The troubled UK Border Agency has built up a backlog of 276,460 immigration and asylum cases that it has not resolved, MPs have revealed.

Keith Vaz, chair of the Commons home affairs committee, said the figure, comprising all outstanding UKBA cases, had been calculated for the first time.

"This backlog is now equivalent to the entire population of Newcastle upon Tyne. It will take years to clear," said Vaz. "The agency seems to have acquired its own Bermuda Triangle. It is easy to get in but near impossible to keep track of anyone, let alone get them out."

The backlog includes 150,000 people in the "migration refusal pool" – those who were refused leave to stay in Britain and the agency has lost track of – plus 101,500 cases in the "controlled archive", which covers asylum claimants who applied before 2007 but whom UKBA lost contact with and is trying to trace.

The rest of the backlog includes 21,000 asylum cases, including some that remain unresolved after years, and 3,900 foreign national prisoners living in the community and awaiting deportation.

The damning committee report, says it is totally unacceptable that there are so many cases yet to be sorted out by the UKBA. The MPs say the agency must act immediately to find ways to locate those people in the backlog and resolve their cases. They have also criticised the time it takes – an average of 74 days in 2011 – to deport foreign prisoners after their sentence ends.

The committee says UKBA is failing to work with the prison service to ensure deportation proceedings begin at the time of sentencing.

The committee's quarterly monitoring report on UKBA's performance says that the government's decision to provide clearer guidance for the courts on article eight "right to family life" deportation cases should lead to a much higher proportion of foreign national prisoners being removed.

Figures given to the Sunday Telegraph under the Freedom of Information Act show 250 foreign prisoners were allowed to stay in Britain last year on article 8 "family life" grounds without their claims being challenged in court.

The committee recommends other sweeping changes in the operation of the immigration system, including the removal of students from the government's target to reduce net migration from 250,000 a year to below 100,000.

The MPs also say the government's aim of reducing the 260,000 student visas issued each year by 25% will not benefit Britain. They argue that excluding students from net migration figures – as campaigners have argued for on grounds that they are only here temporarily – would help maintain the UK's £7.9bn a year earnings from international students.

The MPs say it would also meet the Conservatives' pledge to reduce net migration to the tens of thousands.

Vaz said: "This will enable the government to encourage students to come to the UK whilst maintaining their position on curbing immigration."

The MPs also express alarm that 24% of UKBA staff continued to receive bonuses in 2010-11 despite the agency's poor performance. Senior staff should repay bonuses they received after the committee's recommendation last year that they should not be paid, the MPs said.

The immigration minister, Damian Green, said the report "raises some legitimate concerns about issues we are aware of and are already working to tackle".