This paper considers the steady-state solution of Markov chains of
M/G/1 type. We first derive the matrix product-form solution of the
steady-state probability. This formula is considered as a natural
generalization of the matrix-geometric solution of quasi
birth-and-death processes to Markov chains of M/G/1 type. Based on
this formula, we study the asymptotics of the tail distribution. For
the light-tailed case, we show a sufficient condition for the
geometric asymptotics of the tail distribution, employing the Markov
key renewal theorem. Contrary to the previous works, some periodic
characteristics of transitions in upper levels are taken into account
explicitly and a new geometric asymptotic formula is
established. Furthermore, for the heavy-tailed case, we show a
subexponential asymptotics formula for the tail distribution under a
mild condition.