Limiting performance of a one-unit system under various repair models

We are mainly interested in the limiting availability of a one-unit repairable system supported by one spare and maintained through repair of failed units or preventive maintenance of recalled units, under a perfect repair policy. In Chapter 1, we investigate a one-unit system aided by an identical spare unit and serviced by a facility that performs perfect repair on a failed unit or preventive maintenance on a recalled unit. We extend the results in the existing literature to the most general distributional assumption by obtaining the system up time, the system down time and the limiting availability. Chapter 2 is the study of a one-unit system supported by an identical spare unit. It is perfectly repaired by an in-house repair person, if doable within a random or deterministic patience time, or else by a visiting expert. We study four models in terms of limiting availability and limiting profit per unit time, using semi-Markov processes (SMP), when all distributions are exponential. We show that a deterministic patience time is preferable to a random patience time, and characterize conditions under which the expert should repair multiple failed units (rather than only one failed unit) during each visit. In Chapter 3 we generalize the results of Chapter 2 to the case of arbitrary continuous life and repair time distributions. Our technique involves extending the limiting probability theorem of semi-Markov processes to that of extended semi-Markov processes. In Chapter 4 we try to generalize the results of Chapter 2 by employing the traditional Laplace transformation technique. While we can derive a formal expression of the limiting availability, obtaining the profit per unit time seems intractable with this approach. We demonstrate that our approach in Chapter 3 is superior to the Laplace transformation approach. In Chapter 5 we introduce the line digraph approach which methodically converts the continuous time stochastic process (CTSP) into an SMP (albeit on a different state space). Thereafter, standard limiting theorems for an SMP yield the steady state probabilities, which can be related back to those of the original CTSP. The line digraph approach is applicable to many other stochastic models