Queueing models with service can be applied in the area of performance studies of polling systems in data communication and switching systems. We provide a new approach for analysis on the cycle time of the multiqueue system consisting of statistically identical stations where at most one message is served for any station in a polling cycle. We assume an independent Posson arrival at each station. The message service time and switchover time are assumed to be constant. Based on the dependence between the stations, cycle time distribution is derived. Then, this result is validated by a computer simulation.