Let E be a graph-directed set associated with a di-graph G. Let µ be a Markov-type measure on E. Assuming a separation condition for E, we determine the exact convergence order of the Lr-quantization error for µ. This result provides us with accurate information on the asymptotics of the quantization error, especially when the quantization coefficient is infinite.Let J i , 1 ≤ i ≤ N , be non-empty compact subsets of R t with J i = int(J i ) for all 1 ≤ i ≤ N , where B and int(B) respectively denote the closure and interior in R t of a set B ⊂ R t . Let |A| denotes the diameter of a set A ⊂ R t . Without loss of generality, we assume that |J i | = 1 for all 1 ≤ i ≤ N.

We further study the asymptotics of quantization errors for two classes of in-homogeneous self-similar measures µ. We give a new sufficient condition for the upper quantization coefficient for µ to be finite. This, together with our previous work, leads to a necessary and sufficient condition for the upper and lower quantization coefficient of µ to be both positive and finite. Furthermore, we determine (estimate) the convergence order of the quantization error in case that the quantization coefficient is infinite.2000 Mathematics Subject Classification. Primary 28A80, 28A78; Secondary 94A15.