  We develop a stochastic inventory system which accounts for the limited
patience of backlogged customers. While limited patience is a feature that is
closer to the nature of unmet demand, our model also unifies the classic
backlogging and lost-sales inventory systems which are special cases of the one
we propose. We establish the uniform (asymptotic) optimality of the base-stock
policy when both demand and patience distributions are known. When the
backlogged demands become unobservable, we introduce a novel policy family that
operates without backlogged demands information, and prove that it can approach
the cost efficiency of the optimal policy in the system when the demand and
patience distributions are known. Finally, we consider an online inventory
control problem in which backlogged demand is unobservable and demand and
patience distributions are also not known, and develop a UCB-type algorithm
that yields a near-optimal policy. The regret bounds given by the algorithm are
provably tight within the planning horizon, and are comparable to the
state-of-the-art results in the literature, even in the face of partial and
biased observations and weaker system ergodicity.
