Wow, simple instructions are difficult to follow....
This depends upon the jurisdiction and can be contradicted by special statutes at the will of the legislature, but I'll refer to New York for the purposes of my post.
In general, the standard that has held for the Police applies to EMS and fire as well.
The standard for negligance is always the same and requires:
1) An affirmative duty to act;
2) Breach of that duty;
3) Injury; and
4) a causal link between the breach of duty and the harm suffered.
If any of these elements are missing, the claim will fail. If these elements are satisfied, for the negligent conduct of a municipal agency, we move on to the special relationship test. Only if this final test is satisfied may a plaintiff collect against a municipal agency:
(1) an assumption by the municipality, through promises or actions, of an affirmative duty to act on behalf of the party who was injured; (2) knowledge on the part of the municipality's agents that inaction could lead to harm; (3) some form of direct contact between the municipality's agents and the injured party; and (4) that party's justifiable reliance on the municipality's affirmative undertaking.”
Laratro v. City of New York, 8 N.Y.3d 79, 83, 861 N.E.2d 95, 96-97 (2006)
Discussion of these elements in full detail could take days, but if any one fails, the test fails. To address 3 and 4 quickly, direct contact means direct contact with the victim who has suffered harm. A phone call will suffice, but the test will fail for a patient who is unconscious unless a DIRECT family member makes contact on their behalf. Justifiable reliance implies that there is some other means of rescue that was not undertaken because of the mistaken belief that the initially contacted agency is responding. I haven't dug deep into the case law, but off hand I don't think that there are too many actions that you can take to put out a house fire besides call the FD. A patient, on the other hand, might get themselves to the hospital or secure a private ambulance.
Hope that helps.
ETA: to satisfy the causal analysis one would have to prove that but for the negligent actions, if prolonged response time is such, the harm would not have been suffered. Alternatively, it can be characterized under the lost chance doctrine and depending on the jurisdiction the lost chance may dictate the applicable damages. A good example of this is the misdiagnosed cancer patient who would have had an 80% chance of survival but now has a 30% chance. This stuff is highly dependent on the jurisdiction though.
