As soon as Arthur arrived at the Baxter Building (and announced himself to gain access to the restricted top floors via elevator), a flying robot was waiting for him.
"King Orin of Atlantis! Welcome to the Baxter Building, home of the Fantastic Four! Doctor Richards was not expecting you and is currently locked inside his laboratory. But if you would be kind enough to wait, he has informed me that his experiment will be finished in 615 seconds. Please, follow me, Your Majesty." HERBIE announced and waited to see if Arthur would follow before leading him back through the halls of the Baxter Building to the entrance of Reed's lab, currently sealed with a red light on above the door.
While they waited, HERBIE played elevator music, before he realized that he should be doing something else, "Oh! Would you care for some refreshments while we wait, Your Highness?"
If Arthur was thirsty, HERBIE would bring whatever he requested to drink while they waited.
Exactly five minutes and fifteen seconds after HERBIE's initial announcement, the light turned green and the door to the lab hissed open with a cloud of smoke and Reed rushed out, still wearing his lab coat and goggles, and quickly moved to shake Arthur's hand in friendly greeting.
You didn't keep royalty waiting if you could at all help it. They were high strung like that, in Reed's experience. Particularly Atlantean royalty-- of the princely variety.
"King Orin, it's a pleasure to see you again. Thank you for your patience. For safety's sake, some experiments cannot be interrupted. Please come in. " He said, gesturing for Arthur to step into his lab, the smoke having (mostly) dissipated.