The Collaborative International Dictionary of English v.0.48:

Doze \Doze\ (d[=o]z), v. i. [imp. & p. p. Dozed (d[=o]zd); p.
   pr. & vb. n. Dozing.] [Prob. akin to daze, dizzy: cf. Icel.
   d[=u]sa to doze, Dan. d["o]se to make dull, heavy, or drowsy,
   d["o]s dullness, drowsiness, d["o]sig drowsy, AS. dw[=ae]s
   dull, stupid, foolish. [root]71. Cf. Dizzy.]
   To slumber; to sleep lightly; to be in a dull or stupefied
   condition, as if half asleep; to be drowsy.
   [1913 Webster]

         If he happened to doze a little, the jolly cobbler
         waked him.                               --L'Estrange.
   [1913 Webster]