Stewart, G. W.Queues of M/G/1 type give rise to infinite embedded Markov chains whose transition matrices are upper block Hessenberg. The traditional algorithms for solving these queues have involved the computation of an intermediate matrix G. Recently a recursive descent method for solving block Hessenberg systems has been proposed. In this paper we explore the interrelations of the two methods. (Also cross-referenced as UMIACS-TR-95-37)en-USTechnical Report