mttf: Mean Time To Failure (MTTF) Function

In corentin-dev/smmR: Simulation, Estimation and Reliability of Semi-Markov Models

Mean Time To Failure (MTTF) Function

Description

Consider a system S_{ystem} starting to work at time k = 0. The mean time to failure (MTTF) is defined as the mean lifetime.

Usage

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

Arguments

x	An object of S3 class smmfit or smm.
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 mean time to failure 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 work 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.

Let T_D denote the first passage time in subset D, called the lifetime of the system, i.e.,

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

The mean time to failure (MTTF) is defined as the mean lifetime, i.e., the expectation of the hitting time to down set D,

MTTF = E[T_{D}]

Value

A matrix with \textrm{card}(U) = s_{1} rows, and with columns giving values of the mean time to failure for each state i \in U, 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.

I. Votsi & A. Brouste (2019) Confidence interval for the mean time to failure in semi-Markov models: an application to wind energy production, Journal of Applied Statistics, 46:10, 1756-1773

corentin-dev/smmR documentation built on April 14, 2023, 11:36 p.m.