  In the infinite servers queue with Poisson arrivals real life practical
applications, the busy period and the busy cycle probabilistic study is of main
importance. But it is a very difficult task. In this text, we show that by
solving a Riccati equation induced by this queue transient probabilities
monotony study as time functions, we obtain a collection of service length
distribution functions, for which both the busy period and the busy cycle have
lengths with quite simple distributions, generally given in terms of
exponential distributions and the degenerate at the origin distribution.
