Reliability analysis of circular markov repairable systems with spatial dependence

Aim. The introduction of the full paper reviews a number of papers on reliability models with dependent units in the open literature and proposes what we believe to be a new model mentioned in the title. The core of Section 1 consists of: (1) we present the assumptions of the circular Markov repairable systems with spatial dependence; (2) we put forward the Markov processes associated with the 4-unit and 5-unit circular Markov repairable systems with spatial dependence and define their states as indicated in Fig.1 and Fig. 2. Section 2 analyzes the state transitions of the 4-unit and 5-unit systems and obtains the state transition rate matrices corresponding to them. Section 3 discusses the instantaneous and asymptotic availabilities of the 4-unit and 5-unit systems, which are given by equations (3)-(5). Section 4 presents the numerical examples of the 4-unit and 5-unit systems. The curves of instantaneous availabilities are shown in Fig. 5 and Fig. 6. We obtain the conclusions that: (1) the system with independent units is more reliable than the circular Markov repairable system with spatial dependence; (2) the 5-unit circular Markov repairable system is more reliable than the 4-unit system.