Stuart,
You mention you are using a single Service Desk server, how many users are there during peak usage?
One thing to do would be to run the command "pdm_webstat" on the server from a command prompt during the peak usage to see how many people are connecting (please note, if you have Interval logging enabled for "Web Session Count" this information can be logged automatically every few minutes).
I'd also wonder if perhaps it's not so much the number of sessions that is causing the problem, but perhaps the database queries are getting backlogged and therefore creating a large queue, and slowing down the system. You may also want to run the command "pdm_vdbinfo" during the same time and see how many "Num Requests Pending" you have, and if the number remains high for a long period of time that may indicate an issue. (note, this too can be configured via Interval logging to log the info automatically every few minute as "Virtual DB Info").
Just as a total guess I'd say you may be hitting a resource issue where the webengine is getting loaded with too many sessions or an alternate theory the database agents are being overwhelmed with queries and being backlogged.