Jobs can be setup set up to automatically e-mail the job's owner when the job has completed. This is enabled on a per-job basis. It is also possible to send e-mail to any other address. This is specified in the job's address field. Emails are sent from the Supervisor, so it must have that ability enabled.
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E-mail job owner when the job is done.
% qbsub --flags mail ls
Email anon@server.coma specific address
% qbsub --flags mail --address anon@server.com ls