For Markov renewal processes of GI/M/1 type, the paper defines a double transform R(z,s), a generalization of Neuts’ rate matrix R, and establishes an interesting duality theory. That result expresses R(z,s) explicitly in terms of the double transform G(z,s) of Neuts for a suitably defined dual process of M/G/1 type. For the GI/PH/1 queue, the G(z,s) matrix arising in the duality theorem is identified as the matrix of the dual PH/G/1 queue with a time reversed representation of the PH-distribution. The duality theorem unifies the results on the two paradigms and enables one to obtain many results concerning one from those of the other by simple arguments.
