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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.