  The past decade has witnessed many interesting algorithms for maintaining
statistics over a data stream. This paper initiates a theoretical study of
algorithms for monitoring distributed data streams over a time-based sliding
window (which contains a variable number of items and possibly out-of-order
items). The concern is how to minimize the communication between individual
streams and the root, while allowing the root, at any time, to be able to
report the global statistics of all streams within a given error bound. This
paper presents communication-efficient algorithms for three classical
statistics, namely, basic counting, frequent items and quantiles. The
worst-case communication cost over a window is $O(\frac{k} {\epsilon} \log
\frac{\epsilon N}{k})$ bits for basic counting and $O(\frac{k}{\epsilon} \log
\frac{N}{k})$ words for the remainings, where $k$ is the number of distributed
data streams, $N$ is the total number of items in the streams that arrive or
expire in the window, and $\epsilon < 1$ is the desired error bound. Matching
and nearly matching lower bounds are also obtained.
