Computing an exact solution for the differential equations, with differences, system that allows the determination of the M|M|m|m system transient probabilities is a hard task. The respective complexity grows with m. The computations are extremely fastidious and the length and the fact that the expressions obtained are often approximate, and not exact, will not allow the transient probabilities behavior as time functions characterization. To overcome these problems, in this work it is analyzed how that system can supply approximate values to the M|M|m|m queue system, through matrix calculations. It is also presented an asymptotic method to solve the system that becomes possible in many cases to obtain simple approximated expressions for those probabilities using the M|M|∞ transient probabilities, very well-known and very much easier to study.

This work is partially financed by national funds through FCT - Fundação para a Ciênciae Tecnologia, I.P., under the project FCT UIDB/04466/2020. Furthermore, the author thanks the Iscte-IUL and ISTAR-IUL, for their support.

M|M|m|m,M|M|∞,transient probabilities

• Mathematics - Natural Sciences
• Computer and Information Sciences - Natural Sciences