Eigenvalue expression for mean queue length of BMAP/G/1 queue

A counting process of arrivals according to a batch Markovian arrival process (BMAP) is described by eigenvalues and eigenvectors of the infinitesimal generator of an underlying process. An explicit form of an eigenvalue expression for mean queue lengths of a BMAP/G/1 at departure epochs and at arbitrary time is derived. The difference between the mean queue lengths of a BMAP/G/1 and an M/G/1 is obtained.