How can we convince current customers to switch to support contracts?

We have developed an SaaS product that has been running with some success. Lately we have experienced a spike of customer requests at the end of the month, which is not related to problems with the software, but with human error and business irregularities on the client side. Because of this, there are significant periods where our people are just responding to customer requests and can't do much else.

We have decided that the best way to deal with this is to charge customers for the service. We now need advice on how to convince old customers to start paying for a service that they have been getting for free. We have considered charging on a case-by-case basis, so to recognize the difference between 5-10 minute requests and hour long support work, but that means a lot of billing management for not much extra. The alternative would be support contracts. What is a good way to solve this conundrum?