Two weeks ago, Pin Zhu Zheng, who says she worked 69 hours a week behind a steam table at a Chinatown restaurant on Centre Street, presented herself at a New York State office to report what seemed to be flagrant lawbreaking by her former bosses.

“The first day of the month, they pay $1,500 cash,” Ms. Zheng, 55, said in an interview on Thursday. “Everyone got the same.”

That works out to about $5 an hour for a six-day workweek that ran from 9:30 a.m. to 9 p.m.; the law requires a minimum wage of $7.25 an hour for the first 40 hours a week. After that, workers must be paid time and a half, or a minimum of $10.88 an hour.

“The Labor Department person told me that I had to wait a year for the follow-up,” Ms. Zheng said through a translator.

But what good is a minimum wage law if it takes forever to enforce it? Complaints with the State Labor Department about wage and hour violations were stacked 14,000 high at the end of July, according to documents obtained by the Urban Justice Center through a freedom of information request.

In May 2012, the records showed, 44 percent of the cases had been open for more than a year, said David Colodny, a lawyer with the center.

Carlos Rodriguez, 28, said he made $4.40 an hour in a pizza franchise on the Upper East Side of Manhattan, cutting vegetables, cleaning, unloading trucks by day and making deliveries at night. “We paid for the uniform, the hat, the T-shirt, the pants, the shoes,” Mr. Rodriguez said.

Did he speak to the boss about the pay rate?

“He said if you don’t like it, just go,” Mr. Rodriguez said.

So he did. “I went to the Labor Department in 2007 and told them they robbed me of 20 hours of pay per week,” Mr. Rodriguez said. “They took my information. They said we are going to call at the end of every month to bring you up to date.”

Not once, he said, did they call, but he called religiously. “Every end of the month, they said the same thing: ‘We have a lot of cases,’ ” Mr. Rodriguez said.

The state believes it can cut the time of the investigations to nine months by looking at three years of wage history, rather than six, as had been the practice, said Alphonso David, who works for Gov. Andrew M. Cuomo as the deputy secretary for civil rights. The investigations will not get bogged down in hard-to-resolve factual questions from the past, he said, and the result will be quicker, surer justice.

The national minimum wage came into existence because a social reformer named Frances Perkins told Franklin D. Roosevelt that she would join his cabinet only if he agreed to a floor on the hourly pay rate, and a ceiling on the number of hours in a workweek. Roosevelt agreed. The law was fought almost as fiercely as the new health care act.

In March, the State Legislature passed a bill raising the minimum wage by 75 cents to $8 an hour by the end of 2013, then to $8.75 by the end of 2014, and $9 by the end of 2015. To win the agreement of Republicans in the Senate, the law includes a tax credit for companies that keep or hire teenagers at the minimum wage. The governor’s office estimated the credit’s cost at $45 million.

The law included no funds for faster investigation of wage and hour violations, and the budget for that has been flat for several years, said JoAnn Lum, executive director of the National Mobilization Against Sweatshops.

Mr. David said the state was getting bogged down by looking back six years and would no longer do so.

“That is ridiculous,” said Deborah Axt, an executive director of Make the Road New York, an organization that advocates better conditions for immigrants and low-income people. The group had urged the state in 2007 to use the full six years allowed by the statute of limitations to pursue the wage claims. “You are cutting off half your ability to recover wages under the law,” she said.

Yu Dan Wong, 46, said she filed a complaint in 2006 that the fashion manufacturer she worked for had paid her $450 for a six-day, 54-hour week, a wage that covered none of the required overtime. Her job was to check garment quality at about 30 factories in Brooklyn, Queens and Chinatown. “I kept calling until 2010,” she said. “I never heard.”

Were the workers in those factories paid minimum wage?

The question was translated for her by Wing Lam, the director of the Chinese Staff and Workers Association.

Both of them roared with laughter.

“Nobody!” Ms. Wong said.

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