Unified characteristic numbers and solutions of equations for birth and death processes with barriers
The authors deal with birth and death processes on the state space \(E = \{0, 1, 2, \dots \}\). A birth and death process is a Markov jump process which can jump from any state \(i\) only to neighboring states \(i-1\) and \(i+1\). The state 0 here is special, it is called a barrier. The barrier can be classified into four kinds: (reflection) When the process jumps into 0, it stays there for some (exponentially distributed) time and then jumps back to 1; (absorption) When the process jumps into 0, it remains there forever; (leaping reflection) When the process jumps into 0, it stays there for some exponentially distributed time and then dies (or, in other words, jumps to a fictitious state `\(\infty\)'); (quasi-leaping reflection) This is a combination of reflection and leaping reflection: after staying some time at 0, the process can either jump back to 1, or die. For the first, second and fourth barriers, the characteristic numbers of different forms (e.g., natural scale, canonical measures, etc.) have been introduced earlier in the literature. In the present paper the authors present unified characteristic numbers for birth and death processes with barriers which can serve for all types of barriers. The main purpose is to solve certain equations related to the Markov processes (e.g. \(Qu=\lambda u\), where \(Q\) is the jump rate matrix), and the solutions are given in terms of the unified characteristic numbers. These equations and their solutions are closely related to the probability construction problem of birth and death processes with barriers.
scientific article; zbMATH DE number 5788804
scientific article