption of unlimited repair and momentary replacements, the solution matches the one earlier obtained in a simplified form in [24]. The limit values of the probabilities of inoperable critical and non-critical states were found. They allow concluding that, in case of unlimited repair, the growth of the size of SPTA causes the probability of a critical state of insufficient SPTA to tend to and become zero. Additionally, the probability of an inoperable state associated with the replacement of a failed item with an equivalent from the SPTA that takes a certain time is defined by a stationary unavailability of the alternating repair process. The general solution of the problem allows formalising the SPTA sufficiency coefficient. The required number of SPTAs is identified by progressively increasing the number of SPTAs until the probability of inoperable critical states is below the defined probability of SPTA shortage. An example of finding the required number of SPTAs is given