On optimal polling policies

In a single server polling system, the server visits the queues according to a routing policy and while at a queue serves some or all of the customers there according to a service policy. A polling (or scheduling) policy is a sequence of decisions on whether to serve a customer, idle the server, or switch the server to another queue. The goal of this paper is to nd polling policies that stochastically minimize the unnnished work and the number of customers in the system at all time. This optimization problem is decomposed into three subproblems: determine the optimal action (i.e., serve, switch, idle) when the server is at a nonempty queue; determine the optimal action (i.e., switch, idle) when the server empties a queue; determine the optimal routing (i.e., choice of the queue) when the server empties a queue and decides to switch. Under fairly general assumptions, we show for the rst subproblem that optimal policies are greedy and exhaustive, i.e., the server should neither idle nor switch when it is at a nonempty queue. For the second subproblem, we prove that in symmetric polling systems patient policies are optimal, i.e., the server should stay idling at the last visited queue whenever the system is empty. When the system is slotted, we further prove that non-idling and impatient policies are optimal. For the third subproblem, we establish that in symmetric polling systems optimal policies belong to the class of Stochastically Largest Queue (SLQ) policies. A SLQ policy is one that never routes the server to a queue known to have a queue length that is stochastically smaller than that of another queue. This result implies, in particular, that the policy that routes the server to the queue with the largest queue length is optimal when all queue lengths are known and that the cyclic routing policy is optimal in the case that the only information available is the previous decisions.