The Truncated Birth-Death Process

Abstract

A truncated birth-death process is a temporally homogeneous Markov process {X(t): 0 ≤ t < ∞} on a finite state space S̅= {-1, 0, 1,...., N, N+1}, say, with transition probability functions

$$ {P_{ij}}(t) = \Pr \left\{ {X\left( {t + s)} \right) = j\left| {X\left( s \right)} \right. = i} \right\} $$

((10.1.1))

which satisfy the conditions

$$ {P_{ - 1.j}}(t) = {\delta _{ - 1,j}}\quad j \in \overline S ,t \geqslant 0, $$

((10.1.2))

$$ {P_{N + 1,j}}(t) = {\delta _{N + 1,j}}\quad j \in \overline S ,t \geqslant 0 $$

((10.1.3))

and for i ∈ S = {0, 1,..., N},

$$ \begin{array}{*{20}{c}}{{P_{i,i + 1}}(t) = {\lambda _i}t + o(t)}\\{{P_{ii}}(t) = 1 - \left( {{\lambda _i} + {\mu _i}} \right)t + o(t)} \\{{P_{i,i - 1}}(t) = {\mu _i}t + o(t)} \\\end{array} $$

((10.1.4))

as t → 0, where λ_i and µ_i, i ∈ S, are non-negative constants. Throughout this chapter we assume λ_i > 0 for i ∈ S\{N} and µ_i > 0 for i ∈ S\{0}.