How to Avoid “Machine Creep?”

I work in the support department at a private university. Our support technicians are currently responsible for over 200 PCs each. Every time a department adds a new machine, they want to take the old one and redeploy it elsewhere. Numbers continue to increase, machines continue to age, and the workload is increasing geometrically.
We’ve tried to institute a per-machine maintenance support fee, but have failed to get departmental agreement on the issue.
Any suggestions for avoiding ‘machine creep?’ It would be interesting to hear how others have creatively resolved what must be a very common problem. Thank you.