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Transient Markov arrival processes. (English) Zbl 1030.60067
A family of transient Markov arrival processes (MAPs) is considered and its basic properties are analyzed. Transient MAP is defined on the basis of a continuous-time Markov chain \(\varphi(t) \in \{0,1,\dots,m\}\), \(t \in R^+\), described by constant transition rates \((D^*)_{i,j}\), \(i,j \in \{0,1,\dots,m\}\). The state \(\{0\}\) is absorbing, i.e. \((D^*)_{0,j}=0\), \(j \neq 0\). In any other state \(\{1,\dots,m\}\) a new event can occur with constant rates \((D^*_1)_{i,j}\) depending on the current transition \(i \to j\), events can happen even if \(\varphi(t)\) does not change (\(i\to i\)). The total number of events up to time epoch \(t\) is denoted by \(N(t)\). In absorbing state \(\{0\}\) no more events can occur, so, it is said that the catastrophe occurs when the Markov chain achieves this state. Special examples of MAPs are the Poisson process (\(m=1\)) and the Markov modulated Poisson process for which \(D_1^*\) is a diagonal matrix. Some other examples and possible applications are discussed.
Let \(T_n\) denote the time epoch at which the \(n\)th event occurs. For the MAP the lifetime of the process \(L\), the time \(V\) until the catastrophe occurs and the total number of events \(K\) are defined as follows \[ L=\sup\{T_n:T_n<\infty\},\quad V=\inf\{t\geq 0:\varphi(t)=0\},\quad K=\lim_{t \to \infty} N(t). \] Distributions of \(L\), \(V\), \(K\) are derived. In addition, quasistationary MAPs are especially considered and some estimations are obtained in this case.

MSC:
60J22 Computational methods in Markov chains
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J27 Continuous-time Markov processes on discrete state spaces
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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