Robert Mulligan, chief justice of the state trial court, pledged Thursday to use every possible resource to make sure the cases — expected to number in the thousands — are dealt with expeditiously. The business hours of courts may need to be extended because of the sheer number of cases, Mulligan said.

Prosecutors and defense lawyers pressed for the special courts, described by some judges as unprecedented in Massachusetts.

The precise mechanics of the special courts remain to be worked out. That includes determining which judges will hear the cases in district court and superior court in each county.

The administration of Governor Deval Patrick asked all of the agencies involved in investigating the scandal, and working to repair the damage wrought, to estimate how much their work will cost. The governor plans to ask the Legislature for funds to cover those costs, which could end up in the millions of dollars.

The Suffolk district attorney’s office has estimated it will need at least $1 million to revisit the thousands of cases its prosecutors brought using test results generated by the chemist at the heart of the scandal, Annie Dookhan.

Suffolk District Attorney Daniel F. Conley said his office was already swamped before the drug lab “disaster,” which he said was escalating by the day.

“Now, we are looking at our annual caseload increasing by half in one fell swoop,” Conley said. “We are struggling to keep up with the new cases pouring in while trying to prevent the normal work of the office from grinding to a halt.

“Specialized sessions will hopefully bring some order to the chaos and ensure that the most urgent cases get first priority,” Conley said. “It makes sense to do this and do it quickly.”

Jay Gonzalez, administration and finance secretary for the Patrick administration, sent a memo Thursday seeking work and cost estimates to more than a dozen agencies, including district attorneys, the attorney general, the courts, the public defenders, the state departments of public safety and public health, as well as David Meier, designated by Patrick last week to review the thousands of potentially tainted criminal cases.

Dookhan’s admission to State Police about her alleged misdeeds, made public Wednesday, has sparked profound doubts among prosecutors and defense lawyers about the 60,000 drug samples she handled over her nine years at the lab, run at the time by the Department of Public Health.

So far, authorities have found that more than 1,100 inmates in state prisons and county jails were convicted based on evidence analyzed by Dookhan. Judges have freed, reduced the bail, or suspended the sentences of at least 20 defendants.

No criminal charges have yet been filed in the lab case, but Attorney General Martha Coakley is conducting a wide-ranging criminal investigation.

Patrick, speaking to reporters Thursday, said he expected charges to be brought. Asked if he thought Dookhan should face prison time, Patrick responded: “First of all, I’m not a prosecutor. The attorney general is investigating. The State Police is doing exactly what they should be doing. The attorney general has to make a decision about charges. I fully expect and indeed I hope that there are charges.”

Dookhan, who spoke to State Police assigned to Coakley’s office in August without a lawyer, still has not hired one, according to law enforcement officials. She has declined to speak to reporters.

The 100-page State Police report characterizes a lab rife with suspicion about Dookhan’s shoddy work habits and unusually high output. Supervisors largely stood by her for more than a year, the report said, despite evidence that Dookhan had lied on her resume and removed drug evidence without authorization.

As a result, the convictions and pending charges of thousands of suspects hang in the balance.

“It’s important to expedite the resolution of these cases,” said Michael Traft, whose client Manuel J. Abreu was serving a 17-year sentence based on drug samples tested by Dookhan until he was released on bail Tuesday. “There was serious misconduct that compromised the integrity of a large number of convictions, and we need to address this as soon as possible.”