Birth and Death Processes

Abstract

By a set of birth-death parameters, we shall mean a sequence $\left\{ {\left( {{\lambda _n},{\mu _n}} \right);n = 0,1,2,...} \right\}$ of pairs of numbers such that ${\lambda _n} >$ for all $n \geqslant 0,{\mu _n} > 0$ for all $n \geqslant 1,$, and ${\mu _0} \geqslant 0$. In this chapter, Q will represent the birth and death q-matrix of (3.2.1) given by

$$Q = \left( {\begin{array}{*{20}{c}} { - ({\lambda _0} + {\mu _0})}&{{\lambda _0}} \\ {{\mu _1}}&{ - ({\lambda _1} + {\mu _1})} \\ 0&{{\mu _2}} \\ 0&0 \\ \vdots & \vdots \end{array}\begin{array}{*{20}{c}} 0&0& \cdots \\ 0&0& \cdots \\ {{\lambda _2}}&0& \cdots \\ { - ({\lambda _3} + {\mu _3})}&{{\lambda _3}}& \cdots \end{array}} \right)$$

((1.1))

, where $\left\{ {\left( {{\lambda _n},{\mu _n}} \right);n = 0,1,2,...} \right\}$ is a set of birth-death parameters. Note again that Q is conservative if and only if μ₀ = 0, and that if μ ₀ > 0, we are allowing the process to jump from state 0 directly to an absorbing state which, given the context here, is most conveniently labeled as — 1.