Stochastic Analysis of a Three-Unit Non-Identical Repairable System with Simultaneous Working of Cold Standby Units

Abstract

Here, stochastic analysis of a repairable system of three units has been carried out by taking one unit in operation (called main unit) and two identical units (called duplicate units) in cold standby. The switch device is used to convert the standby units into operative mode. A single server is hired to handle repair activities of the units who visits the system instantly whenever needed. The repair done by the server is perfect and thus the repaired unit follows the same lifetime distribution as the original. The constant failure rates are considered for both main and the duplicate units while their repair time distributions are taken as arbitrary. Some important reliability measures including mean sojourn times (MST), transition probabilities (TP), mean time to system failure (MTSF), availability, expected number of repairs for both kinds of units separately, expected number of visits by the server and busy period analysis of the server due to repair are determined using semi-Markov process (SMP) and regenerative point technique (RPT). The arbitrary values of the parameters are considered to examine the behaviour of some significant reliability measures through graphs. The possible application of the system model can be visualized in a power supply system of a house where a set of solar panels are kept in spare for their simultaneously working when main power supply is discontinued.

A System with Two Spare Units, Two Repair Facilities, and Two Types of Repairers

“…Using the Laplace transform technique, they claim to derive some reliability measures, including availability and profit, for arbitrary failure-and repair time distributions; however, they do not derive analytic expressions for the reliability measures but only calculate the results under exponential distributions. Kadyan et al [15] extend the work by giving priority of operation to the main unit and showing that when the repair rate of the main unit increases, so does the system reliability.…”

“…Kadyan et al [14] study a one-unit redundant system having a single operating (called main) unit supported by two identical units (called duplicate) on cold standby which are nonidentical to the main unit. Upon failure of the main unit, both duplicate units are put on operation, while a single repairer perfectly repairs the failed main unit.…”

Stochastic analysis of a non-identical three units repairable system with operational priority to simultaneously working of identical units

“…“Let ϕ i (t) be the cdf of the first passage time from regenerative state i to a failed state. Regarding the failed state as absorbing state, we have the following recursive relations for ϕ i (t):” (Kadyan et al. , 2020a)…”

“…In reliability literature systems with identical standby units and non-identical standby units are quite popular (Bhardwaj and Singh, 2015; Kumar and Malik, 2018; Kumar and Kadyan, 2012) but no one talked of a system with non-identical simultaneous working units which was considered by Kadyan et al. (2020a, b) but preference was given to the single unit for operation and no priority respectively while in many situations preference is given to the simultaneously working units as described above so it becomes necessary to study a system with operational priority to simultaneous working units. As a similar system has been analysed by Kadyan et al.…”

“…(2020a, b) but preference was given to the single unit for operation and no priority respectively while in many situations preference is given to the simultaneously working units as described above so it becomes necessary to study a system with operational priority to simultaneous working units. As a similar system has been analysed by Kadyan et al. (2020a) where operational priority was given to transformer so it becomes necessary to make a comparative analysis of profit of the systems being used under two different operational policies.…”