Service platforms must determine rules for matching heterogeneous demand (customers) and supply (workers) that arrive randomly over time and may be lost if forced to wait too long for a match. Our objective is to maximize the cumulative value of matches, minus costs incurred when demand and supply wait. We develop a fluid model, that approximates the evolution of the stochastic model, and captures explicitly the nonlinear dependence between the amount of demand and supply waiting and the distribution of their patience times, also known as reneging or abandonment times in the literature. The fluid model invariant states approximate the steady-state mean queue-lengths in the stochastic system, and, therefore, can be used to develop an optimization problem whose optimal solution provides matching rates between demand and supply types that are asymptotically optimal (on fluid scale, as demand and supply rates grow large). We propose a discrete review matching policy that asymptotically achieves the optimal matching rates. We further show that when the aforementioned matching optimization problem has an optimal extreme point solution, which occurs when the patience time distributions have increasing hazard rate functions, a state-independent priority policy, that ranks the edges on the bipartite graph connecting demand and supply, is asymptotically optimal. A key insight from this analysis is that the ranking critically depends on the patience time distributions, and may be different for different distributions even if they have the same mean, demonstrating that models assuming, e.g., exponential patience times for tractability, may lack robustness. Finally, we observe that when holding costs are zero, a discrete review policy, that does not require knowledge of inter-arrival and patience time distributions, is asymptotically optimal.

翻译：摘要：服务平台必须制定规则，以匹配随机到达且若等待过久可能流失的异质性需求（顾客）与供给（工作者）。我们的目标是最大化匹配的累计价值，减去需求与供给等待产生的成本。我们构建了一个流体模型，该模型近似随机模型的演化过程，并明确刻画了需求与供给等待量与其耐心时间分布（文献中也称为放弃时间或退出时间）之间的非线性依赖关系。流体模型的不变状态近似于随机系统中稳态平均队列长度，因此可被用于构建一个优化问题，其最优解提供了需求与供给类型之间的匹配速率，且这些速率在渐近意义下最优（在流体尺度上，当需求与供给速率趋于无穷大时）。我们提出了一种离散审查匹配策略，该策略能渐近地实现最优匹配速率。我们进一步证明，当上述匹配优化问题具有最优极值点解时（这发生在耐心时间分布具有递增风险率函数的情况下），一种对连接需求与供给的二部图边进行排序的状态无关优先级策略是渐近最优的。此项分析的关键洞见在于：排序关键依赖于耐心时间分布，即使均值相同，不同分布下的排序也可能不同，这表明为简化分析而假设（例如）耐心时间服从指数分布的模型可能缺乏稳健性。最后，我们观察到，当持有成本为零时，一种无需了解到达间隔时间与耐心时间分布的离散审查策略是渐近最优的。