maintainability {smmR}

Maintainability Function

Description

For a reparable system S_{ystem} for which the failure occurs at time k = 0, its maintainability at time k \in N is the probability that the system is repaired up to time k, given that it has failed at time k = 0.

Usage

maintainability(x, k, upstates = x$states, level = 0.95, klim = 10000)

Arguments

x

An object of S3 class smmfit or smm.

k

A positive integer giving the period [0, k] on which the maintainability should be computed.

upstates

Vector giving the subset of operational states U.

level

Confidence level of the asymptotic confidence interval. Helpful for an object x of class smmfit.

klim

Optional. The time horizon used to approximate the series in the computation of the mean sojourn times vector m (cf. meanSojournTimes function) for the asymptotic variance.

Details

Consider a system (or a component) S_{ystem} whose possible states during its evolution in time are E = \{1,\dots,s\}. Denote by U = \{1,\dots,s_1\} the subset of operational states of the system (the up states) and by D = \{s_1 + 1,\dots, s\} the subset of failure states (the down states), with 0 < s_1 < s (obviously, E = U \cup D and U \cap D = \emptyset, U \neq \emptyset,\ D \neq \emptyset). One can think of the states of U as different operating modes or performance levels of the system, whereas the states of D can be seen as failures of the systems with different modes.

We are interested in investigating the maintainability of a discrete-time semi-Markov system S_{ystem}. Consequently, we suppose that the evolution in time of the system is governed by an E-state space semi-Markov chain (Z_k)_{k \in N}. The system starts to fail at instant 0 and the state of the system is given at each instant k \in N by Z_k: the event \{Z_k = i\}, for a certain i \in U, means that the system S_{ystem} is in operating mode i at time k, whereas \{Z_k = j\}, for a certain j \in D, means that the system is not operational at time k due to the mode of failure j or that the system is under the repairing mode j.

Thus, we take (\alpha_{i} := P(Z_{0} = i))_{i \in U} = 0 and we denote by T_U the first hitting time of subset U, called the duration of repair or repair time, that is,

T_U := \textrm{inf}\{ n \in N;\ Z_n \in U\}\ \textrm{and}\ \textrm{inf}\ \emptyset := \infty.

The maintainability at time k \in N of a discrete-time semi-Markov system is

M(k) = P(T_U \leq k) = 1 - P(T_{U} > k) = 1 - P(Z_{n} \in D,\ n = 0,\dots,k).

Value

A matrix with k + 1 rows, and with columns giving values of the maintainability, variances, lower and upper asymptotic confidence limits (if x is an object of class smmfit).

References

V. S. Barbu, N. Limnios. (2008). Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications - Their Use in Reliability and DNA Analysis. New York: Lecture Notes in Statistics, vol. 191, Springer.