The mean sojourn time is the mean time spent in each state.
meanSojournTimes(x, states = x$states, klim = 10000)A vector of length \(\textrm{card}(E)\) giving the values of the mean sojourn times for each state \(i \in E\).
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).
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:
$$m_{i} = E[T_{1} | Z_{0} = j] = \sum_{k \geq 0} (1 - P(T_{n + 1} - T_{n} \leq k | J_{n} = j)) = \sum_{k \geq 0} (1 - H_{j}(k)),\ i \in E$$