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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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