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 \([0, k]\).
reliability(x, k, upstates = x$states, level = 0.95, klim = 10000)An object of S3 class smmfit or smm.
A positive integer giving the period \([0, k]\) on which the reliability should be computed.
Vector giving the subset of operational states \(U\).
Confidence level of the asymptotic confidence interval. Helpful
for an object x of class smmfit.
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.
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).
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:
$$R(k) = \sum_{i \in U} P(Z_0 = i) P(T_D > k | Z_0 = i) = \sum_{i \in U} \alpha_i P(T_D > k | Z_0 = i)$$
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.