Setups in polling models: does it make sense to set up if no work is waiting?

We compare two versions of a symmetric two-queue polling model with switchover times and setup times. The SI version has State-Independent setups, according to which the server sets up at the polled queue whether or not work is waiting there; and the SD version has State-Dependent setups, according to which the server sets up only when work is waiting at the polled queue. Naive intuition would lead one to believe that the SD version should perform better than the SI version. We characterize the difference in the expected waiting times of these two versions, and we uncover some surprising facts. In particular, we show that, regardless of the server utilization or the service-time distribution, the SD version performs (i) the same as, (ii) worse than, or (iii) better than its SI counterpart if the switchover and setup times are, respectively, (i) both constants, (ii) variable (i.e. non-deterministic) and constant, or (iii) constant and variable. Only (iii) is consistent with naive intuition.