The argument nobody wins
Almost every busy outpatient department in India runs two queues without meaning to. There is the appointment list the doctor works from, and there is the crowd at the desk. A walk-in with a genuine reason gets slotted in; the person who booked three days ago watches it happen. By eleven o’clock the front desk is negotiating rather than registering.
This is usually described as a discipline problem. It is more often a data problem: the two groups exist in different places, so there is no single ordering that anyone can point at.
What one queue actually changes
When a walk-in and a booked patient both receive a token from the same sequence, the ordering stops being anybody’s opinion. The receptionist is no longer defending a decision. She is reading a screen, and so is the patient.
It also makes the day measurable in a way two queues never are. How many people were seen, how long the queue actually was, and where it stalled all become answerable questions. With two queues you have two partial answers that do not add up, which is why most clinics do not measure it at all.
Priority is a rule, not an exception
One queue does not mean first-come-first-served for everybody. A genuinely urgent patient still goes first, an elderly patient with a two-hour journey may still be accommodated, and a follow-up review may take five minutes rather than fifteen.
The difference is that those become stated rules applied to one list, rather than unstated exceptions to two. A rule can be explained to a waiting room. An exception has to be defended, repeatedly, by whoever is standing at the desk.
Where vitals fit
The other reason clinics keep two queues is that patients disappear between them, to a vitals room, to the cashier or back to the waiting area, and reappear in a different order than they left.
That is worth fixing directly rather than working around. If the system records where a patient is in the sequence, a step out of the queue does not lose their place. Which is, in the end, the whole point: the queue is not the row of chairs, it is the record of who arrived when.