  We study Online Linear Programming (OLP) with batching. The planning horizon
is cut into $K$ batches, and the decisions on customers arriving within a batch
can be delayed to the end of their associated batch. Compared with OLP without
batching, the ability to delay decisions brings better operational performance,
as measured by regret. Two research questions of interest are: (1) What is a
lower bound of the regret as a function of $K$? (2) What algorithms can achieve
the regret lower bound? These questions have been analyzed in the literature
when the distribution of the reward and the resource consumption of the
customers have finite support. By contrast, this paper analyzes these questions
when the conditional distribution of the reward given the resource consumption
is continuous, and we show the answers are different under this setting. When
there is only a single type of resource and the decision maker knows the total
number of customers, we propose an algorithm with a $O(\log K)$ regret upper
bound and provide a $\Omega(\log K)$ regret lower bound. We also propose
algorithms with $O(\log K)$ regret upper bound for the setting in which there
are multiple types of resource and the setting in which customers arrive
following a Poisson process. All these regret upper and lower bounds are
independent of the length of the planning horizon, and all the proposed
algorithms delay decisions on customers arriving in only the first and the last
batch. We also take customer impatience into consideration and establish a way
of selecting an appropriate batch size.
