Dynamic control of a tandem system with abandonments

The goal of this paper is to provide a model that is an extension of classic scheduling problems for a tandem queueing system by including customer impatience. In such scenarios, the server(s) must balance abandonments from each phase of service with the need to prioritize higher reward customers. This presents an interesting challenge since the trade-off between the cost of abandonments and revenue maximization is not at all clear. As a result of customer abandonments, traditional solution techniques are not available. In particular, uniformization is not possible since the transition rates are unbounded. We do our analysis in continuous time, using the continuous-time Markov decision process framework to discern simple relationships of the value functions depending on the starting state. We then use sample path arguments to analyze the structure of the optimal policy. We provide conditions under which an optimal policy is nonidling and never splits servers (except to avoid idling). We then consider the single-server model as a proxy for the decision problem when there are more customers at each station than there are servers. In this case, we provide conditions under which it is optimal to prioritize each station.