Some decomposition results for a class of vacation queues

We analyze the MAP/PH/1 vacation system at arbitrary times using the matrix-analytic method, and obtain decomposition results for the RR and GG matrices. The decomposition results reduce the amount of computational effort needed to obtain these matrices. The results for the GG matrix are extended to the BMAP/PH/1 system. We also show that in the case of the Geo/PH/1 and M/PH/1 systems with PH vacations both the GG and RR matrices can be obtained explicitly.