Fitsmallbusiness.com is a review site. We publish reviews of help desk software and other small business software. We don’t provide the help desk software ourselves.
All help desks allow you to assign tickets to a particular agent and track the ticket’s status. However, for your purposes, it sounds like a help desk software with Service Level Agreement (SLA) tracking would be the best choice. With our recommended help desk, Freshdesk, you can set different target response and resolution times for different groups of customers. Click the link to learn more about this feature.
Zendesk also has SLA tracking.
Hope that helps!