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Ok, well how long until inactive?

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This is again depedent on the number of leads the police have. Even in your "no clues" scenario (which is very unlikely), there are still quite a few compulsory investigative procedures that are followed in a homicide. These can take up to 6-months or a year to resolve depending on the number of investigators and the resources of the department/agency.

As far as when the perpetrator can give a "sigh of relief" goes - I would say never. Sometimes letting a case go inactive can allow new leads to pop up solely because it hasn't been receiving a lot of attention.