| reliability {smmR} | R Documentation |
Reliability Function
Description
Consider a system S_{ystem} starting to function at time
k = 0. The reliability or the survival function of S_{ystem}
at time k \in N is the probability that the system has functioned
without failure in the period [0, k].
Usage
reliability(x, k, upstates = x$states, level = 0.95, klim = 10000)
Arguments
x |
An object of S3 class |
k |
A positive integer giving the period |
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 reliability 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 reliability or the survival function at time k \in N of a
discrete-time semi-Markov system is:
R(k) := P(T_D > k) = P(Zn \in U,n = 0,\dots,k)
which can be rewritten as follows:
R(k) = \sum_{i \in U} P(Z_0 = i) P(T_D > k | Z_0 = i) = \sum_{i \in U} \alpha_i P(T_D > k | Z_0 = i)
Value
A matrix with k + 1 rows, and with columns giving values of
the reliability, 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.