  In the study of order estimation of the Riemann zeta-function $ \zeta(s) =
\sum_{n=1}^\infty n^{-s} $, solving Lindel\"{o}f hypothesis is an important
theme. As one of the relationships, asymptotic behavior of mean values has been
studied. Furthermore, the theory of the mean values is also noted in the double
zeta-functions, and the mean values of the Euler-Zagier type of double
zeta-function and Mordell-Tornheim type of double zeta-function were studied.
In this paper, we prove asymptotic formulas for mean square values of the
Barnes double zeta-function $ \zeta_2 (s, \alpha ; v, w ) = \sum_{m=0}^\infty
\sum_{n=0}^\infty (\alpha+vm+wn)^{-s} $ with respect to $ \mathrm{Im}(s) $ as $
\mathrm{Im}(s) \rightarrow + \infty $.
