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