  We study the problem of matching agents who arrive at a marketplace over time
and leave after d time periods. Agents can only be matched while they are
present in the marketplace. Each pair of agents can yield a different match
value, and the planner's goal is to maximize the total value over a finite time
horizon. First we study the case in which vertices arrive in an adversarial
order. We provide a randomized 0.25-competitive algorithm building on a result
by Feldman et al. (2009) and Lehman et al. (2006). We extend the model to the
case in which departure times are drawn independently from a distribution with
non-decreasing hazard rate, for which we establish a 1/8-competitive algorithm.
  When the arrival order is chosen uniformly at random, we show that a batching
algorithm, which computes a maximum-weighted matching every (d+1) periods, is
0.279-competitive.
