  We consider the allocation of limited resources to heterogeneous customers
who arrive in an online fashion. We would like to allocate the resources
"fairly", so that no group of customers is marginalized in terms of their
overall service rate. We study whether this is possible to do so in an online
fashion, and if so, what a good online allocation policy is.
  We model this problem using online bipartite matching under stationary
arrivals, a fundamental model in the literature typically studied under the
objective of maximizing the total number of customers served. We instead study
the objective of maximizing the minimum service rate across all groups, and
propose two notions of fairness: long-run and short-run.
  For these fairness objectives, we analyze how competitive online algorithms
can be, in comparison to offline algorithms which know the sequence of demands
in advance. For long-run fairness, we propose two online heuristics (Sampling
and Pooling) which establish asymptotic optimality in different regimes (no
specialized supplies, no rare demand types, or imbalanced supply/demand). By
contrast, outside all of these regimes, we show that the competitive ratio of
online algorithms is between 0.632 and 0.732. For short-run fairness, we show
for complete bipartite graphs that the competitive ratio of online algorithms
is between 0.863 and 0.942; we also derive a probabilistic rejection algorithm
which is asymptotically optimal in the total demand.
  Depending on the overall scarcity of resources, either our Sampling or
Pooling heuristics could be desirable. The most difficult situation for online
allocation occurs when the total supply is just enough to serve the total
demand.
  We simulate our algorithms on a public ride-hailing dataset, which both
demonstrates the efficacy of our heuristics and validates our managerial
insights.
