  Motivated by bursty bandwidth allocation and by the allocation of virtual
machines to servers in the cloud, we consider the online problem of packing
items with random sizes into unit-capacity bins. Items arrive sequentially, but
upon arrival an item's actual size is unknown; only its probabilistic
information is available to the decision maker. Without knowing this size, the
decision maker must irrevocably pack the item into an available bin or place it
in a new bin. Once packed in a bin, the decision maker observes the item's
actual size, and overflowing the bin is a possibility. An overflow incurs a
large penalty cost and the corresponding bin is unusable for the rest of the
process. In practical terms, this overflow models delayed services, failure of
servers, and/or loss of end-user goodwill. The objective is to minimize the
total expected cost given by the sum of the number of opened bins and the
overflow penalty cost. We present an online algorithm with expected cost at
most a constant factor times the cost incurred by the optimal packing policy
when item sizes are drawn from an i.i.d. sequence of unknown length. We give a
similar result when item size distributions are exponential with arbitrary
rates. We also study the offline model, where distributions are known in
advance but must be packed sequentially. We construct a soft-capacity PTAS for
this problem, and show that the complexity of computing the optimal offline
cost is $\#\mathbf{P}$-hard. Finally, we provide an empirical study of our
online algorithm's performance.
