Is paying an employee money to expedite the delivery of a service considered a bribe?

It is not permissible for you to give money to a worker in exchange for expediting the delivery of a phone or anything else, and it is not permissible for him to take it if he receives a salary from the state, because it falls under the category of forbidden gifts to workers. The prohibition is further confirmed if that action would prioritize you over others, as it is considered bribery, and both the briber and the bribed are in the Hellfire. However, if you strongly believe that you will not obtain your right except by paying the money, then there is no blame upon you at that time, and the sin falls upon the one who takes it.