Time Needed to Buy Tickets

There are n people in a queue. Person i needs tickets[i] tickets, buying one per second. After buying a ticket, they go to the end of the line instantly; if they’re done, they leave.
Return the total seconds until the person originally at index k finishes buying all their tickets.

Key formula (O(n)):
Let t = tickets[k]. Then the answer is

$$\sum_{i=0}^{k} \min(tickets[i],\, t)\;+\;\sum_{i=k+1}^{n-1} \min(tickets[i],\, t-1).$$

People at or before k get served t times while k is present; those after k get at most t-1 turns before k’s last ticket.