  We study online weighted bipartite matching of reusable resources where an
adversarial sequence of requests for resources arrive over time. A resource
that is matched is 'used' for a random duration, drawn independently from a
resource-dependent distribution, after which it returns and is able to be
matched again. We study the performance of the greedy policy, which matches
requests to the resource that yields the highest reward. Previously, it was
known that the greedy policy is 1/2 competitive against a clairvoyant benchmark
that knows the request sequence in advance. In this work, we improve this
result by introducing a parameter that quantifies the degree of reusability of
the resources. Specifically, if p represents the smallest probability over the
usage distributions that a matched resource returns in one time step, the
greedy policy achieves a competitive ratio of $1/(2-p)$. Furthermore, when the
usage distributions are geometric, we establish a stronger competitive ratio of
$(1+p)/2$, which we demonstrate to be tight. Both of these results align with
the known results in the two extreme scenarios: p = 0 corresponds to
non-reusable resources, where 1/2 is known to be tight, while p = 1 corresponds
to every resource returning immediately, where greedy is the optimal policy and
hence the competitive ratio is 1. Finally, we show that both results are robust
to approximations of the greedy policy. Our work demonstrates that the
reusability of resources can enhance performance compared to the non-reusable
setting, and that a simple greedy policy suffices when the degree of
reusability is high. Our insights contribute to the understanding of how
resource reusability can influence the performance of online algorithms, and
highlight the potential for improved performance as the degree of reusability
increases.
