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Non-homogeneous continuous-time Markov and semi-Markov manpower models. (English) Zbl 0913.60056

The authors consider a manpower system with \(m\) states \(S_{1},\ldots,S_{m}\), and an additional state \(S_{m+1}\) denoting the state of having left the system. Let \(\alpha_{fj}\) be the constant rate of transition from \(S_{f}\) to \(S_{j}\) at any duration \(t\) and \(p_{fj}(t)\) be the transition probability that an individual in \(S_{f}\) at zero duration is in \(S_{j}\) at \(t\). In matrix notation \(P(t)=\{p_{fj}(t)\}\), \(W=\{\alpha_{fj}\}\) we have \(P(t)=\exp(Wt)\). Here the authors assume that the length of stay in a grade is exponential. The authors also consider the generalized non-homogeneous model in which the calendar time is divided into time windows by change points. The model parameters may change only at these change points but remain constant between them. Let us denote by \(\alpha_{fj}^{(w)}\delta t\) the probability of transition to \(S_{j}\) at \((t+\delta t)\) if individual stay in \(S_{f}\) at \(t\) within the \(w\)th time window. The authors obtain maximum likelihood estimator for \(\alpha_{fj}^{(w)}\). The estimation methods employ a competing risks approach and allow for left truncated and right censored data at the change points. In semi-Markov non-homogeneous models maximum likelihood estimators are given for the hazard and survivor functions describing length of stay in any grade of the manpower system.

MSC:

60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
91D35 Manpower systems in sociology
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