  Although many well-known algorithms can solve the bipartite matching problem
instance efficiently, it remains an open question how one could estimate the
expected optimal matching distance for arbitrary numbers of randomly
distributed vertices in a $D$-dimensional $L^p$ space (referred to as a random
bipartite matching problem, or RBMP). This paper proposes an analytical model
with closed-form formulas (without statistical curve-fitting) that estimate
both the probability distribution and expectation of the optimal matching
distance of RBMP. Simpler asymptotic approximations of the formulas are also
developed for some special cases. A series of Monte-Carlo simulation
experiments are conducted to verify the accuracy of the proposed formulas under
varying conditions. These proposed distance estimates could be key for
strategic performance evaluation and resource planning in a wide variety of
application contexts. To illustrate their usefulness, we focus on mobility
service systems where matches must be made between customers and service
vehicles that are randomly distributed over time and space. We show how the
proposed distance formulas provide a theoretical foundation for the empirically
assumed Cobb-Douglas matching function for taxi systems, and reveal conditions
under which the matching function can be suitable. Our formulas can also be
easily incorporated into optimization models to select taxi operation
strategies (e.g., whether newly arriving customers shall be instantly matched
or pooled into a batch for matching). Agent-based simulations are conducted to
verify the predicted performance of the demand pooling strategy for two types
of e-hailing taxi systems. The results not only demonstrate the accuracy of the
proposed model estimates under various service conditions, but also offer
valuable managerial insights for service operators to optimize their
strategies.
