JIRA comes with a workflow designer which allows to design and keep track of execution of pretty much any process (similar to what a BPM solution would do but cheaper). In the new or existing workflows you can define new "statuses" which you can use as "Queues" to handle your process status or requests.
In principle, JIRA was designed to follow software development projects, but we are also using it as a solution for our call centers to follow on "Learning Administration System" processes.
It allows you to extend and define any new entities (record types), define sub-tasks, links between records as well as define new fields, screens and customize your UI based on your needs.
It also integrates very well with other applications thanks to its Restful API. In my case I integrated with JIRA from Webmethods EAI middleware as well as from SQL Server 2014 SSIS.
JIRA has a marketplace where many plug-ins are available (some for free) to extend any missing features, as well as it comes with an SDK to create your own one if needed. In fact, this is one of the biggest potentials from my point of view.
Among others Atlassian (JIRA's company) has two powerful plug-ins
- Service Desk (to control SLAs in top of your processes) - actually not a plug-in a product on top of JIRA.
- JIRA Agile which allows to follow Soft Dev Projects using SCRUM or KANBAN methodology.