In this paper we present a detailed analysis of a single server Markovian queue with impatient customers. Instead of the standard assumption that customers perform independent abandonments, we consider situations where customers abandon the system simultaneously. Moreover, we distinguish two abandonment scenarios; in the first one all present customers become impatient and perform synchronized abandonments, while in the second scenario we exclude the customer in service from the abandonment procedure. Furthermore, we extend our analysis to the M/M/c queue under the second abandonment scenario. For these models we carry out an extensive analysis including the stationary, the busy period and the conditional sojourn time distributions deriving exact formulas and iterative algorithmic schemes. We also obtain explicit results under various limiting regimes that demonstrate the effect of the level of synchronization on the performance of the systems.