  Motivated by display advertising on the internet, the online stochastic
matching problem is proposed by Feldman, Mehta, Mirrokni, and Muthukrishnan
(FOCS 2009). Consider a stochastic bipartite graph with offline vertices on one
side and with i.i.d. online vertices on the other side. The algorithm knows the
offline vertices and the distribution of the online vertices in advance. Upon
the arrival of each online vertex, its type is realized and the algorithm
immediately and irrevocably decides how to match it. In the vertex-weighted
version of the problem, each offline vertex is associated with a weight and the
goal is to maximize the total weight of the matching.
  In this paper, we generalize the model to allow non-identical online vertices
and focus on the fractional version of the vertex-weighted stochastic matching.
We design fractional algorithms that are $0.718$-competitive and
$0.731$-competitive for non i.i.d. arrivals and i.i.d. arrivals respectively.
We also prove that no fractional algorithm can achieve a competitive ratio
better than $0.75$ for non i.i.d. arrivals. Furthermore, we round our
fractional algorithms by applying the recently developed multiway online
correlated selection by Gao et al. (FOCS 2021) and achieve $0.666$-competitive
and $0.704$-competitive integral algorithms for non i.i.d. arrivals and i.i.d.
arrivals. Our results for non i.i.d. arrivals are the first algorithms beating
the $1-1/e \approx 0.632$ barrier of the classical adversarial setting. Our
$0.704$-competitive integral algorithm for i.i.d. arrivals slightly improves
the state-of-the-art $0.701$-competitive ratio by Huang and Shu (STOC 2021).
