A single server with a finite queue and items of different types

A single server system with $r$ waiting facilities ($1\le r\le\infty$) and $k$ independent poisson inputs with intensities $\lambda_1,\dots,\lambda_k$ is considered; the service time of items has an independent exponential distribution with an average $1/\nu_i$, $i=1,2,\dots,k$.
For the case of an absolutely reliable server and also for, the case, when the server can fail and repair, recurrent relations for final probabilities of the system are derived. Expressions for different characteristics of the system which are expressed in especially simple form at $r=\infty$ are derived. NSC of existence of final probability at $r=\infty$ are given.