| mttf {smmR} | R Documentation |
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 |
upstates |
Vector giving the subset of operational states |
level |
Confidence level of the asymptotic confidence interval. Helpful
for an object |
klim |
Optional. The time horizon used to approximate the series in the
computation of the mean sojourn times vector |
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