  We consider an online resource allocation problem where multiple resources,
each with an individual initial capacity, are available to serve random
requests arriving sequentially over multiple discrete time periods. At each
time period, one request arrives and its associated reward and size are drawn
independently from a known distribution that could be resource-dependent. Upon
its arrival and revealing itself, an online decision has to be made on whether
or not to accept the request. If accepted, another online decision should also
be made to specify on assigning which resource to serve the request, then a
certain amount of the resource equal to the size of the request will be
consumed and a reward will be collected. The objective of the decision maker is
to maximize the total collected reward subject to the capacity constraints. We
develop near-optimal policies for our problem under two settings separately.
When the reward distribution has a finite support, we assume that given each
reward realization, the size distributions for different resources are
perfectly positive correlated and we propose an adaptive threshold policy. We
show that with certain regularity conditions on the size distribution, our
policy enjoys an optimal $O(\log T)$ regret bound, where $T$ denotes the total
time periods. When the support of the reward distribution is not necessarily
finite, we develop another adaptive threshold policy with a $O(\log T)$ regret
bound when both the reward and the size of each request are
resource-independent and the size distribution satisfies the same conditions.
