This can be done with Automation for JIRA. Simply use the "Assign Issue" action and select "Assign to a member of a list or group". You will be then offered choice of "Load Balanced", "Round-robin" and "Random".
Load balanced assigning executes the given JQL and then looks for the users with lowest number of tickets assigned to them. This is great for Service Desks.
I have written this up in more details at:
I have multiple projects that use variations of the same base workflow. The variations depend on the requirements of the project or issue type. The variations mostly come in the form of new statuses ...
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