What's the meaning of obsequious (from Theodore Dreiser's urban novel Sister Carrie)?

The adjective obsequious describes someone who shows too great a willingness to serve or obey; someone who is overly eager to please. Synonyms include fawning and servile.

In Sister Carrie, by Theodore Dreiser, Carrie and Mrs. Vance walk down fashionable Broadway in New York:

The street was full of coaches. Pompous doormen in immense coats, shiny brass belts and buttons, waited in front of expensive salesrooms. Coachmen in tan boots, white tights, and blue jackets waited obsequiously for the mistresses of carriages who were shopping inside.