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.

翻译：我们研究具有未知泊松到达的边权重在线随机广义指派问题。在该模型中，考虑一个包含离线容器和在线物品的二分图，其中每个离线容器关联一个D维容量向量，每个在线物品关联一个D维需求向量。在线到达从一组在线物品类型中采样生成，这些类型遵循独立但不一定同分布的泊松过程。每个泊松过程的到达率未知。每个在线物品要么被放入离线容器中（这将扣除所分配容器的容量向量并产生收益），要么被拒绝。决策必须在其到达时立即且不可撤销地做出。我们的目标是在不违反容量约束的情况下最大化分配总收益。我们通过利用既存的离线数据（称为历史数据）和顺序揭示的在线数据，提出一种基于样本的多阶段算法。我们建立了以竞争比衡量的性能保证。在D=1且所有容量与需求等于1的简化场景中，我们证明该竞争比取决于历史数据量的大小及规划期内每个在线物品类型的最小到达次数，由此分析历史数据规模和泊松到达模型对算法性能的影响。进一步将算法推广至一般多维与多需求场景，并给出其参数化性能保证。基于所建立的参数化形式，进一步分析容量（需求）维度对算法性能的影响。最后，通过数值实验验证了我们算法的有效性。