  Service platforms must determine rules for matching heterogeneous demand
(customers) and supply (workers) that arrive randomly over time and may be lost
if forced to wait too long for a match. We show how to balance the trade-off
between making a less good match quickly and waiting for a better match, at the
risk of losing impatient customers and/or workers. When the objective is to
maximize the cumulative value of matches over a finite-time horizon, we propose
discrete-review matching policies, both for the case in which the platform has
access to arrival rate parameter information and the case in which the platform
does not. We show that both the blind and nonblind policies are asymptotically
optimal in a high-volume setting. However, the blind policy requires frequent
re-solving of a linear program. For that reason, we also investigate a blind
policy that makes decisions in a greedy manner, and we are able to establish an
asymptotic lower bound for the greedy, blind policy that depends on the
matching values and is always higher than half of the value of an optimal
policy. Next, we develop a fluid model that approximates the evolution of the
stochastic model and captures explicitly the nonlinear dependence between the
amount of demand and supply waiting and the distribution of their patience
times. We use the fluid model to propose a policy for a more general objective
that additionally penalizes queue build-up. We run numerous simulations to
investigate the performance of the aforementioned proposed matching policies.
