On Variation of Single Birth Processes.

Suppose { X( t); t ≥ 0} is a single birth process with birth rate q _ii +1 ( i ≥ 0) and death rate q _ij ( i > j ≥ 0). It is proved in this paper that (i) if there exists a constant c ≥ 0 such that b( i) − a( i) + ci is nondecreasing with respect to i and a( i) + u( i) − ci ≥ 0 ( i ≥ 0), thenor (ii) if there exists a constant c ≥ 0 such that b( i) − a( i) + ci is non-increasing with respect to i and a( i) + u( i) − ci ≤ 0 ( i ≥ 0), then Here $$ b{\left( i \right)} = q_{{ii + 1}} ,\;a{\left( 0 \right)} = 0,\;a{\left( i \right)} = {\sum\limits_{j = 1}^i {jq_{{ii - j}} } }{\left( {i \geqslant 1} \right)},\;u{\left( 0 \right)} = u{\left( 1 \right)} = 0 $$ and $$ u{\left( i \right)} = \frac{1} {2}{\sum\limits_{j = 1}^i j }{\left( {j - 1} \right)}q_{{ii - j}} {\left( {i \geqslant 2} \right)}. $$ This result covers the results for birth-death processes obtained in [7]. [ABSTRACT FROM AUTHOR]