availability: Availability Function
In corentin-dev/smmR: Simulation, Estimation and Reliability of Semi-Markov Models
availability 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 smmfit or smm.
k
A positive integer giving the time at which the availability
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 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.
corentin-dev/smmR documentation built on April 14, 2023, 11:36 p.m.
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| availability | 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.
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