Having many followers on twitter is a great start, but do you how many accounts your followers are following? Your tweets could be getting buried deep in the endless myriad of tweets floating out there on the internet (As of early 2013, there are over 400 million tweets a day).
Tweriod helps fight this battle my giving you statistics on the online behavior of your followers. Armed with this data, you can time your tweets during the hours when your followers are most likely to be active and therefore increasing the likelihood that they will be seen. Coupled with a timing tool like futuretweets, you can fully automate your tweets and the time which they will get the greatest exposure.
There are different purchase options but a free service does exist to test out. Here is a screenshot from a free report of @shapingsherpas:
Premium options include more automated services and analysis of more followers. There is both monthly pricing and one off ad hoc reports so in terms of running trial premium reports there is very little to lose.