failureRate: Failure Rate Function

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

Failure Rate Function

Description

Function to compute the BMP-failure rate or the RG-failure rate.

Consider a system S_{ystem} starting to work at time k = 0. The BMP-failure rate at time k \in N is the conditional probability that the failure of the system occurs at time k, given that the system has worked until time k - 1.

The RG-failure rate is a discrete-time adapted failure rate, proposed by D. Roy and R. Gupta. Classification of discrete lives. Microelectronics Reliability, 32(10):1459–1473, 1992. We call it the RG-failure rate and denote it by r(k),\ k \in N.

Usage

failureRate(
  x,
  k,
  upstates = x$states,
  failure.rate = c("BMP", "RG"),
  level = 0.95,
  epsilon = 0.001,
  klim = 10000
)

Arguments

x	An object of S3 class smmfit or smm.
k	A positive integer giving the period [0, k] on which the BMP-failure rate should be computed.
upstates	Vector giving the subset of operational states U.
failure.rate	Type of failure rate to compute. If failure.rate = "BMP" (default value), then BMP-failure-rate is computed. If failure.rate = "RG", the RG-failure rate is computed.
level	Confidence level of the asymptotic confidence interval. Helpful for an object x of class smmfit.
epsilon	Value of the reliability above which the latter is supposed to be 0 because of computation errors (see Details).
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 failure rate 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.

The BMP-failure rate at time k \in N is the conditional probability that the failure of the system occurs at time k, given that the system has worked until time k - 1.

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.

For a discrete-time semi-Markov system, the failure rate at time k \geq 1 has the expression:

\lambda(k) := P(T_{D} = k | T_{D} \geq k)

We can rewrite it as follows :

\lambda(k) = 1 - \frac{R(k)}{R(k - 1)},\ \textrm{if } R(k - 1) \neq 0;\ \lambda(k) = 0, \textrm{otherwise}

The failure rate at time k = 0 is defined by \lambda(0) := 1 - R(0), with R being the reliability function (see reliability function).

The calculation of the reliability R involves the computation of many convolutions. It implies that the computation error, may be higher (in value) than the "true" reliability itself for reliability close to 0. In order to avoid inconsistent values of the BMP-failure rate, we use the following formula:

\lambda(k) = 1 - \frac{R(k)}{R(k - 1)},\ \textrm{if } R(k - 1) \geq \epsilon;\ \lambda(k) = 0, \textrm{otherwise}

with \epsilon, the threshold, the parameter epsilon in the function failureRate.

Expressing the RG-failure rate r(k) in terms of the reliability R we obtain that the RG-failure rate function for a discrete-time system is given by:

r(k) = - \ln \frac{R(k)}{R(k - 1)},\ \textrm{if } k \geq 1;\ r(k) = - \ln R(0),\ \textrm{if } k = 0

for R(k) \neq 0. If R(k) = 0, we set r(k) := 0.

Note that the RG-failure rate is related to the BMP-failure rate by:

r(k) = - \ln (1 - \lambda(k)),\ k \in N

Value

A matrix with k + 1 rows, and with columns giving values of the BMP-failure rate or RG-failure rate, 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.

R.E. Barlow, A.W. Marshall, and F. Prochan. (1963). Properties of probability distributions with monotone hazard rate. Ann. Math. Statist., 34, 375-389.

D. Roy and R. Gupta. (1992). Classification of discrete lives. Microelectron. Reliabil., 32 (10), 1459-1473.

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