This article delves into modeling a retrial \(\varvec{k}\) -out-of- \(\varvec{(\mathfrak {a}+\mathfrak {b}+\mathfrak {c})}\) : \(\varvec{G}\) repairable system which incorporates various standby modes using \(\varvec{MRSPNs}\) . Such a system consists of \(\varvec{(\mathfrak {a}+\mathfrak {b}+\mathfrak {c})}\) units and is considered operational provided that at least \(\varvec{k}\) units are functioning. The unit population is composed of \(\varvec{\mathfrak {a}}\) active components, \(\varvec{\mathfrak {b}}\) warm standby components, and \(\varvec{\mathfrak {c}}\) cold standby components. The proposed model considers corrective and preventive maintenance. When repairing a failing unit, the repairer may encounter breakdowns (active breakdowns). A failed unit is immediately repaired if the repairer is available; otherwise, if the repairer is repairing a failed unit, out of order or under preventive maintenance, the failed unit enters an \(\varvec{FCFS}\) orbit and waits for repair service after a certain duration. The main stationary probabilities for the system are obtained by adopting the \(\varvec{MRSPN}\) model and its underlying Markov regenerative process, followed by deriving the availability and various performance measures in the steady-state. To get the mathematical expressions that describe how reliably a system operates over time and how long it is expected to function before encountering its first failure, we utilize the Markov renewal equation along with the Laplace transform technique. Through a numerical example, we exhibit how system parameters affect the performance measures and the reliability indices.
