  In this paper, we study the following batch scheduling model: find a schedule
that minimizes total flow time for $n$ uniform length jobs, with release times
and deadlines, where the machine is only actively processing jobs in at most
$k$ synchronized batches of size at most $B$. Prior work on such batch
scheduling models has considered only feasibility with no regard to the flow
time of the schedule. However, algorithms that minimize the cost from the
scheduler's perspective -- such as ones that minimize the active time of the
processor -- can result in schedules where the total flow time is arbitrarily
high \cite{ChangGabowKhuller}. Such schedules are not valuable from the
perspective of the client. In response, our work provides dynamic programs
which minimize flow time subject to active time constraints. Our main
contribution focuses on jobs with agreeable deadlines; for such job instances,
we introduce dynamic programs that achieve runtimes of O$(B \cdot k \cdot n)$
for unit jobs and O$(B \cdot k \cdot n^5)$ for uniform length jobs. These
results improve upon our modification of a different, classical dynamic
programming approach by Baptiste. While the modified DP works when deadlines
are non-agreeable, this solution is more expensive, with runtime $O(B \cdot k^2
\cdot n^7)$ \cite{Baptiste00}.
