Iarnrod Eireann is hoping to run normal "or close to normal services" on most of its routes today.
Munster and suburban routes to Kildare and Dundalk remain the areas most affected by the refusal of members of the Irish Locomotive Drivers' Association to operate new rosters.
Over a dozen ILDA members reported for work yesterday and the company expects to provide at least one extra train on the Dublin-Tralee line.
Improvements are also anticipated on the Dublin-Galway and Dublin-Limerick lines.
The Minister for Public Enterprise, Ms O'Rourke, told the Dail yesterday that there was "no prospect of intervention by the State's industrial relations machinery" in the dispute.
"The company's position is that it has negotiated an agreement with the recognised trade unions representing drivers and it is now proceeding to implement that agreement.
"It is not prepared to negotiate with the Irish Locomotive Drivers' Association and considers that to do so would be unlawful having regard to a recent High Court decision."
Ms O'Rourke appealed to ILDA members "to immediately begin co-operating with the introduction of the new rostering arrangements.
"It is totally unacceptable that rail users should be deprived of services by this unofficial and unnecessary action."
She called on all drivers to look again at the benefits of the `New Deal for Locomotive Drivers' and asked them to consider the damage their action is causing "to the image of the railway."
Iarnrod Eireann expects that today only two of its seven trains will run between Dublin and Cork.
Two trains will run from Dublin to Tralee but only one in the opposite direction.
Three of the four trains due to operate between Dublin and Limerick are expected to do so. Normal services are expected on the Dublin-Galway line.
Two of three trains will operate on the Dublin-Westport line, with bus transfers between Athlone and Westport.
Three out of four trains will operate on the Dublin-Waterford line and services are expected to be normal on the Dublin-Sligo, Dublin-Rosslare and Dublin-Belfast lines.
The company expects to run 70 per cent of services on the Dundalk and Drogheda suburban routes, and 50 per cent on the Kildare service.
The DART and Maynooth lines are expected to operate normally, but severe disruption is expected between Cork and Cobh.
The ILDA, which is not responding to inquiries from The Irish Times because it is dissatisfied with coverage of the dispute, is understood to have issued a statement yesterday criticising media coverage generally.
It still has an appeal to be heard by the Supreme Court on union recognition.
There was speculation yesterday that the ILDA might allow members to work the new agreement under protest while awaiting the outcome of the court case.
However, most observers did not believe a resolution was likely this week.