The System.Diagnostics assembly is part of the Silverlight CLR framework, but it only includes classes related to debugging, the Process class is not available. Is there any
If you are using Silverlight inside an ASPX page you could use HtmlDocument.Window.Invoke to call a JavaScript function which could in turn call a static method within your page (using WebMethod attribute)
The article here describes the JS/ASPX bridge well.
Kindness,
Dan
NB: Not sure I agree with what you are trying to do; just want to help you do it :)