A Semi-Markov Model with Geometric Renewal Processes

Jingqi Zhang (1) , Mitra Fouladirad (2, 3) , Nikolaos Limnios (4)

1 LM2S - Laboratoire Modélisation et Sûreté des Systèmes
2 AMU - Aix Marseille Université
3 M2P2 - Laboratoire de Mécanique, Modélisation et Procédés Propres
4 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne

We consider a repairable system modeled by a semi-Markov process (SMP), where we include a geometric renewal process for system degradation upon repair, and replacement strategies for non-repairable failure or upon N repairs. First Pérez-Ocón and Torres-Castro studied this system (Pérez-Ocón and Torres-Castro in Appl Stoch Model Bus Ind 18(2):157–170, 2002) and proposed availability calculation using the Laplace Transform. In our work, we consider an extended state space for up and down times separately. This allows us to leverage the standard theory for SMP to obtain all reliability related measurements such as reliability, availability (point and steady-state), mean times and rate of occurrence of failures of the system with general initial law. We proceed with a convolution algebra, which allows us to obtain final closed form formulas for the above measurements. Finally, numerical examples are given to illustrate the methodology.

Mots clés: Semi-Markov process; Geometric process; Reliability; Semi-Markov process; semi-Markov process geometric process reliability; semi-Markov process; geometric process; reliability

Domaines: Informatique [cs]; Mathématiques [math]

Jingqi Zhang, Mitra Fouladirad, Nikolaos Limnios. A Semi-Markov Model with Geometric Renewal Processes. Methodology and Computing in Applied Probability, 2023, 25 (4), pp.85. ⟨10.1007/s11009-023-10060-z⟩. ⟨hal-04543367⟩