  The busy period for a queue is cast as the area swept under the random walk
until it first returns to zero, $B$. Encompassing non-i.i.d. increments, the
large-deviations asymptotics of $B$ is addressed, under the assumption that the
increments satisfy standard conditions, including a negative drift. The main
conclusions provide insight on the probability of a large busy period, and the
manner in which this occurs:
  I) The scaled probability of a large busy period has the asymptote, for any
$b>0$, \lim_{n\to\infty} \frac{1}{\sqrt{n}} \log P(B\geq bn) = -K\sqrt{b},
\hbox{where} \quad K = 2 \sqrt{-\int_0^{\lambda^*} \Lambda(\theta) d\theta},
\quad \hbox{with $\lambda^*=\sup\{\theta:\Lambda(\theta)\leq0\}$,} and with
$\Lambda$ denoting the scaled cumulant generating function of the increments
process.
  II) The most likely path to a large swept area is found to be a simple
rescaling of the path on $[0,1]$ given by, [\psi^*(t) =
-\Lambda(\lambda^*(1-t))/\lambda^*.] In contrast to the piecewise linear most
likely path leading the random walk to hit a high level, this is strictly
concave in general. While these two most likely paths have very different
forms, their derivatives coincide at the start of their trajectories, and at
their first return to zero.
  These results partially answer an open problem of Kulick and Palmowski
regarding the tail of the work done during a busy period at a single server
queue. The paper concludes with applications of these results to the estimation
of the busy period statistics $(\lambda^*, K)$ based on observations of the
increments, offering the possibility of estimating the likelihood of a large
busy period in advance of observing one.

Probability