Censoring technique applied to a MAP/G/1 queue with set-up time and multiple vacations (Q717725)

The paper deals with the \(MAP/G/1\) queue with general distributed set-up times and general distributed multiple vacation times. Introducing the supplementary variables (elapsed vacation-, set-up- and service time up to \(t\) depending on the state of the server) for the resulting continuous time Markow process the differential equations for the density of the steady state distribution with boundary conditions are considered. The transition matrices are transformed to a block-Toeplitz or block-repeating structure, establishing a connection between the solution of the boundary equations and the solution of a Markow chain of \(M/G/1\) type. The solution of Markow chain can be expressed in terms of the censoring technique (embedded Markow chain) and the RG-factorization. This leads to a solution of the boundary equations, being crucial for solving the system of differential equations. Expressions for various performance measures (stationary queue length, virtual waiting time, busy period) are provided.