The mean sojourn time is the mean time spent in each state.
meanSojournTimes(x, states = x$states, klim = 10000)An object of S3 class smmfit or smm.
Vector giving the states for which the mean sojourn time
should be computed. states is a subset of \(E\).
Optional. The time horizon used to approximate the series in the computation of the mean sojourn times vector \(m\) (cf. meanSojournTimes function).
A vector of length \(\textrm{card}(E)\) giving the values of the mean sojourn times for each state \(i \in E\).
Consider a system (or a component) \(S_{ystem}\) whose possible states during its evolution in time are \(E = \{1,\dots,s\}\).
We are interested in investigating the mean sojourn times 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 \(\{Z_k = i\}\).
Let \(T = (T_{n})_{n \in N}\) denote the successive time points when state changes in \((Z_{n})_{n \in N}\) occur and let also \(J = (J_{n})_{n \in N}\) denote the successively visited states at these time points.
The mean sojourn times vector is defined as follows:
$$m_{i} = E[T_{1} | Z_{0} = j] = \sum_{k \geq 0} (1 - P(T_{n + 1} - T_{n} \leq k | J_{n} = j)) = \sum_{k \geq 0} (1 - H_{j}(k)),\ i \in E$$