The Analyst Queue capability requires the REST Web Service API Role to be available to the users.
The Web Service API Role should have at least View access level for below three function accesses in CA Service Desk Manager (CA SDM):
Modify access level on the above three is also OK, but a View is mandatory to be able to access the Analyst Queue capability.
Lets pick an example here, for Service Desk Staff Access Type in CA SDM. Login to CA SDM GUI as an admin -> Administrator tab -> Security and Role Management on the left pane -> Access Types -> select Service Desk Staff Access type on the right. In the Service Desk Staff Access details window, select Roles tab --> Note down the REST Service API Role name there.
Now identify that role in the Attached Role list section down below on the same details page. Click on it, it opens the appropriate Role detail window. Select Additional Information -> Function Access tab now. Go to the second page in the list there. You'll find Administration, Security and Reference.
Ensure all the three are set to at least View, if not Modify. Save.
Let the user logout and login to test his access to Analyst Queue capability now.