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Within the context of stochastic probing with commitment, we consider theonline stochastic matching problem; that is, the one sided online bipartitematching problem where edges adjacent to an online node must be probed todetermine if they exist, based on known edge probabilities. If a probed edgeexists, it must be used in the matching (if possible). We study this problem inthe generality of a patience (or budget) constraint which limits the number ofprobes that can be made to edges adjacent to an online node. The patienceconstraint results in modelling and computational efficiency issues that arenot encountered in the special cases of unit patience and full (i.e.,unlimited) patience. The stochastic matching problem leads to a variety ofsettings. Our main contribution is to provide a new LP relaxation and a unifiedapproach for establishing new and improved competitive bounds in threedifferent input model settings (namely, adversarial, random order, and knowni.i.d.). In all these settings, the algorithm does not have any control on theordering of the online nodes. We establish competitive bounds in thesesettings, all of which generalize the standard non-stochastic setting whenedges do not need to be probed (i.e., exist with certainty). All of ourcompetitive ratios hold for arbitrary edge probabilities and patienceconstraints:  (1) A $1-1/e$ ratio when the stochastic graph is known, offline vertices areweighted and online arrivals are adversarial.  (2) A $1-1/e$ ratio when the stochastic graph is known, edges are weighted,and online arrivals are given in random order (i.e., in ROM, the random ordermodel).  (3) A $1-1/e$ ratio when online arrivals are drawn i.i.d. from a knownstochastic type graph and edges are weighted.  (4) A (tight) $1/e$ ratio when the stochastic graph is unknown, edges areweighted and online arrivals are given in random order.

Bipartite Stochastic Matching: Online, Random Order, and I.I.D. Models