How do you politely dismiss delinquent clients?

I've implemented a policy that states if you are late three times, we require a deposit equal to one month's invoice. This gives clients slack if they forgot to update their new card or having a tough month. I process payments on the last day of the month for that month's service. Any clients who are late do not receive service until their bill is paid. The bulk of my billing is automatic payment, and I do have some older clients who use checks. This policy would work for either.