  Consider a family of random walks $S_n^{(a)}=X_1^{(a)}+\cdots+X_n^{(a)}$ with
negative drift $\mathbf E X_1^{(a)}=-a<0$ and finite variance
$\mbox{var}(X_1^{(a)})=\sigma^2<\infty$.Let $M^{(a)}=\max_{n\ge 0} S_n^{(a)}$
be the maximums of the random walks. The exponential asymptotics $\mathbf
P(aM^{(a)}>x)\sim e^{-2x/\sigma^2}$, as $a\to 0$, were found by Kingman and are
known as heavy traffic approximation in the queueing theory. For subexponential
random variables the large deviation asymptotics for $\mathbf P(M^{(a)}>x)\sim
\frac{1}{a}\overline F^I(x)$ hold for fixed $a$ as $x\to\infty$. In this paper
we present asymptotics for $\mathbf P(M^{(a)}>x)$, which hold uniformly on the
whole positive axis, as $a\to 0$. Thus, these uniform asymptotics include both
the regime of normal and large deviations. We identify the regions where
exponential or subexponential asymptotics hold. Our approach is based on
construction of corresponding super/sub - martingales to obtain sharp upper and
lower bounds.
