A gaming system which includes a bonus event associated with a bonus event time period or bonus event time window. The bonus event utilizes a plurality of bonus event component sequences, wherein each bonus event component sequence is associated with a plurality of time periods or windows which collectively (when accounting for any overlaps) forms the bonus event time period. In these embodiments, upon the triggering of the bonus event, the gaming system provides one or more awards in association with each bonus event component sequence. That is, for the single occurrence of a bonus event triggering event, the gaming system determines a plurality of winners for a plurality of different time periods that are each associated with the bonus event.