This paper considers the steady-state solution of
Markov chains of M/G/1 type. We first show an alternative
formula for the steady-state solution in terms of the sum
of the convolutions of certain matrix functions. This
formula is closely related to an LCFS discrete-time
single-server queue and remedies some theoretical flaws
in the conventional M/G/1 paradigm. Besides, it is
suitable for studying both the asymptotics of the tail
distribution. Employing the Markov key renewal theorem for
the light-tailed case, we show the geometric asymptotics
of the tail distribution. Further, for the heavy-tailed
case, we show a sufficient condition under which the tail
distribution has the subexponential asymptotics.