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We have recently proposed quantized gossip algorithms which solve the consensus and averaging problems on directed graphs. In this paper we focus on the convergence time of these algorithms; specifically, we provide upper bounds on the mean time taken for convergence on complete graphs, as functions of the number of nodes. To this end, we investigate the shrinking time of the smallest interval that contains all states for the consensus algorithm, and the decay time of a suitable Lyapunov function for the averaging algorithm. The investigation leads us to characterizing the convergence time by the hitting time in certain special Markov chains, from which we derive polynomial upper bounds.