Necessary and sufficient conditions for an irreducible Markov chain of G/M/1 type to have an invariant measure that is matrix-geometric are given. For example, it is shown that such an invariant measure exists when a(z), the generating function corresponding to transitions in the homogeneous part of the chain, is either an entire function or a rational function. This generalises a recent result of Latouche, Pearce and Taylor, who showed that a matrix-geometric invariant measure always exists for level-independent quasi-birth-and-death processes. Conditions ensuring the uniqueness of such an invariant measure up to multiplication by a positive constant are also given. Examples of G/M/1 type Markov chains with no matrix-geometric invariant measure and with more than one distinct matrix-geometric invariant measure are presented. As a byproduct of our work, it is shown in the transient case that if det{zI – a(z)} = 0 has a solution in the exterior of the closed unit disk, then the solution of smallest modulus there is real and positive.
