  This paper considers the level-increment (LI) truncation approximation of
M/G/1-type Markov chains. The LI truncation approximation is useful for
implementing the M/G/1 paradigm, which is the framework for computing the
stationary distribution of M/G/1-type Markov chains. The main result of this
paper is a subgeometric convergence formula for the total variation distance
between the original stationary distribution and its LI truncation
approximation. Suppose that the equilibrium level-increment distribution is
subexponential, and that the downward transition matrix is rank one. We then
show that the convergence rate of the total variation error of the LI
truncation approximation is equal to that of the tail of the equilibrium
level-increment distribution and that of the tail of the original stationary
distribution.
