Where I used to live, I was in a city of ~160k. We had the city police, and theoretically the county sheriff could also patrol there (but rarely did unless it was a special enforcement like when they were doing a blitz to get everyone using a cell phone while driving). The suburbs also sometimes had police agencies and others only had the sheriff dept. Suburbs with police departments tended to get much quicker responses on minor crimes.
In the city, response time varied *widely*. Sometimes, it'd take a few hours to get the police to respond (non-emergency). After awhile during my nightmare of a divorce, the police were *very* quick to respond to my calls for service. Then when I got my case bumped up to a detective, within 24 hrs of completing my statement (so that he could then do all the other paperwork necessary to get the warrant), he had arrested my ex (who was violating my order of protection by stalking me). They were also great when one of my neighbors (rental property) had a dog that was horribly abused. It was a young pit bull that they allowed to breed (although I later learned that the size was more typical of a 6mo dog of that breed, so apparently that was the dog's first heat, too) and then tied up in the back porch by a leash, and that dog jumped out of the window on more than one occasion (kind of hanging there, the leash didn't allow it to put feet on the ground. Well, calls to Animal Control didn't get an immediate response (they were the agency I was first directed to), so the next time the dog ended up hanging by the neck from the window, I called the police, and they came, a young female officer took the dog away, and they took care of the 'owner' of the dog, too as the puppies were all just on the floor in the back porch, no one had ever cleaned up the poop, and some of them had died. A month after that, Animal Control called me to get more info.
Where I live now is a much smaller location--a city of ~16k surrounded by farmland. There is also a sheriff's department for the county and they are also within city limits sometimes. We also have state police due to the interstates. The city police are very quick to respond, but thankfully I have had almost zero need for their direct services.