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.

This work is financed by national funds through FCT - Fundação para a Ciência e Tecnologia, I.P., under the project FCT UIDB/04466/2022. Furthermore, the author thanks the ISCTE-IUL and ISTAR-IUL, for their support.

• Matemáticas - Ciências Naturais
• Ciências da Computação e da Informação - Ciências Naturais