Decision Processes with Monotone Hysteretic Policies

Queueing models play an important role in the optimization of dynamic processes, especially in serving problems. For example, dynamic processor or storage allocation for computers, working intensity in a shop or in a service industry, and staff policy in the order-dependent industry may be modeled by optimizing the service levels of a queue. The goal is to minimize the cost that occurs from serving and from customers waiting in the queue. Changing the service levels creates switching costs, e.g., for resource reallocation, personnel increase or reduction. When switching costs are considered, the optimal service level depends on the current service level as well as on the number of customers in the queue. Of special interest are the structural properties of the optimal policies because structured policies are easier to determine and to implement. We give conditions for two properties of optimal policies: (1) monotone policies where the service level increases with the number of customers, and (2) hysteretic policies that change the service level only under some switching condition.