| Abstract: | In this paper, we establish an explicit form of matrix decompositions for the queue length distributions of the MAP/G/1 queues under multiple and single vacations with N-policy. We show that the vector generating function Y (z) of the queue length at an arbitrary time and X (z) at departures are decomposed into Y (z) = p idle (z)ζ Y (z) and X (z) = p idle (z)ζ X (z) where p idle (z) is the vector generating function of the queue length at an arbitrary epoch at which the server is not in service, and ζ Y (z) and ζ X (z) are unidentified matrix generating functions. |
