With a click, the call came through.

It was from a doctor. Routine scans had unexpectedly revealed that a patient of his was in danger because of a brain aneurysm. The man, who had already left the doctor’s office, wasn’t answering his phone. The doctor needed an ambulance to get him to the hospital before the aneurysm could rupture.

In San Francisco’s 911 emergency call center, Natalie Eliceteche, 41, hastily switched among five computer displays, inputting a string of commands to dispatch help. The seconds it takes her to answer and dispatch a call can mean the difference between life and death. The faster she can get an ambulance to the man, the better his chances of survival.

Eliceteche has spent 14 years answering 911 calls like this one. But in recent years, she said, conditions at the center have worsened, and there aren’t enough dispatchers on duty to answer calls. On this night, she was able to get the patient’s information to medics, and he made it safely to the hospital.

But through the evening, the display above her desk kept flashing, registering more calls waiting. Small screens scattered around the center alerted her and other dispatchers to the incoming emergencies. With a click, Eliceteche would answer the next one. Click: a stabbing. Click: a pocket dial — false alarm. Click: a suicide attempt.

The screen — signaling the backlog of calls — flashed red. The delay in answering those calls has the potential to leave people facing health emergencies or other dangers without the help they need in the time they need it.

National guidelines say that 90 percent of 911 calls should be answered within 10 seconds, a standard San Francisco has not met since early 2012. The state standard says 95 percent of calls should be picked up within 15 seconds.

At San Francisco’s 911 call center — where hiring freezes have left staffing short and emergency call volume has surged — only about 79 percent of the 1.27 million emergency calls received in 2015, the latest year for which data are available, were answered within the 10-second standard, a Chronicle analysis shows. Nonemergency calls trended lower, with an average of 57 percent of calls answered within the recommended one-minute mark.

Some other cities, like Oakland, split responsibilities for emergency calls between two centers. The Police Department fields law-enforcement emergencies, while the Fire Department responds to medical and fire emergencies. To compare, in 2015, Oakland received 72,803 medical and fire calls while San Francisco handled 135,229 medical and fire calls.

Calls to San Francisco’s call center jumped from 919,908 in 2007 to 1.26 million in 2015, paralleling a swell in the city’s population.

For years, the city has criticized its own call center on its public safety scorecard website for failing to meet targets. But it has been sluggish in its efforts to improve its operation, one city supervisor says.

To handle the increased calls, mandatory overtime for dispatchers has been increased. The Department of Emergency Management had 146 dispatchers as of Dec. 15, leaving it short 34 full-time employees. But that roster is misleading because it includes people on leave and new hires not yet trained to dispatch calls.

In fact, about 120 dispatchers are available to work at any given time, workers say. That’s far short of the 180 or so that would make up a full staff, according to both department spokesman Francis Zamora and the dispatchers union, Service Employees International Union 1021, which is renegotiating its contract with the city.

San Francisco’s dispatchers are working 12- to 14-hour shifts. Their overtime — pay for any shift beyond eight hours — cost the city more than $3.51 million in 2016, nearly triple the total in 2011.

The center’s open positions have gone unfilled even as employees retire or take extended medical leave for work-related injuries, according to interviews with more than a dozen dispatchers. Five employees are out on long-term medical leave, and 10 have retired in the past two years, Zamora said. But hiring new employees isn’t easy or quick — it takes nine months to a year to train a replacement, and only about 60 percent complete their training.

“The reality is, answering standards are not being met and are not going to be met anytime soon,” said dispatcher Lynette Galarza, 36. “The fact of the matter is, you don’t pay as much attention if it’s your 12th hour of work. Things start to slip through the cracks. I wouldn’t be comfortable with one of my family members calling 911.”

While City Hall has been aware of the center’s staffing shortage, it has not acted quickly to alleviate it.

A citywide hiring freeze that was a result of the 2008 recession blocked the Department of Emergency Management from recruiting and training more than a single class of new dispatchers annually through 2013. Before that, multiple classes were offered each year.

In 2014, a controller’s report concluded that the dispatch center was actually overstaffed — by 2.2 to 5.3 people per hour on average, based on historical call volumes and processing times. But that report was based on data gathered before 2011, when call volume began increasing, Zamora said, and it didn’t account for extra staffing needed during major events such as the Super Bowl or a natural disaster.

The department is training multiple classes per year again; applications for its next class open Monday. But after testing, interviews and a background check, those trainees won’t make it into the classroom until September.

“Unfortunately ... our workload has increased, and we don’t have more people coming in,” said Rob Smuts, who became deputy director of the department’s Division of Emergency Communications in 2014, two years after call response times — the time it takes to answer a call — began to plummet.

“Those things together added up to a real crunch,” he said. “We’ve addressed part of the issue through an increase in overtime. But all the overtime we could spend wasn’t sufficient to keep staffing as high as we would like. ... It just takes a little while to get new bodies trained and through the process.”

The huge increase in 911 calls stems from a booming population, greater cell phone use and an increased number of calls regarding the city’s homeless population, city data indicate. From 2012 to 2014, for example, the number of police responses generated as a result of a 911 call jumped 11 percent, and about half the extra calls were related to homelessness, a suspicious person, an auto boost or an unknown complaint, according to a city report published last year.

The number of calls San Francisco’s center receives is remarkable, said Stewart McGhee, fire and medical services division manager for the Oakland Fire Department.

“There’s nothing in the Bay Area of any appreciable size that would be a good comparison,” McGhee said. “There’s been an annual increase in 911 calls every year in Oakland and across the region. But it is nowhere near what San Francisco has seen.”

