The analysis of waiting times in polling systems in which the stations are polled according to a general service-order table is discussed. Such systems can be used to represent token-bus local area networks in which the routing of the token is fixed. Stations are given higher priority by being listed more frequently in the table, or by receiving service according to the exhaustive service discipline. The polling system is modeled by a single-server multiqueue system in discrete time. Nonzero switchover times between the queues are assumed. An extension of the principle of work conservation to systems with nonzero switchover times leads to an exact expression for a weighted sum of the mean waiting times at the various queues. By using a limiting procedure, the discrete-time results are translated to continuous-time results. The special case of polling in a star network is discussed and compared to polling in a corresponding network with strictly cyclic service order.>