In many Internet scale replicated system, not all replicas can be dealt with in the same way, since some will be in greater demand than others. In the case of weak consistency algorithms, we have observed that updating first replicas having most demand, a greater number of clients would gain access to updated content in a shorter period of time. In this work we have investigated the benefits that can be obtained by prioritizing replicas with greater demand, and considerable improvements have been achieved. In zones of higher demand, the consistent state is reached up to six times quicker than with a normal weak consistency algorithm, without incurring the additional costs of the strong consistency.