  We study online task assignment problem with reusable resources, motivated by
practical applications such as ridesharing, crowdsourcing and job hiring. In
the problem, we are given a set of offline vertices (agents), and, at each
time, an online vertex (task) arrives randomly according to a known
time-dependent distribution. Upon arrival, we assign the task to agents
immediately and irrevocably. The goal of the problem is to maximize the
expected total profit produced by completed tasks. The key features of our
problem are (1) an agent is reusable, i.e., an agent comes back to the market
after completing the assigned task, (2) an agent may reject the assigned task
to stay the market, and (3) a task may accommodate multiple agents. The setting
generalizes that of existing work in which an online task is assigned to one
agent under (1).
  In this paper, we propose an online algorithm that is $1/2$-competitive for
the above setting, which is tight. Moreover, when each agent can reject
assigned tasks at most $\Delta$ times, the algorithm is shown to have the
competitive ratio $\Delta/(3\Delta-1)\geq 1/3$. We also evaluate our proposed
algorithm with numerical experiments.
