  We examine a new class of channel coding strategies, and hypothesis tests
referred to as almost-fixed-length strategies that have little flexibility in
the stopping time over fixed-length strategies. The stopping time of these
strategies is allowed to be slightly large only on a rare set of sample paths
with an exponentially small probability. We show that almost-fixed-length
channel coding strategies can achieve Burnashev's optimal error exponent.
Similarly, almost-fixed length hypothesis tests are shown to bridge the gap
between hypothesis testing with fixed sample size and sequential hypothesis
testing and improve the trade-off between type-I and type-II error exponents.
