Transient solutions of some multiserver queueing systems with finite spaces

The purpose of this paper is to provide explicit transient solutions for the multiserver queueing system Geom(n)/Geom(n)/c/N+c. The method proposed here can also be used for obtaining transient solutions of Markov chains having the transition matrix of Hesselberg type. To support this, we also consider a more complex model such as GI/M/c/N+c. In our analysis, we use eigenvalues and generalized eigenvectors of transition probability matrices. Since we use the Jordan canonical form from linear algebra, the method is good even if the eigenvalues are repeated. Numerical procedures for computations involved in various examples are also provided.