We study several generalizations of the Online Bipartite Matching problem. We consider settings with stochastic rewards, patience constraints, and weights (considering both vertex- and edge-weighted variants). We introduce a stochastic variant of the patience-constrained problem, where the patience is chosen randomly according to some known distribution and is not known in advance. We also consider stochastic arrival settings (i.e. the nature in which the online vertices arrive is determined by a known random process), which are natural settings that are able to beat the hard worst-case bounds of adversarial arrivals.

We design black-box algorithms for star graphs under various models of patience, which solve the problem optimally for deterministic or geometrically-distributed patience, and yield a 1/2-approximation for any patience distribution. These star graph algorithms are then used as black boxes to solve the online matching problems under different arrival settings. We show improved (or first-known) competitive ratios for these problems. We also present negative results that include formalizing the concept of a stochasticity gap for LP upper bounds on these problems, showing some new stochasticity gaps for popular LPs, and bounding the worst-case performance of some greedy approaches.