  We study an edge-weighted online stochastic \emph{Generalized Assignment
Problem} with \emph{unknown} Poisson arrivals. In this model, we consider a
bipartite graph that contains offline bins and online items, where each offline
bin is associated with a $D$-dimensional capacity vector and each online item
is with a $D$-dimensional demand vector. Online arrivals are sampled from a set
of online item types which follow independent but not necessarily identical
Poisson processes. The arrival rate for each Poisson process is unknown. Each
online item will either be packed into an offline bin which will deduct the
allocated bin's capacity vector and generate a reward, or be rejected. The
decision should be made immediately and irrevocably upon its arrival. Our goal
is to maximize the total reward of the allocation without violating the
capacity constraints.
  We provide a sample-based multi-phase algorithm by utilizing both
pre-existing offline data (named historical data) and sequentially revealed
online data. We establish its performance guarantee measured by a competitive
ratio. In a simplified setting where $D=1$ and all capacities and demands are
equal to $1$, we prove that the ratio depends on the number of historical data
size and the minimum number of arrivals for each online item type during the
planning horizon, from which we analyze the effect of the historical data size
and the Poisson arrival model on the algorithm's performance. We further
generalize the algorithm to the general multidimensional and multi-demand
setting, and present its parametric performance guarantee. The effect of the
capacity's (demand's) dimension on the algorithm's performance is further
analyzed based on the established parametric form. Finally, we demonstrate the
effectiveness of our algorithms numerically.
