  The infinite servers queue with Poisson arrivals state transient
probabilities, considering the time origin at the beginning of a busy period,
mean and variance monotony as time functions is studied. These studies, for
which results it is determinant the hazard rate function service time length,
induce the consideration of two differential equations, one related with the
mean monotony study and another with the variance monotony study, which
solutions lead to some particular service time distributions, for which those
parameters present specific behaviors as time functions.
