  In this article we focus on the partial sum $S_{n}=X_{1}+\cdots+X_{n}$ of the
subcritical branching process with immigration
$\{X_{n}\}_{n\in\mathbb{N_{+}}}$, under the condition that one of the offspring
$\xi$ or immigration $\eta$ is regularly varying. The tail distribution of
$S_n$ is heavily dependent on that of $\xi$ and $\eta$, and a precise large
deviation probability for $S_{n}$ is specified. (i)When the tail of offspring
$\xi$ is lighter than immigration $\eta$, uniformly for $x\geq x_{n}$,
$P(S_{n}-ES_{n}>x)\sim c_{1}nP(\eta>x)$ with some constant $c_{1}$ and sequence
$\{x_{n}\}$, where $c_{1}$ is only related to the mean of offspring; (ii) When
the tail of immigration $\eta$ is not heavier than offspring $\xi$, uniformly
for $x\geq x_{n}$,$P(S_{n} ES_{n}>x)\sim c_{2}nP(\xi>x)$ with some constant
$c_{2}$ and sequence $\{x_{n}\}$, where $c_{2}$ is related to both the mean of
offspring and the mean of immigration.
