At the end of May, Inspector Hines was canvassing for witnesses to the killing of a 15-year-old boy. He came upon a taxi driver who, if not likely to be a crucial witness, still could have helped describe the scene. But the man declined to help, saying the police had failed to help him after he reported three robberies.

“He said, ‘I don’t really see it’s my job to help you,’” Inspector Hines said.

Part of what has slowed response times has been the closing of local police stations and the consolidation of response teams that began in the early years of austerity. Only 10 stations in the West Midlands now have a public counter, down from 48 in 2010, and 14 local stations have been shut completely.

Sgt. Richard Cooke, the chairman of the West Midlands Police Federation, remembered an assault case in Walsall, an industrial town northwest of Birmingham. A woman called the emergency number to say that her son, who had a history of drug abuse, was trying to kick in her door.

The nearest officers were in Wolverhampton, about 10 miles away. By the time they arrived, the woman had been attacked and her son had fled.

“That’s happening on a regular basis,” Sergeant Cooke said.

‘Their Gang Is Bigger Than Our Gang’

For many police departments, as well as for Britain’s still influential tabloids, the answer to the deteriorating service seems simple: more officers. Patterns of taxi robberies and home break-ins are being missed because so many cases are shelved without so much as a cursory investigation, they say. Victims stop calling the police. Criminals are emboldened.

“When the criminals realize that their gang is bigger than our gang,” Constable Chant said, “they start exploiting that.”