| availability {smmR} | R Documentation |
Availability Function
Description
The pointwise (or instantaneous) availability of a system
S_{ystem} at time k \in N is the probability that the system
is operational at time k (independently of the fact that the system
has failed or not in [0, k)).
Usage
availability(x, k, upstates = x$states, level = 0.95, klim = 10000)
Arguments
x |
An object of S3 class |
k |
A positive integer giving the time at which the availability should be computed. |
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 availability 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 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.
The pointwise (or instantaneous) availability of a system S_{ystem}
at time k \in N is the probability that the system is operational
at time k (independently of the fact that the system has failed or
not in [0, k)).
Thus, the pointwise availability of a semi-Markov system at time
k \in N is
A(k) = P(Z_k \in U) = \sum_{i \in E} \alpha_i A_i(k),
where we have denoted by A_i(k) the conditional availability of the
system at time k \in N, given that it starts in state i \in E,
A_i(k) = P(Z_k \in U | Z_0 = i).
Value
A matrix with k + 1 rows, and with columns giving values of
the availability, 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.