Even as the Department of Emergency Management’s budget has grown, 911 staffing hasn’t kept pace with the calls. This fiscal year, the department was earmarked to receive $93.9 million in city funding, a 13.3 percent increase from $82.9 million in fiscal year 2015. Most of the increase will go to a radio replacement project, personnel costs and capital costs related to long-term space planning for its facilities. About $1 million was allocated for three training classes of up to 15 students in fiscal year 2016, and two more in fiscal year 2017.

The city hired 15 dispatchers in 2015 and plans to hire 40 more trainees in the next two fiscal years, said Deirdre Hussey, spokeswoman for Mayor Ed Lee. But since about 2 in 5 of those students won’t complete their training, the city will need to find money for more classes.

“We’re confident that with (some) investments we will have the dispatchers we need to meet the increasing demand,” Hussey said.

Others aren’t so certain. Supervisor Aaron Peskin, who chairs the Board of Supervisors’ Government Audit and Oversight Committee, said City Hall penny-pinching is putting lives at risk.

“I don’t know what is worse: the unacceptable amount of time it takes 911 to respond to emergency calls or the unacceptable amount of time it’s taken the city to address this serious safety problem,” Peskin said. “Response time is a perennial issue that should have been dealt with years ago, and our 911 call volume has only continued to increase.

“As I was reminded recently, even 20 seconds can mean the difference between life and death,” said Peskin, who arrived at Lombard and Powell streets last month shortly after a 13-year-old stopped breathing near a Muni stop and almost died.

Answering times won’t improve until more dispatchers are brought on staff, said Charles Cullen, past president of the California chapter of the National Emergency Number Association and a technical services director for the Palo Alto Police Department. He helped lead the national organization that creates standards for emergency services.

“If you don’t have the staffing, it’s really hard to get your numbers up to standard,” Cullen said. “When there is a lot of mandatory overtime, people burn out and get fatigued. You’re constantly dealing with issues and problems. It’s all about having people there to answer the phones, because every second counts.”

The city plans to upgrade its 911 phone system this winter, replacing an outmoded version in use since 2000. Zamora, the Department of Emergency Management spokesman, said that would help prevent repetitive-motion injuries and decrease the number of call takers needing extended medical leave.

But many dispatchers and union leaders say the city’s efforts are not enough.

“A lot of people are just burnt to the ground,” said dispatcher and SEIU Local 1021 President Burt Wilson. “There are people working 16 hours a day, and we are still coming up short. They are missing calls and missing certain details. This is not the best service that we can provide to the public.”

The dispatch center, with lime-green walls and American flags hanging near the ceiling, is located in a dimly lit building near Turk and Laguna streets. It serves a city and county of 860,000 people and fields calls for police, fire and medical emergencies.

It also handles calls for nonemergency services. On the Fourth of July last year, some residents ringing the nonemergency line — (415) 553-0123 — got a busy signal. That line, which can take in and hold up to 20 calls at one time, overloaded after an influx of complaints about firecrackers and noise. Many calls were never answered.

“A lot of people aren’t getting the help they need,” said dispatcher Sean Dryden. “If there was a crime, and the suspect is running away, the one or two minutes that someone who has that information is waiting on hold makes a difference. The police could be driving right past the suspect. I get a horrible feeling when our bell has been ringing nonstop, and we can see how many people have been holding. That is wasted time.”

But even callers who get through to 911 quickly might have to wait for help. Police are supposed to respond to high-priority calls within four minutes, but have been taking an average of four minutes and 57 seconds from receiving a report from the 911 dispatch center to arriving on scene, according to a September 2016 report by the city controller. In 2009, they were about 70 seconds faster.

Police blame the same problems that plague the 911 call center — more emergency calls and short staffing — for their delays. Since 2010, there has been a 38 percent increase in high-priority calls for police help. Attrition within the police force has also increased. The city plans to hire as many as 400 officers within the next two fiscal years.

Fire Department ambulance response times are better. The department had major problems in recent years, but ambulances are now arriving at life-threatening emergencies within 10 minutes 91.1 percent of the time, surpassing the goal of 90 percent. In 2013, the goal was hit only 73 percent of the time. The city established working groups to brainstorm solutions and found funding to replace the department’s aging fleet of ambulances and hire more paramedics.

“Generally speaking, we don’t have a problem responding to top-priority calls,” said Smuts of the Division of Emergency Communications. “The middle and lower calls, (and) staffing levels at the Police Department impact how long it takes officers to get on those. We see that impact when we have pending calls, and people keep calling back asking for help.”

Duplicate calls spiked from 2012 to 2014, meaning that dispatchers are handling the same information multiple times, which slows the system. Supplemental calls from bystanders about the same incident also have increased.

“You get people that have been calling all day,” said dispatcher Eliceteche, signing off a call. “They call, call, call. It slows everything down. But it’s just the way it is.”

On a recent Friday at the city’s call center, dispatchers hunched over their computers and answered a stream of incoming calls.

Among them was Patrizia Marcuzzi, who has worked 11 years as a dispatcher, helping people on the worst days of their lives. She has handled calls about shootings and stabbings, car accidents and rapes.

One time, she said, a man overdosed in a Burger King parking lot off Van Ness Avenue. That one stayed in her mind for a long time, Marcuzzi said. She had nightmares about the call, the man’s girlfriend sobbing into the phone while trying to resuscitate him. She doesn’t always find out what happens to the victims she tries to help.

“They take a toll on everybody here,” she said, pausing to answer an incoming emergency call. “When you think you’ve heard everything, you haven’t. There’s some calls that affect you differently. You hear someone crying because the person with them isn’t breathing. It’s heart-wrenching.”

She typed in a chain of commands on her computer. Above her workstation, the screen counting unanswered emergencies ticked higher.

Lizzie Johnson is a San Francisco Chronicle staff writer. Email: ljohnson@sfchronicle.com Twitter: @LizzieJohnsonnn