drowsing

(verb) sleep lightly or for a short period of time. Synonyms: snooze, drowse, doze.

(verb) be on the verge of sleeping. Synonyms: drowse. Example: "The students were drowsing in the 8 am class."

(adj) half asleep. Synonyms: drowsy, drowsing(prenominal), dozy. Examples: "Made drowsy by the long ride." "It seemed a pity to disturb the drowsing (or dozing) professor." "A tired dozy child." "The nodding (or napping) grandmother in her rocking chair."