Unlike the private hospitals, QM has an anaesthetist on duty in the hospital 24 hours a day. I was very grateful for the fact that they did not force or try to suggest an epidural until I asked for it (I was warned that this may be the case in private hospitals as the anaesthetist is not always available). In the event, I was given an epidural within about 10 mins of asking for it. More importantly, if there is a real emergency requiring a general anaesthetic, at least you can have confidence that the QM will be able to cope with this. The private hospitals may not have an anaesthetist on site. I believe that the quota for epidurals is for the number of patients receiving epidurals at any one time because they require extra monitoring. I have not heard of anyone who was refused one for this reason, although I suppose it is possible if they are very busy. I had my epidural in the evening.
Try not to be too worried by the staff at the Tsan Yuk, I found that once admitted to QM the staff there were far more flexible and accommodating.