I saw two usages for this phrase:
1: do you have any time available? no, I'm booked solid (for a person)
2: the restaurant is booked solid! (for a place)
is that equivalent to overbooked?
When can we use this phrase?
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No, it is not equivalent to "booked solid"
Imagine a person's schedule, it starts as "empty" spaces, and then when someone has scheduled all possible appointments it is a "solid" mass of appointments. So "booked solid" means they have every possible appointment time booked.
Implies that they have booked more appointments than they have time for. If you are "booked solid", but still book even more appointments, you become "overbooked"
You can use "booked solid" any time you have no more available appointments. You can also use it more casually when you are very close to having no additional time, to imply that you are very busy.
"Booked solid" means that all available time slots or seats or other resources are allocated. This phrase may be used literally for a person or organization that meets customers with scheduled visits, like a doctor or a lawyer, or that has some other specific resource, like seats in a theater or on a plane. It is also used somewhat less literally to mean that a person or group are very busy, that they have no available time for other tasks.
"Overbooked" means that commitments have been made beyond what can be delivered. For example, if there are 100 seats on a plane for a certain flight, and the airline has sold 101 tickets, they are overbooked. This could happen through a mistake, like the same seat is accidentally sold to two customers. Or the organization may be trying to optimize use of resources, knowing that some customers will cancel. For example, airlines have figured out that on the average, about 10% of passengers who made reservations for a flight will not show up -- they'll just decide they don't want to make this trip after all, or more often they arrive at the airport late. So -- and I don't know the actual numbers here, just making up numbers -- say they figure out that 99% of the time, at least 3% of the passengers won't show up. So they'll accept reservations for 103% of the available seats, knowing that with cancels and no-shows they'll almost certainly be able to get everyone who actually does show up on the plane. They deliberately overbook. Every now and then they get fewer no-shows than expected and they have to offer people an incentive to take a later flight.
So "booked solid" means 100% of available resources committed. "Overbooked" means more than 100%.