Optimal control of a head-of-line processor sharing model with regular and opportunity customers

Motivated by a workload control setting, we study a model where two types of customers are served by a single server according to the head-of-line processor sharing discipline. Regular customers and opportunity customers are arriving to the system according to two independent Poisson processes, each requiring an exponentially distributed service time. The regular customers will queue, incurring some holding costs. On contrary, an opportunity customer has to be taken into service directly, or is lost otherwise. There can be at most one opportunity customer in the system. The server can work on both one regular and one opportunity customer at the same time, where one can decide on how the server speed is split out. Moreover, one can decide whether to accept or reject an opportunity customer, incurring penalty costs for the latter. In this way, one has partial control about the workload in the system. We formulate the model as a Markov decision problem. We prove that the optimal policy, minimizing the expected discounted long-run cost, has a monotone structure in the number of regular customers. That is, the optimal policy for accepting an opportunity customer is described be a threshold, and the fraction of the server's attention devoted to the opportunity customer is a monotone decreasing function. Further, we generalize our model to the case where opportunity customers will queue as well, and to the case where also regular customers can be rejected.