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：我们研究了边加权在线随机\textit{广义分配问题}，其中在线到达过程服从\textit{未知}的泊松分布。在此模型中，我们考虑一个包含离线存储箱和在线物品的二分图，每个离线存储箱关联一个$D$维容量向量，每个在线物品关联一个$D$维需求向量。在线到达物品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，且每个泊松过程的到达率未知。每个在线物品要么被分配至某个离线存储箱（这将扣除该箱的容量向量并产生奖励），要么被拒绝。决策须在物品到达时立即做出且不可撤销。我们的目标是在不违反容量约束的前提下最大化总分配奖励。我们提出了一种基于样本的多阶段算法，同时利用预先存在的离线数据（称为历史数据）和顺序揭示的在线数据，并以竞争比衡量其性能保障。在$D=1$且所有容量与需求均为1的简化设定下，我们证明该竞争比取决于历史数据规模以及规划周期内每种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。我们将算法进一步推广至一般多维多需求设定，并给出其参数化性能保障。基于该参数化形式，进一步分析了容量（需求）维度对算法性能的影响。最后，我们通过数值实验验证了算法的有效性。